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Dynamics of Rigid Bodies Problem Sheet 2: Rectangular Coordinates: Kinematics of Particles Francis S. Dela Cruz September 2020

Contents 1 Supplementary Problems

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2 Problem Set

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3 Assignment

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4 Answers for the Supplementary Problems and Problem Set 4.1 For Supplementary Problems . . . . . . . . . . . . . . . . . . . 4.2 For Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Further Reading and Problem Solving

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6 Figures

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Supplementary Problems 1. (Pytel’s Problem 12.2) When an object is tossed vertically upward on the surface of a planet (1), the ensuing motion in the absence of atmospheric resistance can be described by 1 x = − gt2 + v0 t 2 1

where g and v0 are constants. (a) Derive the expressions for the velocity and acceleration of the object. Use the results to show that v0 is the initial speed of the body and that g represents the gravitational acceleration. (b) Derive the maximum height reached by the object and the total time of flight. (c) Evaluate the results of Part (b) for v0 = 90 km/h and g = 9.8m/s2 (surface of the earth). 2. (Pytel’s Problem 12.4) The position of a particle that moves along the x-axis is given by x = t3 − 3t2 − 45t m where t is the time in seconds. Determine the position, velocity, acceleration, and distance traveled at t = 8 s. 3. (Pytel’s Problem 12.6) A body is released from rest at A and allowed to fall freely (2). Including the effects of air resistance, the position of the body as a function of the elapsed time is x = v0 (t − t0 + t0 e−t/t0 ) where v0 and t0 are constants. (a) Derive the expression for the speed v of the body. Use the result to explain why v0 is called the terminal velocity. (b) Derive the expressions for the acceleration a of the body as a function of t and as a function of v. 4. (Pytel’s Problem 12.8) A particle moves along the curve x2 = 12y, where x and y are measured in millimeters. The x-coordinate varies with time according to x = 4t2 − 2 mm where the time t is in seconds. Determine the magnitudes of the velocity and acceleration vectors when t = 2 s.

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Problem Set 1. (Pytel’s Problem 12.12) The coordinates of a particle undergoing plane motion are x = 15 − 2t2 m

y = 15 − 10t + t2 m 2

where t is the time in seconds. Find the velocity and acceleration vectors at (a) t = 0 s; and (b) t = 5 s. 2. (Pytel’s Problem 12.14) An automobile goes down a hill that has the parabolic cross section shown in (4). Assuming that the horizontal component of the velocity vector has a constant magnitude v0 , determine (a) the expression for the speed of the automobile in terms of x; and (b) the magnitude and direction of the acceleration 3. (Pytel’s Problem 12.16) When a taut string is unwound from a stationary cylinder, the end B of the string generates a curve known as the involute of a circle. If the string is unwound at the constant angular speed ω, the equation of the involute is x = R cos ωt + Rωt sin ωt

y = x = R sin ωt + Rωt cos ωt

where R is the radius of the cylinder. Find the speed of B as a function of time. Show that the velocity vector is always perpendicular to the string.

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Assignment 1. (Pytel’s Problem 12.10) The elevator A is lowered by a cable that runs over pulley B (3). If the cable unwinds from the winch C at the constant speed v0 , the motion of the elevator is p x = (v0 t − b)2 − b2 Determine the velocity and acceleration of the elevator in terms of the time t. 2. (Pytel’s Problem 12.20) The spatial motion of a particle is described by x = 3t2 + 4t

y = −4t2 + 3t

z = −6t + 9

where the coordinates are measured in m and the time t is in seconds. (a) Determine the velocity and acceleration vectors of the particle as functions of time.(b) Verify that the particle is undergoing plane motion (the motion is not in a coordinate plane) by showing that the unit vector perpendicular to the plane formed by v and a is constant. 3

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Answers for the Supplementary Problems and Problem Set

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For Supplementary Problems

1. Look for Pytel’s Dynamic of Rigid Bodies - Answer to Even-Problems, 12.2 2. Look for Pytel’s Dynamic of Rigid Bodies - Answer to Even-Problems, 12.4 3. Look for Pytel’s Dynamic of Rigid Bodies - Answer to Even-Problems, 12.6 4. Look for Pytel’s Dynamic of Rigid Bodies - Answer to Even-Problems, 12.8

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For Problem Set

1. Look for Pytel’s Dynamic of Rigid Bodies - Answer to Even-Problems, 12.12 2. Look for Pytel’s Dynamic of Rigid Bodies - Answer to Even-Problems, 12.14 3. Look for Pytel’s Dynamic of Rigid Bodies - Answer to Even-Problems, 12.16

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Further Reading and Problem Solving

Please visit our reference books: Andrew Pytel, Jaan Kiusalaas - Engineering Mechanics - Dynamics SI Version (3rd Edition) Russell C. Hibbeler - Engineering Mechanics - Dynamics (14th Edition) Try to solve their problems whenever you have free time. (Physics is ♥)

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Figures 4

Figure 1: P12.2

Figure 2: P12.6

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Figure 3: P12.10

Figure 4: P12.14

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Figure 5: P12.16

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