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ST. FRANCIS INSTITUE OF TECHNOLOGY MT. POINSUR, BORIVALI (W), MUMBAI

QUESTION BANK ENGINEERING MECHANICS (STATICS) FE SEM 1 Academic Year: 2014-15

Prepared By: Engineering Mechanics Dept.

1.1 COPLANAR FORCES – Class Work Questions 1. A concurrent system of forces is shown in the Fig-1. Find the resultant passing through the origin. = 189.5N,  = 38.8˚]

Fig – 1

[R

Fig – 2

2. Forces 7KN, 10KN, 10KN and 3KN respectively act at one of the angular point of regular pentagon toward the other four points taken in order. Find their resultant completely. [R = 25.10 KN, = 46.59°] 3. Three forces shown in the fig - 3 plus two additional forces one in “U” direction and one in “V” direction, combine produce a zero resultant. Determine the magnitude of the forces. [U = 14.14N,  = 45°, V= 14.14 N,  = 45°]

Fig – 3 Fig – 4 4. Three coplanar forces act at a point on a bracket as shown in fig – 4. Determine the value of angle α such that the resultant of the system is vertical. Also find the resultant of the system.[α = 36.87°, R = 80N] 5. Find the force F4 so as to give the resultant of the force system as shown in fig – 5. [F4 = 1219.5 N,  = 40.73°]

Fig – 5 Fig – 6 6. To pull a hook from an inclined surface, total pull required is 800N, perpendicular to the inclined surface. Three forces are applied on the hook for the purpose. Two of them are shown in the fig – 6. Find the third force. [F3 = 350.95 N,  = 69.45°]

7. A force of 1000 N is to be resolved into two components along line a – a and b – b as shown in fig – 7. If the component along line b – b is 350N. Find the angle α and the component along a – a. [α = 14.33°, Fa-a = 1216.39N]

Fig – 7

Fig – 8

8. Find the resultant of forces as shown in the fig – 8 [R = 5 kN at 2.1m from 4 kN force] 9. Find the resultant of forces as shown in the fig – 9 and locate with respect to “O”, radius is 1 m. 80 N left wards, Y = 2.078m]

Fig – 9

[R =

Fig – 10

10. A right bar is subjected to a system of parallel forces as shown in fig -10. Reduce this system to (1) A single force (2) A single force moment system at “A” (3) A single force moment system at B. [ (1) R = 60 N at 0.783 m from A (2) 60 N ( ) 47 N-m (3) 60 N ( ) 25 N-m] 11. Reduce the system of forces and couples to the simplest system at point A. [R = 48.26 N,  = 4.64º, MA = 36.2 Nm]

Fig – 11

Fig – 12

12. Determine the magnitude, direction & point of application of the resultant of the given coplanar force system. [R = 25 kN at 2.8 m from O upwards] Refer fig – 12. 13. The forces acting on 1m length of dam are shown in fig – 13. Determine the resultant force acting on the dam; calculate the point of intersection of the resultant with the base. [R = 137.12 kN,  = 79.91°]

Fig – 13

Fig – 14

14. Find the resultant of forces acting on the bell crank lever as shown in fig – 14. [R = 203.86 N, 424.87 N-cm, d = 2.084 cm from B]

MB =

Fig – 14

15. Replace the force system shown in fig – 15, by two parallel forces at B and D [6.25 kN downward at B and 8.75 kN at D downward.]

Fig – 15

Fig – 16

16. Resolve the force F equal to 900N acting at B into (a) parallel components at O and A. Ref fig – 16. [1800N up and 2700N down]

1.1.1 COPLANAR FORCES - Assignment Questions 1. (a). Define and explain Law of transmissibility of force. (b). State and prove varignon’s theorem or law of moments. 2. A hook is being pulled from a wall. Three forces are applied on the hook as shown in fig – 1. Find the resultant pull on hook. [R = 136.25N, = 30.40˚]

Fig – 1

Fig – 2

3. If the resultant of the forces shown in the Fig – 2 acts due east, away from the point and its magnitude is 1000N, find the value of F and . [F = 1452.58 N,  = 3.42˚] 4. Find the resultant of forces as shown in fig – 3. [R = 300N up at 9.25 m from 1st force.]

Fig – 3 5. Replace the loading on the frame by a force and moment at point A. Ref fig – 4 [R = 929.95N,  = 77.47˚, MA = 3200 N-m]

Fig – 4

1.1.2 COPLANAR FORCES – Tutorial Questions 1. Six forces acting on a particle. Determine the magnitude and direction of their resultant of the forces as shown in fig – 1.

Fig – 1

Fig – 2

2. Five concurrent coplanar forces act on a body as shown in fig - 2. Find the force P & Q such that the resultant of the five forces is zero. 3. The striker of carom board is being pulled by 4 players as shown in fig – 3. The players are sitting exactly at the center of the four sides. Determine the resultant force in magnitude and direction.

Fig – 3

Fig – 4

4. For the fig - 4 shown find resultant force and moment at point A. 5. Find the resultant of forces as shown in fig – 5.

Fig – 5

Fig – 6

6. A right bar is subjected to a system of parallel forces as shown in Fig - 6. Reduce this system to (1) A single force (2) single force moment system at A. (3) Single force moment system at B. 7. Reduce the system of forces and couples to the simplest system at point A. Ref fig – 7

85N Fig – 7

Fig – 8

8. Determine the resultant of the system of forces shown in fig.8. Locate the point where the resultant cuts the base AB.

1.1.3 COPLANAR FORCES - Questions for Practice 1. A ring is pulled by three forces as shown in fig – 1. Find the force F and angle  if the resultant of these three forces is 100N acting in vertical direction. [If R is taken upwards F Cos = 146.08 N, F Sin = 160 N, ∑Fx = 0, ∑Fy = 100N,  = 47.6˚, F = 216.65N. If R is taken downwards, F Cos = 146.08 N, F Sin = -40 N, ∑ Fx = 0, ∑Fy = -100 N,  = 15.31˚, F = 151.49N.]

Fig – 1

Fig – 2

2. Find the resultant of forces acting on the particle p as shown in fig. 2. [500.096 N,  = 88.8°] 3.Determine the resultant of concurrent coplanar forces acting at point O as shown in fig. 3 [R = 404.417 N,  = 73.53°]

Fig 3

Fig – 4

4. Determine the force F in the cable so that the resultant of three concurrent forces acting at point A is vertical. Also find the resultant. Fig – 3 [F = 3.3 KN, R = - 28.23 kN] 5. If the resultant of three forces is acting along the arm OA from O to A, determine the force F1 and its direction. The magnitude of resultant is 500 N. Fig –5. [F1 = 351.16 N, α = 67.74°]

Fig – 5 Fig – 6 6. A force R = 25 KN acting at O has three components FA, FB and FC as shown in fig – 6. If FC = 20 kN, find FA and FB . [FA = 35.04 KN, FB = 33.91 KN] 7. An electric transmission tower is subjected to two coplanar forces as shown in fig.7 a cable is used as guy wire and anchored to the ground as shown. Determine the required tension in the guy wire so that the resultant of the coplanar forces exerted by the three cables will be vertical. Also find the magnitude of the resultant. [T = 232.05kN, R = - 225.67 kN]

Fig – 7

Fig – 8

8. Find the resultant of the force system acting on a body OABC, shown in Fig – 8. Also find the points where the resultant will cut the X and Y axis. What is the distance of resultant from O? [∑Fx = 8 KN (←), Fy = 6 KN (↓), R = 10kN, Ѳ = 36.87˚, Mo = 20 kN.m (anti clockwise), d = 2 m, X = 3.33 m, Y = 2.5m] 9. Determine the resultant of the given system of forces shown in fig – 9. Find the point of application of the resultant on the horizontal bar AD w.r.t. point A. [∑Fx = 86.603 kN (→), ∑Fy = 165 kN (↓), R = 186.347 kN, Ѳ = 62.3˚, MA = 1265 kN.m (clockwise), X = 7.667 m to the right of A]

Fig – 9

Fig – 10 10. Replace the system of forces by a single force. Ref. Fig – 10. Take C1 = 85 Nm, C2 = 65 Nm, C3 = 90 Nm. [R = ∑Fy = 80 kN (↓), MA = 190 Nm (Clockwise)]

1.2 CENTROID – Class Work Questions 1. Find the centroid of the following areas.

[X = 4.243m, Y = 0.908m]

2. Determine the depth of web of a tee-section, such that centroid coincides with AB axis. [h = 55.9 mm] 3. Find the centroid of the area shown in fig- 7. O(0,0), A(75,0), B(175,100), C(75,200), D(0,150),E(0,100)

[77.68,

] units

1.2.1 CENTROID – Assignment Questions 2. Find the centroid of the following areas.

[58.33, 24.76] cm

[73.29, 80.19] mm

1.2.2 CENTROID – Tutorial Questions 1. Find the centroid of the following areas.

[6.98, 12.74] cm

[101.58, 90.34] unit

[12.45, 22.04] mm

2. Find the centroid of the shaded area fig – 5. [107.189, 107.189] mm

Fig – 5

Fig – 6

3. Locate the centroid of the shaded portion of lamina. Take diameter of semi-circle as 90 mm. ref. fig-6 [37.494, 26.540] mm 4. Locate the centroid of the shaded area of lamina shown in fig – 7 [7.981, 10.423]

Fig – 7

5. Find the centroid of the shaded plane area shown in fig – 8. [0.156, 0.156] cm.

Fig – 8

2.1 EQUILIBRIUM - Class work Questions 1. A circular roller of weight 1000 N and Radius 200 mm hangs by a tie rod AB = 400 mm and rests against a smooth vertical wall at C as shown in the fig.1. Determine the tension in the rod and reaction at point C. [T=1154.7N, Rc = 577.35N]

fig.1 fig.2 fig.3 2. A uniform wheel of 600mm dia. and weighing 1000N rests against rectangular block 15cm high lying on a horizontal plane as shown in fig. 2. It is to be pulled over the block by a horizontal force P applied to the end of a string wound round the circumference of the wheel. Find the force P when the wheel is just about to roll over the block. [P = 577.33 N] 3. Two smooth spheres A and B weigh 200N and 100N respectively are resting against two smooth vertical walls and a smooth horizontal floor as shown in fig. 4. The radius of sphere A is 100mm and Radius of sphere B is 50mm. Find the reactions from the vertical walls and horizontal floor. Also find the reaction exerted by each sphere on the other. [Rc = 89.44N, Rd = 300N, Re = 89.44N, R = 134.164N]

fig.4 fig.5 fig.6 4. Find force transmitted by cable BC shown in fig.4. E is frictionless pulley where B and D are weightless rings. [TBC = 565.7N] 5. If the cords suspend the two buckets in equilibrium position shown in fig.5. Determine weight of bucket B if Bucket A has a weight of 60N. [WB = 88.8N] 6. A roller of weight W = 1000N rests on a smooth inclined plane. It is kept from rolling down the plane by string AC. Find the tension in the spring and reaction at the point of contact D. Refer fig.6 [T=732N, RD=896.6N at 45⁰]

fig.7 fig.8 fig – 9 7. Determine the uniform reaction from the surface on cylindrical log as shown in fig.7. The self-weight of log is 1300N/m. [R=4500N/m] 8. Determine the horizontal distant x to which a 5m long inextensible string holding a weight of 3kN can be pulled before the string breaks. The string can withstand a maximum force of 6kN as shown in fig.8. Determine also the required force F. [x=4.33m, F=5.196kN] 9. A crane pivoted at the end B is supported by a guide at A. Determine the reaction produced at A and B by a vertical load W=5kN applied at C. As shown in fig.9 [RB= 8.33kN & θ=36.87⁰, RA=6.67kN] 10. Forces act on a plate ABCD are shown in fig.10. Given that the plate is in equilibrium. Find the force F, the angle ἀ and distance the distance AD.

Fig - 10

2.1.1 EQUILIBRIUM – Assignment Questions 1. A man raises a 12Kg joist of length 4m by pulling the rope. Find the tension in the rope and reaction at A. [T = 98.51 N, HA = 92.57 N, VA = 151.4 N]

Fig .1

Fig .2

2. Two identical rollers each of weight Q=500N are supported by an inclined plane in the vertical wall as shown in the fig.4. Assuming smooth surfaces find the reaction induced at the points of support A, B and C [HC=577.36N, RA=433N & 600, RB=721.7N & 600) 3. A thin ring of mass 3 kg and radius 140 mm is held against a frictionless wall by a string of length 120 mm. Find distance d and tension in the string. Ref. fig.3 [d = 219.09mm, T = 34.925N]

Fig.3

Fig.4

4. Three cylinders are piled up in a rectangular channel as shown in the fig.2. Determine all reactions between contact surfaces. All surfaces are smooth. Assume g=10m/sec2. [R1 =165N, R2=259.27N, R3=965N, R4 = 1000N, R5 =750N, R6 =800N]

2.1.2 EQUILIBRIUM – Tutorial Questions 1. Two spheres A and B of weight 1000N and 750N respectively are kept as shown in the fig – 1. Determine the reactions at all contact points 1, 2, 3 and 4. Radius of A = 400 mm and B = 300 mm. [ R1 = 496.65N, R2 = 1463.26N, R3 = 869.14N, R4 = 573.48N]

Fig.1 Fig.2 2. Two cylinders each of diameter 100 mm and each weighing 200N are placed as shown in fig- 2. Assuming that all contact surfaces are smooth find the reactions at A, B and C. 3. A 40 kg cylinder is held in position on an inclined plane by means of a wire AB as shown in fig- 3. Determine reactions at surface of inclined plane and tension in the wire. [T = 319.544N, R = 436.086N]

Fig.3

Fig.4

4. Determine the forces in various segments of the cable. Vertical forces 200N and 250N acts at point B and D. Ref. Fig – 4. [TDB = 183N, TDE = 224.125N, TBC = 336.60, TBA = 326.79N]

2.1.3 EQUILIBRIUM – Practice Questions 1. Determine the force P applied at 450 to the horizontal just necessary to start a roller 100cm in dia. over an obstruction 25cm high, if the roller weighs 1000N. Also find the magnitude and direction of P when it is minimum. [896.48N and 866N at 600 to horizontal] Ref. fig – 1

Fig – 1

Fig – 2

Fig – 3

2. Three cylinders are piled up in a rectangular channel as shown in the fig.2. Determine all reactions between contact surfaces. All surfaces are smooth. Assume g=10m/sec2. [R1 =165N, R2=259.27N, R3 = 965N, R4 = 1000N, R5 = 750N,R6 = 800N] 3. Find the reactions at A, B, C and D. Neglect friction. Refer fig. 3. [RA=11.754kN, 300, RB=54.123kN, RC=10.18kN, RD=22.441kN]

Fig – 4

Fig – 5

Fig – 6

4. Two identical rollers each of weight Q=500N are supported by an inclined plane in the vertical wall as shown in the fig.4. Assuming smooth surfaces find the reaction induced at the points of support A, B and C [HC=577.36N, RA=433N & 600, RB=721.7N & 600] 5. Two Homogeneous solid cylinders of identical weight of 5000N and radius 400mm are resting against inclined wall and sloping ground as shown in fig.5. Assuming all smooth surfaces, find the reaction at A, B and C of the contact points. [RA=RB=4330.13N, RC=5000N] 6. Determine the reactions at point of contact, 1, 2 & 3 assume smooth surfaces. Take mA=1kg, mB=4kg, rA=1cm, rB=4cm. Refer fig 6. [R1=19.729N, R2=11.604N, R3=32.216N]

Fig - 7

7. Two cylinders having weights WA=2000N, WB =1000N are resting on a smooth inclined plane having inclinations 60⁰ and 45⁰ with the horizontal respectively as shown in the fig.7. They are connected by a weightless ball AB with hinge connections. Find the magnitude of force P required to hold the system in equilibrium. [P=0.536kN]

2.2 EQUILIBRIUM OF BEAMS - Class Work Questions 1. Calculate the support reactions of the beam loaded as shown in figs.

[RB = 11.266 KN, Ay = 6.734 KN]

[BX = 15.32 KN, right, BY = 16.635 kN, RB = 251.72KN]

[RB = 136.03 KN, Ay = 103.9 KN]

[AX = 125.86 KN,right, AY = 71KN,up RB = 22.61KN, RA = 18.225 KN up]

2. Find analytically the support reaction at B and the load P for the given beam if the reaction at support A is Zero. Take M = 20KN.m [RB = 102 KN, up, P = 56 KN down]

3. Find reactions at A and B for a bent beam ABC loaded as shown in fig.

4. A two span beam ABCD is loaded as shown in fig. Calculate reactions RA, RB, RC. [RD = 30 kN up, CY = 30 kN, AX = 10 kN right, RB = 53 kN up, AY = 13 kN down]

5. Find the reactions of the beam as shown in the fig.[RB = 60 kN up, AY = 10 kN up, RB = 60 kN up]

6. Two beams AB and CD are arranged and supported as shown in fig. Compute the reactions due to load of 1000 N acting on AB. [ RD = 750N, AY = 250 kN up, RE =937.5N up, CY = 187.5 N down]

7. Determine the support reactions of the beam loaded as shown below. [HA = 10 kN(→), VA = 110 kN(↑), MA = 680 kN.m ( Anti clockwise)]

8. Find the reaction at A of the following beam loaded as shown below. [MA = 339.17 kNm(Anti cockwise), HA = 28.19 kN (→), VA = 41.26 kN, RA = 49.97 kN at 55.66˚ in 1st Quadrant]

2.2.1 EQUILIBRIUM OF BEAMS – Assignment Questions 1. Determine support reactions for beam AD analytically. Neglect self-weight of beam.[RBy =150kN, RBx = 41.33(← ) kN, RD= 85.89 kN]

2. Determine support reactions for a beam loaded and supported as shown fig below. [RB = 7.216 KN (↑), HA = 5.66 KN (←), VA = 8.44 KN (↑), RA = 10.16 KN, θ = 56.15˚]

3. Determine support reactions for a beam loaded and supported as shown fig below. [RB = 60 KN (↑), RA = 50 KN (↑)]

2.2.2 EQUILIBRIUM OF BEAMS – Tutorial Questions 1. Find the reactions at the supports of the beam AB loaded as shown in fig. [RA = 705kN, RBX = 2.12kN, RBY = 8.62kN]

2. Assuming pulley to be smooth calculate the support reactions of the beam as shown in fig. [HA = 14.14 kN left, Rc = 98.79 kN up, VA= 67.07 kN up, RA = 68.54 kN up, θ = 78.01˚]

3. Determine support reactions for the beam. [RE = 103.65 kN up, HA = 50 kN(→), VA = 82.95 kN(↑), RA = 96.85 kN, θ= 58.92˚]

4. Find reaction components at the internal hinge B and supports A and C for the compound beam shown. [Rc = 589.15 up, By = 4.15kN up, HA = 25 kN Right, VA = 44.15 kN up]

2.2.3 EQUILIBRIUM OF BEAMS – Practice Questions 1. Find the reactions at the supports of the beam AB loaded as shown in fig. [RF = 67.63 kN at 60˚ 2nd Q, HB = 104.53kN(→), VB = 152.14 kN(↑), RB = 184.59 at 55.15˚,1st Q]

2. Find the reactions at the supports of the beam at hinge A and roller B.

3. For the beam shown in fig. find all support reactions. Point C is an internal hinge. [RD = 40 kN(↑), HC = 60 kN (→), VC = 40 kN (↑), RB = 150 kN (↑), RA = 67.08 kN at 26.57˚ 1st Q]

4. Find all support reactions, B is an internal hinge [Rc = 10 kN(↑), VB = 10 kN(↑), MA = 95 k N.m (anti Clock), HA = 0, VA= 80 kN (↑)]

5. For the beam shown in fig point c is an internal hinge. Calculate all support reactions. [RD = 18.475N, HC = 9.24N, VC = 8 N, M = 14 Nm (anti clockwise) VA = 18 N(↑), RA = 20.23 N at 62.82˚ 1st Q]

6. Calculate the reaction at supports D and C. [RD = 4.55 kN(↑), HC = 0.05 kN(→), VC = 0.4 kN (↑)]

Fig - 7 7. Determine the intensity of distributed load W at the end C of the beam ABC for which the reaction at C is zero. Also calculate the reaction at B. Ref. fig. - 7. [W = 3 kN/m, RB = 21.6kN] 8. Determine minimum weight of block required to keep the beam in horizontal equilibrium. Assume smooth pulley Ref. fig. – 8.

Fig – 8

Fig – 9

9. A light beam AB 1.8 m long resting on soil has linearly distributed reaction as shown in fig – 9. Determine distance ‘a’ in meter and corresponding WB in kN/m. [WB = 42 kN/m, a = 0.42m]

2.3 TRUSSES – Class work Questions 1. Referring to the truss in fig – 1 Identify zero force members and forces in other members by method of joints.

Fig – 1

Fig – 2

2. Find the forces in the members CE, CD and DB by method of sections and remaining by method of joints. Ref. fig. 2. 3. Find the forces in members BD, BE and CE by method of section only for the truss shown in fig 3. Also find the forces in other members by method of joints.

Fig – 3 4. Calculate forces in all members by method of joints. Ref. Fig – 4.

Fig – 4

2.3.1 TRUSSES – Assignment Questions 1. Find the magnitude and nature of forces in all the members of truss shown in fig – 1.

Fig – 1 Fig – 2 2. Find the forces in the members CD, CE and DG by method of sections and remaining members by method of joints. 3. Find the forces in the members CD, DE and FD by method of sections and remaining members by method of joints.

2.3.2 TRUSSES – Tutorial Questions 1. A pin jointed truss is loaded and supported as shown; determine the forces in the members BD, CD and CE by method of section and forces in remaining members by method of joints.

2. For the truss loaded as shown in the fig find the forces in the members CE, CF by method of sections only.

3. Referring to the truss shown in fig. below find (i) Reactions at D and C (ii) Zero force members (iii) Forces in members FE, EC and DC by method of section (iv) Forces in other members by method of joints.

3.1 FORCES IN SPACE (Non - coplanar forces) - Class work Questions 1. A force of magnitude 650N passes through point p (0, 3, 0) to point Q (5, 0, 4). Write equation of the force in vector form. [459.6i – 275.7j +367.5k] 2. Express the force F shown in fig – 1 as Cartesian vector and write component of force. [150N, 150N, 212.13N]

Fig – 1 Fig – 2 Fig – 3 3. Represent the force F shown in fig – 2 as a Cartesian vector. [F = 4.33 i + 7.5 j + 5 k] 4. Determine components of 6 kN force and 4.5 kN force as shown in fig – 3. Also find angles  x, y and z with co – ordinate axis. [6 kN = Fx = 2.196, Fy = 5.44, Fz = 1.268, x, y and z= 68.53, 25, 77.8. For 4.5 kN = - 2.369, 2.581, 2.823, 121.76, 55, 51.15] 5. Three forces F1, F2 and F3 act at the origin O. F1 = 70 N acting along OA, where A (2, 1, 3). F2 = 80N acting along OB, where B (-1, 2, 0). F3 = 100 N acting along OC, where C (4, -1, 5). Find the resultant of these concurrent forces. [63.36i + 74.83j + 133.27k] 6. The resultant of three concurrent space forces at A is R = (-788j) N. Find the magnitudes of F1, F2 and F3. [F1 = 153.48N, F2 = 320.3N, F3 = 400.39N] Ref. Fig – 4.

Fig – 4 Fig – 5 7. Knowing that the tension in AC is 20 KN, determine the required value of tension TAB and TAD so that the resultant of the three forces applied at A is vertical and calculate the Resultant. Ref. Fig – 5. [TAB = 37.143 kN, TAD = 61.22 kN, R = 110.09 kN(↓)] 8. A force F = 80 i + 50 j – 60 k passes through a point A (6, 2, 6). Compute its moment about point B (8, 1, 4). [MB = 244.13 units, x , y and z = 130.95, 80.57, 137.5]

9. A force of 500N passes through points whose position vectors are r1 = 10 i – 3j + 12k and r2 = 3i – 2j + 5k. What is the moment of this force about a line in X – Y plane, passing through the origin and inclined at 30° with X – axis? [MFOC = 39.9 N.m. = - 34.55i – 19.95j] 10. A force F acting along MN where M (0, 1, 2) and N (-1, 3, 0) produces a moment of 80 Nm about point P (2, 1, -1). Find the magnitude of the force. [25.3N] 11. A force of 20 kN acts at point A(3, 4, 5)m and has its line of action passing through B(5, -3, 4)m. Calculate the moment of this force about a line passing through ST where S(2, -5, 3) m and T(-3, 4, 6)m. [MST = 14.02 kN.m = 6.61i – 11.78j + 3.93k] 12. A square foundation supports four loads as shown in fig – 6. Determine magnitude, direction and points of application of resultant of four forces. Ref fig –6 [ R = - 80 kN, Point ( 1.75, 0, 1.5)m]

Fig – 6 Fig – 7 13. Replace the given force system by a force and couple at the origin. Magnitude of the forces are F1= 40√2 N, F2 = 50 N, F3 = F4 = 40 N. Ref. Fig – 7. [R = -40j – 30k N, ∑MO = -120i Nm.] 14. Find the resultant of the force system given in fig – 8. Take F1= 100N, F2 = 20 N, F3 = F4 = 40 N. C1 = 20 N.m, C2 = 40 N.m, C3 = 80 N.m. [R = (-76.64i + 11.1j + 8k) N.MO =(-72i – 220j – 150.7k)N.m]

Fig – 8

Fig – 9

15. A vertical load of 1000 N is supported by three bars as shown in fig – 9. Find the forces in each bar. Points C, O and D are in X – Z plane while B is 1.5 m above this plane. [TBA = 327.77 N, TAC = 406.56 N, TDA = 445.4 N]

3.1.1 FORCES IN SPACE – Tutorial Questions 1. A pole is held in place by three cables. If the force of each cable acting on the pole is as shown in Fig – 1 determine the resultant. [F1 = - 317.6i – 282.4j – 423.5k, F2 = - 173.1i – 230.8j – 277k. F3 = 127.7i + 191.5j – 766.2k. R = - 16.8i – 321.7j – 1466.7k]

Fig – 1 Fig – 2 2. Find the resultant and couple moment about the origin of system of forces shown in Fig – 2. Take F1 = 20 kN, F2 = 50 kN, F3 = 30 kN and F4 = 40 kN and OA = 4m, OC = 5m and OE = 3m. [R = 25.54i + 58.78j + 49.21k. MO = - 130.29i – 168.22j + 13.72k] 3. A machine component is subjected to the forces as shown in fig – 3, each of which is parallel to one of the coordinate axes. Replace these forces by an equivalent force couple system at origin. [R = - 300i – 240j + 25k. MO = 14.25i – 9j + 0k]

Fig – 3

Fig – 4

4. A transmission tower is held by three wires at B, C and D. If T AB = 2000N, T AD = 1400N and T AC = 1600N, Find components of forces acting at B, C and D. Ref fig – 4. [At B = X = 368.42N, Y = 1894.74N, Z = - 526.315N. At D = 22.82, 1135.65, 788.64. At C = - 877.91, 1316.87, - 234.11] 5. A force of 80 N acts through point A (2, 0, 11) towards point B (6, 8, 3). Determine the moment of this force about (i) X- axis (ii) Y- axis (iii) Z – axis. [ F = 26.66i + 53.33j – 53.33k, MOX = ( - 586.63i)N.m. MOY = (399.92j) N.m. MOZ = ( - 106.67k) N.m.]

3.1.2 FORCES IN SPACE – Assignment Questions 1. A wire is connected by a bolt at A. If tension in wire is 3 kN, determine (i) Components of force acting at A (ii) The angles x, y and z. Ref fig – 1. [Fx = - 1.272, Fy =2. 545, Fz = 0.95 kN, 115.087°, 31.97°, 71.54°]

Fig – 1 Fig – 2 2. A square foundation supports four loads as shown in fig – 2. Determine magnitude, direction and points of application of resultant of four forces. Ref fig – 2 [R = - 400 kN, Point ( 1.75, 0, 1.5)m] 3. Determine the tension in the cables AB, AC and AD. If weight of cylinder is 160 kg Ref fig -3 [1316.435N, 1892.95N, 1211.63N]

Fig – 3

Fig – 4

4. A box of size 3 × 4 × 2m is subjected to three forces as shown in fig – 4. Find in vector form the sum of moments of the three forces about diagonal OB. [MOB = 16.548i + 8.274j + 12.411k]

5. Force P1 = 10N acts along AB. Another force P2 = 5N acts along BC. The co-ordinates of points are A (3, 2, - 1) B (8, 5, 3); C (- 2, 11, - 5). Determine (i) The resultant of p1 and p2 in vector form (ii) The moment of resultant about point D (1, 1, 1). Co – ordinate distances are in meters. [R = 7.81N. MD = 32.989 N-m]

3.2 FRICTION - Class Work & Practice Questions 1. A wooden block rests on a horizontal plane as shown in fig.1. Determine the force P required. (a) Pull it (b) Push it. Weight of the block is 100N µ=0.4. [37.4N, 46.38N]

fig.1

fig.2

2. Determine the minimum value and the direction of force P required to cause motion of a 100kg block to intend upon 30⁰ plane. µ=0.2. Take g=10m/s2. Refer fig.2. [α=11.31⁰, P=660.13N] 3. Find tensions in the cords of the inclined plane system shown in fig.3. Assume that system of blocks is in impending motion. [165.36kN in the cord connecting two blocks, 80.51kN in the cord connecting support]

fig.3 fig.4 4. Two inclined planes AC and BC inclined at 60⁰ and 30⁰ to the horizontal meet at ridge C. A mass of 100kg rest on the inclined plane BC and is tied to a rope which passes over a smooth pulley at the ridge, the other end of rope being connected to a block of W kg mass resting on the plane AC. Determine the least and greatest value of W for the equilibrium of the whole system. Refer fig.4 [Wmin=243.92N, Wmax=973.07N] 5. Find minimum value of F to move block A up the plane. µ=0.2 for all rubbing surfaces. Refer fig.5.[135.62N] 6. Determine the force P to cause motion to impend. Masses of blocks A & B are 9 kg and 4 kg respectively and of sliding friction as 0.25. [P=3.04N, P=27.695N] Refer fig .6.

Fig.5

Fig.6

Fig.7 Fig.8 7. Find the weight Wmin if weight WA is 20kN. Is to be kept in equilibrium with pin connected rod AB in horizontal position. Find also maximum value of WB for the same purpose. Refer fig.7. [Wmin=4.511kN, Wmax=26.365kN] 8. Two blocks A=100N and B=150N are resting on the ground as shown in the fig.8. Find the minimum weight P in the pan so that motion starts. Find whether B is stationary w.r.t ground and A moves or B is stationary w.r.t A. Assume pulley to be mass less and frictionless. [P=27.41N. B is stationary w.r.t A)

Fig.9 Fig.10 9. A non-homogeneous ladder as shown in fig.9 rest against a smooth wall at A and a rough horizontal floor the mass of ladder is 30kg and is concentrated at 2m from bottom. µ=0.3 at floor. Will the ladder stand in 60° position as shown? [Ladder will stand in position] 10. A 100N uniform rod AB is held in position as shown in fig.10. If µ=0.15 at A and B calculate range of value of P for which equilibrium is maintained. [8.29N < P < 80.58N] 11. A weightless ladder of length 8m is resting against a smooth vertical wall and rough horizontal ground as shown in fig.11. The co efficient of friction between the ground and the ladder is 0.25. A man weighing 500N wants to climb up the ladder. Find how much distance along the ladder the man can climb without slip. A second person weighing 800N wants to climb up the same ladder. Would he climb less than the earlier person? Find his distance covered. [X=3.46m]

Fig.11

Fig.12

12. A uniform ladder 3m long weighs 200N, it is placed against a wall 60° with floor as shown in fig.12. Co-efficient of friction between the wall and ladder is 0.3 and that between floor and ladder is 0.4. The ladder in addition to its own weight has to support a man weighing 800N at its top at A. (i) Calculate the horizontal force F to be applied to the floor level to prevent slipping. (ii) If the force F is not applied, what would be the minimum inclination of the ladder with the horizontal so that there is no slipping of it with the man at the top. [F=96.048N, Ѳ=65.75°]

Fig.13 Fig.14 13. A block of mass 150kg each raised by a 10⁰ wedge weighing 50kg under it and by applying a horizontal force at it as shown in the fig.13. Taking µ between all surfaces of contact as 0.3, find what minimum force should be applied to raise the block. [1538.2N] 14. Refer to the fig.14, draw FBD for different bodies and find the minimum value of force F to move the block A up the plane. [134.6 N]

fig.15 fig.16 15. Two 6° wedges are used to push a block horizontally as shown in fig.15. Calculate the minimum force P required to push the block of weight of 10kN. Take µ=0.25. [1.639kN] 16. Referring to the fig.16, µ is 0.25 at the floor, 0.3 at the wall and 0.2 between the blocks. Find the minimum value of a horizontal force applied to the lower block that will hold the system in equilibrium. [81.02N] 17. What force P must be applied to the weightless wedges shown in the fig.17 to start them under the 1000kN block? The angle of friction for all surfaces is 10°. [P=321.06kN]

Fig.17

3.2.1 FRICTION – Assignment Questions 1. A block of weight 200 N rests on a horizontal surface. The coefficient of friction between the block and the horizontal is 0.4. Find the frictional force acting on the block if a horizontal force of 40 N is applied to the block. Ref. fig – 1. [F max = 80N, F = 40N]

Fig.1

Fig.2

2. A support block piece acted upon by two forces as shown in fig.2. The coefficient of friction between the block and inclined are µS=0.35, µK=0.25. Determine the force P required (a) To start the block moving up the plane. (b) To keep it moving up. (c) To prevent it from sliding. [(a) P=780.8N, (b) P=649N, (c) P=80N]

3. Determine the force P to cause motion to impend. Take masses of blocks A and B as 8kg and 4kg respectively and the coefficient of sliding friction is 0.3. The force P and rope are parallel to the inclined plane. Assume frictionless pulley. Ref. fig – 3. [P = 10.99 N]

Fig.3

Fig.4

4. Two identical blocks A and B are connected by a rod, rest against vertical and horizontal planes respectively as shown in fig.15. If sliding impends when Ѳ=45°, determine the µ assuming it to be same at both surfaces. [0.414]

3.2.2 FRICTION – Tutorial Questions 1. Find the value of θ if the block A and B placed as shown in the fig.2 have impending motion. Weight of A=200N, B=200N, µ for all surfaces= 0.25. [θ=28.07⁰]

Fig. 1

Fig.2

2. Two blocks are connected by a horizontal link AB, are supported on two rough planes. The µ for block A is 0.4 and angle of friction for block B is 15⁰. What is the smallest W of block A for which equilibrium of the system can exist. Refer fig.1. [W=1000]

Fig.3

Fig.4

3. The mass of A is 23kg and mass of B is 36 kg. µ is 0.4 between A and B, 0.2 between plane and block B. Assume smooth drum. Determine the minimum mass of m before it impends. Ref fig. 3. [9.2 kg] 4. A ladder of length 7m beam against a wall as shown in fig.6. Assuming that the coefficient of static friction is zero at B. Determine smallest value of µS at A for which equilibrium is maintained. [µ=0.202] 5. The block A as shown in the fig.5 supports a load W=5000N and is to be raised by forcing the wedge B under it. Determine the force P which is necessary to start the wedge under the block. The block and wedge having negligible weight. µ=0.27 for all surfaces. [P=6010.2N]

Fig – 5

ST. FRANCIS INSTITUE OF TECHNOLOGY MT. POINSUR, BORIVALI (W), MUMBAI

QUESTION BANK ENGINEERING MECHANICS (DYNAMICS) FE SEM 1 Academic Year: 2014-15

Prepared By: Engineering Mechanics Dept.

4.1 CURVILINEAR MOTION - Class Work & Tutorial Questions 1. A car is travelling along a circular curve that has a radius of 50m. If its speed is 16m/s and is increasing uniformly at 8m/s2. Determine the magnitude of its acceleration at this instant. [9.5m/s2] 2. A particle moves in x-y plane and its position is given by r = (3t)i + (4t-3t2)j, where r is the position vector of particle in meters at time t sec. Find the radius of curvature of the path and normal and tangential components of acceleration when it crosses X-axis again. [Ρ = 4.167m, at=0, an=6m/s2] 3. A particle moves in X - Y plane with acceleration components ax = -3m/s2 and ay = -16m/s2. If its initial velocity is V0 = 50m/s directed at 300 to the X – axis, compute the radius of curvature of the path at t = 2 sec. [Ρ = 88.84m] 4. A particle moves in X – Y plane with acceleration components a = (-3)I – (16t)j m/s2. If its initial velocity is V0 = 30m/s directed at 30o to the X – axis, compute at t = 2 sec. (1) The radius of curvature of the path. (2) Tangential acceleration. (3) Normal acceleration. [Ρ = 26.15m, an=26.31m/s2, at=18.45m/s2] 5. A rocket follows the path such that its acceleration is given by a = 4i + tj m/s2. At t = 0 it starts from rest. At t = 10 sec. Determine speed of the rocket and radius of curvature of its path. [V=64.03m/s, Ρ=1312.64m] 6. A particle at the position (4, 6, 3) at start is accelerated at a = (4t)I – (10t2)j m/s2. Determine the acceleration, velocity and displacement after 2 sec. [a=40.79m/s2, V = 27.84m/s, s = 14.25m] 7. An airplane travels on a curved path. At point P it has a speed of 360kmph which is increasing at the rate of 0.5 m/s2. Determine at P (1) magnitude of total acceleration (2) angle made by the acceleration vector with positive X – axis. Refer fig.7. [a=0.778m/s2, θ=108o]

Y=x2/3 Y=0.2 x2 km

A

P(4,3.2)km

Y=x2/20

P(3,3)

5m 10m

fig.7

fig.8

fig.9

8. A skier travels with a constant speed of 6m/s along the parabolic path Y = X2/20 shown in the fig.8. Determine the velocity and acceleration at the instant he arrives at A. Neglect the size of the skier. [a=1.273m/s2] 9. A point moves along a path Y = X2/3 with a constant speed of 8m/s. What are the X and Y components of velocity, when X= 3m. What is the acceleration at this point? Refer fig. 9. [Vx=3.578m/s, Vy=7.155m/s, a=3.816m/s2] 10. The Y – coordinate of a particle is given by Y = 4t3-3t, ax = 12t m/s2 and Vx= 4m/s at t = 0. Calculate magnitude of velocity and acceleration of particle when t = 1 sec. [V=13.45m/s, a=26.83m/s2].

11. If x=1-t and y=t2, where x and y are in meters and t is seconds. Determine X and Y components of velocity and acceleration. Also write equation of path. [Vx= -1m/s, Vy= 2t, ax= 0m/s2, ay= 2m/s2, y=(1-x)2] 12. Explain what tangential and normal acceleration is. A particle is moving in X – Y plane and its position is defined by r = (

)i + (

)j. Find radius of curvature when t = 2 sec. [41.667m]

4.1.1 CURVILINEAR MOTION – Assignment Questions 1. A particle starting from rest at the position (5, 6, 2) m accelerates at a = 6t I – 24t2 j + 10 k m/s2. Determine the acceleration, velocity and displacement of the particle at the end of 2 seconds. [a = 97.26m/s2, V = 68.12m/s, S = 36.45m] 2. A rocket follows a path such that its acceleration is given by a = (4i + tj) m/s2 at r = 0, it starts from rest. At t = 10 sec. Determine (i) speed of the rocket (ii) Radius of curvature of its path (iii) Magnitude of normal and tangential components of acceleration. [V=64.03, R= 1312.64, a=10.77,an=3.123,at=10.307]

4.1.2 PROJECTILE MOTION - Class Work & Tutorial Questions 1. A gunman standing on the ground fires from his gun to hit a bird flying at an altitude of 20m from the ground. The angle of projection is 60° up the horizontal. If the bullet hits the bird 2.5 sec. after firing find (i) velocity of the bullet as it left the gun. (ii) The bird killed instantly. Find the time it takes to reach the ground. Neglect the height of gunman. [23.4m/s. t = 2.019sec] 2. It is observed that a skier leaves the platform at A and then hits the ramp at B after 5 sec. calculate the initial speed ‘u’ and launch angle ‘α’. Ref. fig – 1.[α = 36.23°, u = 19.84 m/s]

fig – 1

fig – 2

fig – 3

3. A box is released from a helicopter moving horizontally with constant velocity ‘u’ from a certain height ‘h’ from the ground takes 5 sec. to reach the ground hitting it at an angle of 75° as shown in fig – 2. Determine (i) horizontal distance ‘X’ (ii) The height ‘h’ (iii) The velocity ‘u’ [h = 122.625m, u = 13.14m/s, X = 65.7m] 4. A ball is thrown upwards from a high cliff with velocity 100m/s at an angle of elevation 30°. The ball strikes the inclined ground at right angles. If the inclination of the ground is 30° as shown in fig – 3. Determine (i) the time after which the ball strikes the ground. (ii) Velocity with which it strikes the ground. (iii) Co – ordinates (X,Y) of the point of strike w.r.t. the point of projection. [VB = 173.2 m/s, t = 20.39 sec, B ( 1765.2m,1021.8m)]

5. A ball is thrown with a speed of 12 m/s at an angle of 60˚ with a building strikes the ground 11.3 m horizontally from the foot of the building as shown in fig -4. Determine the height of the building and time.[12.32 m, 1.087s]

Fig –4

fig – 5

fig – 6

6. A projectile ‘P’ is fired at a muzzle velocity of 200 m/s at angle of elevation of 60°. After sometime a missile ‘M’ is fired at 2000m/s and at an angle of elevation of 45° from the same point to destroy the projectile P. Find (i) height (ii) Horizontal distance (iii) time w.r.t. P firing at which the destruction takes place. Ref fig – 5. [1495m, 1500m, 15sec] 7. A player is ready to throw a ball with a maximum possible velocity of 15m/s from a height of 1.5m above the ground level. He wants to hit the wall 16m away from him at its highest point. If the ceiling height of the room is 8m, find the angle of projection of the ball and the height of point where the ball hits the wall. Ref fig – 6. [48.84°, 5042m] 8. A boy throws a ball with an initial velocity u = 24 m/s. Knowing that the boy throws the ball from a distance of 30m from the building, determine (i) The maximum height ‘h’ that can be reached by the ball. (ii) the corresponding angle α. Ref fig – 7 [21.71 m, 62.95˚]

fig – 7

fig – 8

fig – 9

9. Balls of 10mm diameter leave the horizontal trough with an initial velocity (horizontal) u to fall through gap of 80mm size as shown in fig – 8. Calculate the permissible range of velocity u that will enable the balls to enter the gap. 10. The drinking fountain is designed such that the nozzle is located from the edge of the basin as shown in fig – 9. Determine the maximum and minimum speed at which the water can be ejected from the nozzle so that it does not splash over the sides of the basin at B and C which are at the same level. [0.835 ≤ VA ≤ 1.76 m/s] 11. A ball thrown down the incline strikes at a distance of 75 m along. If ball rises to a maximum height of 20 m above the point of projection. Compute the initial velocity and angle of projection with horizontal. Ref fig – 10. [u = 24.38m/s, α = 54.3°]

fig – 10 fig – 11 12. A stunt man wishes to jump across a water pool with his car and lands on a ramp at ‘D’ without jerk. Determine initial velocity and angle ‘θ’ of ramp. Ref fig – 11 [u = 8.85m/s, θ = 36.86°] 13. A jet of water discharging from a nozzle hits a verticle screen placed at a distance of 6 m from the nozzle at a height of 4m. when the screen is shifted by 4m away from the nozzle from its initial position the jet hits the screen again at the same point. Find the angle of projection and velocity of projection of the jet at the nozzle. [α = 46.85°, u = 12.54 m/s]

4.1.3 PROJECTILE MOTION – Assignment Questions 3. A boy throws a ball so that it may just clear a wall 3.6 m high. The boy is at a distance of 4.8 m from the wall. The ball was found to hit the ground at a distance of 3.6 m on the other side of the wall. Find the least velocity with which the ball can be thrown. [ 9.779m/s] 4. An aero plane is flying in horizontal direction with a velocity of 540 km/hr and at a height of 2200 m. When it is vertically above the point A on the ground, a body is dropped from it. The body strikes the ground at point B. Calculate the distance AB (ignore air resistance). Also find velocity at B and time taken to reach B. [t = 21.18s, X = 3176.7m, VB = 256m/s]

5. A ball is thrown from a horizontal level, such that it clears a wall 6 m high, situated at a horizontal distance of 35 m. If the angle of projection is 60˚ with respect to horizontal, what should be the minimum velocity of projection? [20.98m/s]

6. A projectile is aimed at a target on the horizontal plane and falls 12 m short when the angle of projection is 15˚, while it overshoots by 24 m when the angle is 45˚. Find the angle of projection to hit the target. 7. A gunman fires a bullet with a velocity of 100 m/s, 50˚ upwards from the top of a hill 300m high to hit a bird. The bullet misses its target and finally lands on the ground. Calculate (a) the maximum height reached by the bullet above the ground. (b) total time of flight (c) velocity with which the bullet hits the ground. [h = 299.09m, Hmax = 599.09m, t = 18.86 s, v = 126 m/s, θ = 59.30°, Vx = 64.278 m/s→, Vy = 108.4m/s(↓)]

4.1.4 RELATIVE MOTION - Class Work Questions 1. Fig .1 shows two cars A and B at a distance of 35m. Car A is travelling East at a constant speed of 36 Kmph. Car B starts from rest and moves south with a constant acceleration of 1.2 m/s2 determine (i) Position (ii) Velocity (iii) acceleration of car B relative to car A, 6 sec after car A crosses the intersection of roads. [Position of car B after 6sec = 13.4m, VB = 7.2m/s, a B/A = 1.2 m/s2]

fig – 1

fig – 2

fig – 3

2. A monkey is climbing a tree with a velocity of 10 m/s while a dog running towards the tree chasing the monkey with a velocity of 15 m/s. Find the velocity of dog relative to the monkey. [18.03m/s, θ = 33.69° in 4th quadrant] 3. Two trains leave a station in different directions at the same instant. Train A travels at 360 Kmph at 10° west of North. While train B travels at 450kmph at 60° East of North. Find (i) Relative velocity of train A w.r.t. train B (ii) The two trains are how much apart 2 minutes later? [V A/B = -452.22i + 129.53j, rA/B = 15.6799 km] 4. Fig – 2 shows cars A and B at a distance of 35m. Car A moves with a constant speed of 36 Kmph and car B starts with an acceleration of 1.5 m/s2. Determine (i) Position (ii) Velocity (iii) acceleration of car B relative to car A, 5 sec after car A crosses the intersection of roads. [Position of car B after 5sec = 18.75m,VB = 7.2m/s, a B/A = 1.5 m/s2] 5.

The velocities of commuter trains A and B are as shown in fig – 3. Knowing that the speed of each train is constant and that B reaches the crossing 10 min after A has passed through the same crossing, determine. (i) The relative velocity of B w.r.t. A (ii) The distance between the fronts of the engines 3 minutes after A passed through the crossing. [VB/A = 111.366 Kmph and θ = 10.5° in 1st quadrant, 2.96m]

4.1.5 RELATIVE MOTION – Tutorial Questions 1. Two ships move from a port at the same time. Ship A has a velocity of 30 kmph and is moving North 30˚ West, while ship B is moving South – West direction with a velocity of 40 kmph. Determine the relative velocity of A w.r.t B. [55.866 kmph, θ= 13.76˚ NE]

4.1.6 UNIFORM ACCELERATION MOTION AND MOTION UNDER GRAVITY – Class Work Questions 1. Two cars moving in the same directions are 150 m apart. Car A being ahead of car B. At this instant, velocity of A is 3 m/s and constant acceleration is 1.2 m/s2. While the velocity of B is 30 m/s and its uniform retardation is 0.6 m/s2. How many times do the cars cross each other? Find when and where they cross w.r.t. position of A. [t=7.36s and 22.64s, d=54.58m and 375.46m] 2. Two cars A and B travelling in the same direction on same adjacent lanes are stopped at a traffic signal. As the signal turns green car A accelerates at a constant rate of 2 m/s2. Three seconds later car B starts and accelerates at 3.6 m/s2. Find (1) When and where B will overtake A (2) The speed of each car at that time. [SA = 138.77m, VA = 23.56 m/s, VB = 31.61 m/s] 3. Two cars A and B travelling at a constant speed of 170 kmph. The car A is leading car B by 36 m, At t=0. They accelerate at constant rates. Knowing that when B passes A, t = 8 sec and VA = 220 kmph find acc of A & B. [aA = 1.74 m/s2, aB = 2.865 m/s2, d= 433.44 m]. 4. A stone was thrown vertically up from the ground with a velocity of 49 m/s. After 2 sec, another stone was thrown vertically up from the same level. If both stones hit the ground simultaneously, find the velocity with which the second stone was thrown? [t = 10 sec, uB =39.2m/s]. 5. From the top of a tower 49m high a stone is allowed to fall vertically downwards. One second later, a ball is thrown vertically upwards from ground with a velocity of 12.25 m/sec. At what distance above the ground will they cross each other? [t = 3 sec, d = 4.9m] 6. From the top of a tower 30 m high, a stone is thrown vertically up with a velocity of 8m/sec. After how much time will the stone reach the ground? With what velocity does it strike the ground? [t=3.42sec, v = 25.52 m/sec] 7. A stone is dropped into a well and sound of splash is heard 3.3 sec after the stone is dropped. If velocity of sound is 350 m/sec, find depth of well upto water level. [t1 = 3.16 sec, d = 48.93m] 8. A stone is released from top of a tower h meter high. It covers a vertical distance h/5 during its last second of descend. Find height of the tower. [t=8.47 sec, h =439.41m] 9. A stone has fallen a distance of 12m after being dropped from the top of a tower, another stone is dropped from a point 20m below from the top of a tower. If both stones reach the ground together, find height of tower. [VB = 15.34 m/sec, h = 21.33m] 10. A stone is projected from top of a building 120 m high with initial velocity of 25 m/sec. A second stone is projected vertically downwards with the same velocity. Find time taken by each stone to reach the ground. At what height the first stone must be just released from rest in order the two stones may hit the ground simultaneously? [t = 8.12 sec, t =3.016 sec, h = 44.57m] 11. In a flood relief area a helicopter going up with a constant velocity drops first batch of food packets which takes 4 sec to reach ground. No sooner than this batch reaches the ground, second batch of food packets are released which takes 5 sec to reach the ground. From what height the first batch of packets is released? What is the velocity with which the helicopter is moving up? [u = 14.72 m/sec, h = 137.36m].

12. Water dips from a tap at the rate of 5 drops/ sec. determine the vertical separation between two consecutive drops after the lower drop has attained a velocity of 3m/sec. [t = 0.306 sec, S1 = 0.458 m, S2 = 0.055m, S = 0.403m] 13. A balloon rises from the ground with constant acceleration of 1m/sec2. 6 sec later a stone is thrown vertically up from the ground. What must be the minimum initial velocity of the stone to just touch the balloon? [t =1.825 sec, u = 25.71m/sec] 14. A stone is thrown up with a velocity of 25m/sec from ground. On return journey it strikes a glass placed at half the height and it loses 25 % of its velocity in breaking the glass. Find velocity with which it will strike the ground [v = 22.096 m/s] 15. A stone is observed to fall past a 1.25m window in 0.2 sec. Determine (a) the average speed of the stone while it is in view from inside the room. (b) The velocity when it reaches bottom of window (c) the starting position of the stone [average speed = 6.25m/s, t= .54 sec, d = 1.43m (starting position above top of window) v = 7.25 m/sec]

4.1.7 UNIFORM ACCELERATION MOTION AND MOTION UNDER GRAVITY – Assignment Questions 1. A ball is thrown vertically upwards at 30 m/s from the top of a tower 100 m high. Five seconds later another ball is thrown upwards from the base of the tower along the same vertical line at 50 m/s. Find when and where they will meet and their instantaneous velocity then. [t = 6.85 s, Vst ball = 37.52m/s down, V second ball = 31.85 m/s up] 2. A stone is released from the top of the tower, during the last second of its motion, it covers ¼th of the height of the tower. Find the height of the tower. [273.27m]

4.1.8 UNIFORM ACCELERATION MOTION AND MOTION UNDER GRAVITY – Tutorial Questions 8. A point is moving with uniform acceleration. In the 11th and 15th second from the commencement, it moves through 7.2 m and 9.6 m respectively. Find its initial velocity and the acceleration with which it moves. 9. A stone is dropped into a well and 4.5 seconds later a splash is heard. Then the second stone is thrown downward with initial velocity u into the well and splash is heard after 4 seconds. If the velocity sound is constant and it is 330 m/s, determine the initial velocity of second stone. [ 5.23m/s] 10. A burglar’s car had a start with an acceleration of 2 m/s2. A police vigilant came in van to the spot at a velocity of 20m/s after 3.75 sec and continued to chase the burglar’s car with uniform velocity. Find the time in which the police van will overtake the burglar’s car. [t = 5 sec and 15 sec]

4.1.9 VARIABLE ACCELERATION MOTION - Class Work Questions 1. The equation of motion of a particle moving in a straight line is given by S = 18t + 3t2 – 2t3 where s is in meter and t is in sec. Find (i) Velocity and acceleration at start. (ii) Time when particle reaches its maximum velocity. (iii) Maximum velocity of the particle. [V = 18 m/s, a = 6 m/s2, t = 0.5 sec, Vmax = 19.5 m/s] 2. Rectilinear motion of a particle is defined by equation v3 = x2, determine velocity and acceleration at x = 8m [v= 4m, a = 1.33m/s2] 3. A sphere is fired downward into a medium with an initial speed of 27m/s. If it experiences a deceleration a = - 6t m/s2 where t is in sec. Determine the distance travelled before it comes to rest. [x = 54m] 4. A particle starting with an initial velocity and travelling in a straight line has an acceleration of (2t+4) m/s2 where t is time in sec. from start. The distance covered in the first second measured from the starting point is 6.33m. Calculate (i) initial velocity (ii) The distance covered in the third second.[u = 4m/s, x 3rd = 20.33m] 5. The position of a particle which moves along a straight line is defined by the relation x = t3 – 6t2 – 15t + 40. Where x is in meters and t is in seconds. Determine (1) The time at which the velocity will be zero (2) The position and distance travelled by particle at that time. (3) The acceleration at that time (4) The distance travelled by particle from t = 4 sec to t = 6sec. [t=5sec; position=-60m & d5=100m; a=18m/s2; d=18m] 6. The acceleration of a particle is given by the relation a=90 – 6x2 where a is expressed in cm/sec2 and x is cm. if the particle starts with zero initial velocity at position x = 0. Determine, (1) The velocity when x = 5cm. (2) The position where velocity is again zero. (3) The position where the velocity is maximum. [V=20m/s; x=6.71cm; x=3.87m]

4.1.10 VARIABLE ACCELERATION MOTION – Assignment Questions 7. The velocity of the body is defined by the expression v = (6 – 0.03X) m/s where X is expressed in meters. If x = 0 at t = 0, determine (a) the distance travelled by the body when it comes to rest. (b) Acceleration at t = 0 (c) time when x = 100m. [x = 200m, a = - 0.18 m/s2, t = 23.105s] 8. Motion of the particle along a straight line is defined by V3 = 64 S2 where V is in m/sec and S is in m. Determine (a) Velocity when the distance covered is 8 m. (b) Acceleration when the distance covered is 27 m. (c) Acceleration when the velocity is 9 m/sec. [V = 16 m/s, a = 32 m/s2, a = 16 m/s2]

4.1.11 VARIABLE ACCELERATION MOTION – Tutorial Questions 9. A particle moving in positive x – direction has an acceleration a = (100 – 4 V2 ) m/s2 where V is in m/s. determine the time interval and displacement of particle when speed changes from 1 m/s to 3 m/s. [0.0245 sec, 0.0506 m] 10. The velocity of a particle moving along a straight line is given by V = 2t3 + 5t2 where v is in m/s and t in sec. What distance does it travel while its velocity increases from 7 m/s to 99 m/s. [83.33 m]

5.1 INSTANTANEOUS CENTRE OF ROTATION – Class Work Questions 1. Fig – 1 shows a ladder AB = 6 m resting against a vertical wall at A and horizontal ground at B. If the end B of the ladder pulled towards right with a constant velocity VB = 4m/s find (i) ICR of the ladder (ii) angular velocity of the ladder at this instant (iii) Velocity end A of the ladder (iv) Velocity components VCX, VCY of the mid-point C of the ladder. [1.33 rad/sec, 6.927 m/s, VC = 4m/s, 2m/s, 3.446m/s]

Fig – 1

Fig – 2

Fig – 3

2. Velocity of point A on rod is 2 m/s at the instant shown in the fig – 2. Locate ICR and VB on the rod [1m/s at 60°] 3. A cylinder with diameter 36 cm is held between two plates as shown in fig – 3. The upper plate moves to the right with velocity 8 cm/s while the lower plate moves to the left with velocity 4cm/s. Locate the ICR for the cylinder. Find the velocity of centre point C. [12 cm above the bottom plate]

Fig – 4

Fig – 5

4. Crank OA rotates at 60 r.p.m. in clock wise direction. In the position shown  = 40°. Determine angular velocity of B which is constrained to move in a horizontal cylinder. Ref Fig – 4. [ωAB = 1.817 r/s, VB = 1.53 m/s] 5. Fig – 5 shows a collar B which moves upwards with constant velocity of 1.5 m/s. At the instant when  = 50°, determine (i) angular velocity of rod pinned at B and freely resting at A against 25° sloping ground and (ii) Velocity of end A of the rod. [ω = 1.173 r/s, VA = 0.998 m/s]

Fig – 6

Fig - 7

6. In the mechanism shown; rod AB is horizontal, CD is vertical. End A is guided in an inclined slot having slope 3 in 4. If velocity of A is 1.2 m/s up the slot. Determine (i) Angular velocity of AB and CD. (ii) Linear velocity of B. Ref Fig – 7. [ωAB= 1.6 r/s, VB = 1.073 m/s, ω CD = 2.67 r/s.] 7. Rod BDE is partially guided by a roller at D which moves in a vertical track. Knowing that at the instant shown the angular velocity of AB is 5 r/s clockwise determine (i) angular velocity of rod BE (ii) velocity of point E. Ref. Fig – 8 [ωBDE = 2.84 r/s, VE = 1817.69 mm/sec]

Fig – 8 Fig – 9 8. A bar 24 cm long and is hinged to a wall at A. Another bar CD 32 cm long is connected to it by a pin at B such that CB = 12 cm and BD = 20 cm. At the instant shown [AB is perpendicular to CD] the angular velocities of the bars are ω AB = 4 r/s and ω CD = 6 r/s. Determine the linear velocities of point C and D. Note that bar CD is in plane motion. Ref Fig – 8. [VC = 120 cm/sec, VD = 153.675 cm/sec] 9. Locate the instantaneous centre of rotation for the link ABC and determine velocity of points B & C. Angular velocity of rod OA is 15 rad/sec counter clockwise. Length of OA is 200 mm, AB is 400 mm and BC is 150 mm. Ref fig – 9. [ωAB = ωBC = 3.7 r/s, VB = 16 m/s, Vc = 20.6 m/s]

Fig – 10 Fig -11 10. In Fig – 10, the disc rolls without slipping on the horizontal plane with an angular velocity of 10 rpm clockwise. The bar AB is attached as shown in fig. Line OA is horizontal. Point B moves along the horizontal plane. Determine the velocity of Point B at the instant shown. [VB = 1099.35 m/s] 11. A wheel of 2 m diameter rolls without slipping on a flat surface. The Centre of the wheel is moving with a velocity of 4 m/s towards right. Determine the angular velocity of the wheel and velocity of P, Q and R shown on wheel. [ω = 4 r/s, VP = 5.322 m/s, VR = 8 m/s, VQ = 5.6568 m/s]

Fig – 12

Fig – 13

12. A flanged wheel rolls such that its Centre has a velocity of 8 m/s to the left. Find the velocities of points A, B, D and E on the wheel. [VA = 3.6 m/s (right), VB = 5.65 m/s at 45°, VD = 11.6 m/s (left), VE = 8.59 m/s at 62.24°] Ref Fig -12. 13. The trolley shown in Fig – 13 moves to the left along a horizontal pipe at a speed of 2.4 m/s. The angular velocity of 0.5 m disc is 8 r/s anticlockwise. Determine the velocity of point D on the disc. [1.6 m/s (right)]

5.1.1 ICR - Assignmnt Questions 1. A wheel of radius 0.75 m rolls without slipping on a horizontal surface to the right. Determine the velocities of the points P and Q shown in fig – 1 when the velocity of centre of the wheel is 10 m/s towards right. [Vp = 14.137m/s, VQ = 19.995m/s]

Fig – 1

Fig – 2

Fig – 3

2. Rod AB of length 3 m is kept on smooth planes as shown in the fig – 2. The velocity of the end A is 5 m/s. along the inclined plane. Locate the ICR and find the velocity of the end B. [5.24 m/s] 3. For the crank and connecting rod mechanism shown in fig – 3, determine the velocity of cross head P and angular velocity of connecting rod AP. Take OA = 100 mm and AP = 400 mm. [0.574 m/s, 2.057 r/s]

Fig – 4 4. A slider crank mechanism is shown in fig – 4. The crank OA rotates anticlockwise at 100 rad/s. Find the angular velocity of rod AB and the velocity of the slider at B. [ωAB = 43.38 rad/s, VB = 18.43m/s] 5. At the position shown in fig – 5 the crank AB has an angular velocity of 3 r/s clockwise. Find the velocity of the slider C and point D at this instant. [VC = 0.4 m/s, VD = 0.7212 m/s]

Fig – 5

6. A reciprocating engine mechanism is shown in the fig – 6. The length of crank OA is 150 mm and is rotating at 600 rpm. The connecting rod AB is 700 mm. (i) find angular velocity of connecting rod (ii) the velocity of piston B (iii) the velocity of point C on the connecting rod at a distance 20 cm from A, when  = 45°. [ωAB = 9.64 r/s, VB = 7.67 m/s, VC = 8.4 m/s]

Fig -6

7. In the mechanism shown in fig – 9, piston C is constrained to move in a vertical slot. A and B moves on horizontal surface. Rods CA and CB are connected with smooth hinges. If VA = 0.45 m/s to the right. Find velocity of C and B. Also find angular velocity of two rods. [ VC = 0.1875 m/s, VB = 0.25 m/s, ωAC = 1.5 r/s, ωBC = 1.25 r/s]

Fig – 7

5.1.2 INSTANTANEOUS CENTRE OF ROTATION – Tutorial Questions 1.

A rod AB 26 m long leans against a vertical wall. The end A on the floor is drawn away from the wall at the rate of 24 m/s. When the end A of the rod is 10 m from the wall, determine the velocity of the end B sliding down vertically and the angular velocity of the rod AB.[10m/s, 1rad/s]

2. A bar 3 m long slides down the plane shown in fig – 1. The velocity of end A is 3.6 m/s to the right. Determine the angular velocity of AB, velocity of end B and Centre C at the instant shown.[0.9363r/s, 3.733 m/s and 3.3874m/s]

Fig – 1 Fig – 2 3. Block D shown in the fig – 2 moves with a speed of 3 m/s. Determine angular velocity of link BD and AB and velocity of point B at the instant shown. [ωBD = 5.3 r/s, ωAB = 5.3 r/s, VB = 2.12m/s] 4. For the link and slider mechanism shown in fig – 3 (i) Locate the ICR of link AB (ii) The angular velocity of link OA. Take velocity of slide at B = 2500 mm/sec. [ω OA = 6.25 r/s]

Fig – 3 Fig – 4 5. C is a uniform cylinder to which a rod AB is pinned at A and the other end of the rod B is moving along a vertical wall as shown in Fig – 4. If the end B of the rod is moving upwards along the wall with a speed of 3.3 m/s find the angular velocity of wheel and rod assuming that cylinder is rolling without slipping. [ω rod = 2.931r/s, ω Cylinder = 3.175r/s] 6. Rod ABD is guided by wheels at A and B which rolls in horizontal and vertical as shown in fig – 5. Knowing that β = 60° and the wheel at A moves to the left with constant velocity of 30 cm/sec. Determine velocity of point D. [VD = 108.24 cm/sec]

Fig – 5

5.1.3 INSTANTANEOUS CENTRE OF ROTATION – Practice Questions 1. At the instant shown in the fig, The Rod AB is rotating clockwise at 2.5 r/s. If end C of the rod BC is free to move on a horizontal surface find angular Velocity of rod BC and the velocity of Its end point C. Ref fig – 1.[ωBC = 0.7211r/s, Vc = 4.33 m/s]

Fig – 1 Fig – 2 2. Determine the velocity of slider Block C if ωAB = 4 r/s. Ref. Fig – 2 [1.7 m/s] 3. Knowing that at the instant shown in fig – 3, the velocity of collar D is 120 mm/s to the left. Determine (a) Angular velocities of crank AB and rod BE.(b) The velocity of point E. [ωBE = 1.2 r/s, ωAB = 2.7 r/s, VE = 239.46 mm/s]

Fig – 3 Fig – 4 4. At the instant shown in fig – 4 link BC of length 350 mm is horizontal. Link AB, 500 mm long rotates counter clockwise at 4 rad/sec. Determine angular velocity of link BC and velocity of collar ‘C’. [ωBc = 7.805 r/s, Vc = 2449.21 mm/s] 5. Crank AB has a constant velocity 200 rpm counter clockwise. Determine the angular velocity of BD and velocity of collar D. Ref fig – 5. [ωBD = 7.26 r/s, VD = 1.088m/s]

Fig – 5

Fig – 6

Fig – 7

6. At the instant shown in fig 6. Find angular velocity of link BD, Velocity of collar D. E is mid point of link BD, find velocity of point E. [ωBD = 1.5 r/s, VD = 0.36m/s Ve = 0.065m/s] 7. Crank AB has a constant angular velocity of 1.5 r/s counter clockwise, for the position shown in fig – 7, determine a) angular velocity of rod BD b) the velocity of collar D. [0.175 r/s, 66.2 mm/s]

Fig – 8 8. The velocity of the slider block is 4 m/s up the inclined groove. Determine the angular velocity of links AB and BC and velocity of point B at the instant shown in fig – 8. [ωAB = ωBC = 2.83 r/s, VB = 2.83 m/s]

5.1.4 MOTION DIAGRAMS - Class Work Questions 1. Fig – 1 shows a diagram of acceleration versus time for a particle moving along X – axis for a time interval of 0 to 40 seconds. For the same time interval plot (a) the velocity time diagram (b) the displacement time diagram and hence find the maximum speed attained and maximum distance covered by the particle during the interval. [240 m/s and 4800 m]

Fig – 1

Fig – 2

2. Fig – 2 shows a diagram of acceleration versus time for a particle moving along X – axis plot (a) the velocity time diagram (b) the displacement time diagram. Find the speed and distance covered by the particle after 50 seconds. Also find the maximum speed attained and the height at which the speed is attained by the particle.[420 m/s, 9600 m] 3. Fig – 3 shows a diagram of acceleration versus time for a particle moving along X – axis plot (a) the velocity time diagram (b) the displacement time diagram. The motion starts with initial velocity of 5 m/s from starting point.

Fig – 3 Fig – 4 4. Fig – 4 shows a diagram of acceleration versus time diagram for the linear motion. Construct (a) the velocity time diagram (b) the displacement time diagram. Assume the motion starts from rest. 5. The motion of jet plane while travelling along a runway is defined by the V – t graph as shown in Fig – 5. Construct the s – t and a – t graph for the motion. The plane starts from rest.

Fig – 5

Fig – 6

6. V – s graph is given in fig – 6. Find the velocity and acceleration at s = 50 m and s = 150 m. 7. A particle is moving along a straight path variation of its position with respect to time is shown in s – t graph. Draw a- t and v – t curve.

Fig – 7 Fig – 8 8. A particle is moving along a straight path with an acceleration shown in the a – t graph in Fig – 8. Draw v – t and s – t graph. Find maximum velocity in the time interval, distance travelled by the particle in the time interval.

5.1.5 MOTION DIAGRAMS – Assignment Questions 1. Velocity – time graph for a particle moving along a straight line is given above. Draw Displacement – Time and Acceleration – Time graphs. Also find the maximum Displacement of the particle. [S=275m]

Fig – 1

5.1.6 MOTION DIAGRAMS – Tutorial Questions 1. The car starts from rest and travels along a straight track such that it accelerates at a constant rate for 10 seconds and then decelerates at a constant rate. Draw the v – t and s – t graph and determine the time t̀ needed to stop the car. How far has the car travelled? [t̀ = 60 sec, 3000m]

6.1

D’ALEMBERT PRINCIPLE – Class Work Questions

1. An elevator has a downward acceleration of 1 m/s2 , What pressure will be transmitted to the floor of the elevator by a man weighing 500N travelling in the lift? [449 N] 2. A 50 kg block is kept on the top of a 15º sloping surface is pushed down the plane with an initial velocity of 20m/s. If μk = 0.4, determine the acceleration of the block. [a = - 1.25 m/s2] 3. Two blocks A and B are separated by 10 m as shown in fig1 .on a 200 incline plane. If the blocks start moving, find the time t when the blocks collide. Assume µk = 0.3 for block A and plane and µk = 0.10 for block B and plane.

Fig – 1 Fig – 2 4. Two blocks A and B of masses MA = 280 Kg and MB = 420 Kg are joined by an inextensible cable as shown in fig - 2. Assume that the pulley is frictionless and µ= 0.3 between block A and the surface. The system is initially at rest. Determine (a) acceleration of block A, (b) Velocity it has moved 3.5m and (c) velocity after 1.5 sec. 5. A horizontal force P = 600 N is exerted on block A of mass 120 Kg as shown in fig - 3. The µ between block A and the horizontal plane is 0.25. Block B has a mass of 30 Kg and µ between it and the plane is 0.4. The wire between the two blocks makes 300 with horizontal. Calculate the tension in the wire.

6.1.1 D’ALEMBERT PRINCIPLE - Assignment & Tutorial Questions 1. An elevator being lowered into mine shaft starts from rest and attains a speed of 10 m/s with in a distance of 15 meters. The elevator alone has a mass of 500 Kg and it carries a box of mass 600 Kg in it. Find the total tension in cables supporting the elevator, during this accelerated motion. [T = 7128N]

2. A man moves a crate by pushing horizontally against until it sides on the floor. If µs = 0.5, µk = 0.4. With what acceleration does the crate begin the move?. Assume that the force exerted by man at impending motion is maintained when sliding begins. [a = 0.981 m/s2] 3. A motorist travelling at a speed of 70 kmph suddenly applies brakes and halts after skidding 50 m. determine (a) The time required to stop the car (b) The coefficient of friction between the tyres and the road. [t = 5.14 sec, µ = 0.385] 4. Two bodies weighing 300 N and 450 N are hung to the ends of a rope passing over an ideal pulley as shown in fig – 1. With what acceleration the heavier body comes down? What is the tension in the string? [a = 1.962 m/s2, T = 360N]

Fig – 1.

6.2 WORK ENERGY PRINCIPLE - Class work Questions 1. A 3000N block shown in fig – 1 slides down a 50° incline. It starts from rest. After moving 2m it strikes a spring whose modulus is 20N/mm. If the co–efficient of friction between block and incline is 0.2, determine the maximum deformation of the spring. [0.72m]

Fig – 1 Fig – 2 2. The 10 kg slider A moves with negligible friction up the inclined guide. The attached spring has stiffness of 60 N/m and is stretched 0.6 m in position A where the slider is released from rest. The 250N force is constant and the pulley offers negligible resistance to the motion of the cord. Calculate the velocity of the slider as it passes point ‘C’. Ref fig – 2. [0.974 m/s] 3. Fig – 3 shows a collar of mass 20kg which is supported on the smooth rod. The attached springs are undeformed when d=0.5m. Determine the speed of the collar after the applied force of 1000N causes it to displace so that d=0.3m. The collar is at rest when d= 0.5m. [4.6m/s]

Fig – 3

Fig – 4

4. Two springs each having stiffness of 0.5N/cm are connected to a ball B having a mass of 5kg in horizontal position producing initial tension of 1.5N in each spring. If the ball is allowed to fall from rest, What will be its velocity after it has fallen through a height of 15 cm. Ref fig – 4. [1.683m/s] 5. A 2kg collar M is attached to a spring and slides without friction in a vertical plane along the curved rod ABC as shown in fig 5. The spring has an undeformed length of 100mm and its constant is 800 N/m. If the collar is released from rest at A. Determine its velocity (i) as it passes through B (ii) as it reaches C. [2.3332,1.414m/s]

Fig – 5 Fig - 6 6. The 25N collar is released from rest at A and travels along the smooth guide. Determine its speed when its centre reaches point ‘C’. The spring has an unstretched length of 300mm, and point ‘C’ is located just before the end of the curved portion of the rod. Ref fig – 6. [3.796 m/s]

6.2.1 WORK ENERGY PRINCIPLE – Assignment Questions 1 Marbles having a mass of 5 g fall from rest at A through the glass tube and accumulate in the can at ‘C’. Determine the placement ‘R’ of the can from the end of the tube and the speed at which the marbles fall into the can. Neglect the size of the can. Ref. fig – 1 below. [R = 2.828m, V = 7.67 m/s]

Fig – 1 Fig – 2 9. A collar of mass 10 kg moves in vertical guide as shown in fig – 2. Neglecting friction between guide and collar, find its velocity when it passes through position (2), after starting from rest in position (1). The spring constant is 200 N/m and the free length of the spring is 200 mm. [3.811m/s]

6.2.2 WORK ENERGY PRINCIPLE – Tutorial Questions 1 A 7.5 kg mass slides 150 mm from rest on a 25° incline. It hits a spring whose modulus is 1750N/m. If µ = 0.2, determine the maximum compression in the spring. [0.06m]

Fig – 1 Fig – 2 10. The cylinder has a mass of 20kg and is released from rest when h = 0. Determine its speed when h = 3m. The springs each have an unstretched length of 2m. Ref fig – 2. [6.968m/s] 11. A 4kg collar is attached to a spring, slides on a smooth bent rod ABCD. The spring has constant k = 500 N/m and is undeformed when the collar is at ‘C’. If the collar is released from rest at A. Determine the velocity of collar, when it passes through ‘B’ and ‘C’. Also find the distance moved by collar beyond ‘C’ before it comes to rest again. Ref fig – 3. [VB = 5.975m/s, VC = 6.408, x = 0.95m]

Fig – 3

Fig – 4

12. The system shown in fig – 4 is initially at rest. Modulus of the spring is 160N/m. If the spring is initially undeformed calculate the velocity of the two blocks after a displacement of 0.3 m. [V2 = 0.9063 m/s]

6.3 IMPULSE MOMENTUM PRINCIPLE AND IMPACT – Class Work Questions 1. A 2 kg sphere is moving towards left with velocity of 1.8 m/s. It strikes the vertical face of stationary block B of mass 4 kg. A spring K = 5000 N/m is attached to the other face of the block. If e = 0.75. Determine the maximum compression of the spring. Ref. fig. – 1. [26.69 mm]

2.

Fig – 1 Fig – 2 Fig – 3 Block A falls through height H onto block B supported on spring of stiffness K. Assuming the impact to be plastic, calculate maximum compression of the spring over and above that due to static action of block A. WA= WB = 20N, K = 20 N/mm, h = 100mm. Ref. fig -2 [10mm]

3. A bullet of mass 20 gm and moving horizontally with 800 m/s strikes a block of wood of mass 5 kg suspended by a wire 2m long. To what angle with vertical will the block and embedded bullet swing. Ref. fig – 3 [42.18º]. 4.

A boy throws a ball vertically downwards. He wants the ball to rebound from floor and just touch the ceiling of the room which is at a height of 4 m from the ground, if e = 0.8, the velocity with which the ball strikes the floor. Take g = 10 m/s2. [11.18m/s]

5.

A ball is thrown vertically downwards with a velocity of V m/s from a height of 1m so that it hits the ground and just touches the ceiling after impact. If ceiling is 3.5 m high from the ground and e = 0.7 determine V. [10.978 m/s]

6.

A glass ball is dropped on to a smooth horizontal floor which it bounces to a height of 9m. On the second bounce, it attains a height of 6 m. What is the coefficient of restitution between the glass and the floor.

7. A pile hammer, weighing 15 kN drops from a height of 600 mm on a pile of 7.5 kN. How deep does a single blow of hammer drive the pile if the resistance of the ground to pile is 140 kN? Assume plastic impact. 8.

A smooth spherical ball A of mass 10 gms is moving from left to right, with a velocity 2 m/s in a horizontal plane. Another identical ball B travelling in perpendicular direction with a velocity of 6 m/s collides with ball A as shown in fig – 4. Determine velocities of A and B after impact. 9. Two smooth balls A (mass 3 kg) and B (mass 4kg) are moving with velocities 25 m/s and 40 m/s respectively. Before impact, the directions of velocity of two balls are 30 and 60 degrees with the line joining the centers as shown in fig – 5. If e = 0.8, find the magnitude and direction of velocities of the balls after impact.[VA = 24.6 m/s,  = 59.5º, VB = 36.7 m/s, α = 19.3º]

Fig – 4 Fig – 5 10. If e = 0.7. Determine magnitude and direction of velocity of ball after impact. Ref. fig – 6. [34.52m/s,  = 35º]

Fig – 6 Fig - 7 11. A ball is dropped from a height of 5 m on an inclined surface of 30º inclination, find velocity of ball after impact, take e = 0.8 [V = 9.5 m/s  = 35.21º] 12. A ball is dropped from a height of 5 m on an inclined surface of 30˚ inclination. Find the velocity of ball after impact, take e = 0.6. Ref. Fig – 7. [8.463 m/s, θ = 24.18˚]

6.3.1 IMPULSE MOMENTUM PRINCIPLE AND IMPACT – Assignment Questions 1. A ball is dropped from a height of 12 m upon a horizontal slab. If it rebounds to height of 4 m. Find coefficient of restitution.[0.5777] 2. A boy throws a ball vertically downwards from a height of 1.5 m. He wants the ball to rebound from floor and just touch the ceiling of room which is at a height of 4m from ground. If coefficient of restitution e is 0.8, find initial velocity with which the ball should be thrown. [u = 9.6m/s] 3. A ball is dropped on to a smooth horizontal floor from a height of 4m. On the second bounce it attains a height of 2.25m. Find coefficient of restitution between the ball and floor. [0.866]

4. Determine the horizontal velocity uA at which we must throw the ball so that it bounces once on the surface and then lands into the cup at C. Take e between ball and surface as 0.6 and neglect the size of cup. [2.548 m/s]

6.3.2 IMPULSE MOMENTUM PRINCIPLE AND IMPACT – Tutorial Questions 1. Two smooth spheres 1 and 2 having mass of 2 Kg and 4Kg respectively collide with initial velocities as shown in Fig – 1. If e = 0.8, determine the velocities of each sphere after collision. [V1 = 2.923 m/s, V2 = 3.472 m/s]

Fig – 1 Fig – 2 2. A ball is thrown against a wall with a velocity u forming an angle 30˚ with the horizontal. Assuming frictionless conditions and e = 0.5, determine the magnitude and direction of velocity of ball after it rebounds from the wall. Ref. Fig – 2. [0.661u, θ = 49.12˚] 3. Block A is released from rest in the position shown and slides without friction until it strikes the ball B of a simple pendulum. Knowing that coefficient of restitution between A and B is 0.9. Determine (i) the velocity of B immediately after impact (ii) the max. angular displacement of the pendulum. ref fig – 3 [2.5,49.9˚]