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• Sujib Barman

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16

LESSON

Solving Lever Problems One of the oldest machines known to humans is the lever. The principles of the lever are studied in physics. Most people are familiar with the simplest kind of lever, known as the seesaw or teeterboard, often seen in parks. The lever is a board placed on a fulcrum or point of support. On a seesaw, the fulcrum is in the center of the board. A child sits at either end of the board. If one child is heavier than the other child, he or she can sit closer to the center in order to balance the seesaw. This is the basic principle of the lever. In general, the weights are placed on the ends of the board, and the distance the weight is from the fulcrum is called the length or arm. The basic principle of the lever is that the weight times the length of the arm on the left side of the lever is equal to the weight times the length of the arm on the right side of the lever, or WL ¼ wl. See Figure 16-1.

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LESSON 16 Solving Lever Problems

Fig. 16-1.

Given any of the three variables, you can set up an equation and solve for the fourth one. Unless otherwise specified, assume the fulcrum is in the center of the lever. EXAMPLE: Bill weighs 120 pounds and sits on a seesaw 3 feet from the fulcrum. Where must Mary, who weighs 96 pounds, sit to balance it? SOLUTION: GOAL: You are being asked to find the distance from the fulcrum Mary needs to sit to balance the seesaw. STRATEGY: Use the formula WL ¼ wl where W ¼ 120, L ¼ 3, w ¼ 96, l ¼ x. WL ¼ wl 120ð3Þ ¼ 96ðxÞ See Figure 16-2. IMPLEMENTATION: Solve the equation: 120ð3Þ ¼ 96ðxÞ 360 ¼ 96x

Fig. 16-2.

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LESSON 16 Solving Lever Problems

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360 96 x ¼ 1 96 96 3:75 ¼ x Hence she must sit 3.75 feet from the fulcrum. EVALUATION: Check the equation: WL ¼ wl 120ð3Þ ¼ 96ð3:75Þ 360 ¼ 360 The fulcrum of a lever does not have to be at its center. EXAMPLE: The fulcrum of a lever is 3 feet from the end of a 10-foot lever. On the short end rests an 84-pound weight. How much weight must be placed on the other end to balance the lever? SOLUTION: GOAL: You are being asked to find how much weight is needed to balance the lever. STRATEGY: Let x ¼ the weight of the object needed. WL ¼ wl 84ð3Þ ¼ xð7Þ See Figure 16-3.

Fig. 16-3.

LESSON 16 Solving Lever Problems IMPLEMENTATION: Solve the equation: 84ð3Þ ¼ 7ðxÞ 252 ¼ 7x 252 71 x ¼ 1 7 7 36 ¼ x 36 pounds needs to be placed at the 7-foot end to balance the lever. EVALUATION: WL ¼ wl 84ð3Þ ¼ 36ð7Þ 252 ¼ 252 EXAMPLE: Where should the fulcrum be placed on an 18-foot lever with a 36-pound weight on one end and a 64-pound weight on the other end? SOLUTION: GOAL: You are being asked to find the placement of the fulcrum so that the lever is balanced. STRATEGY: Let x ¼ the length of the lever from the fulcrum to the 36-pound weight and (18
x) ¼ the length of the lever from the fulcrum to the 64-pound weight. See Figure 16-4. The equation is WL ¼ wl 36x ¼ 64ð18
xÞ

Fig. 16-4.

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LESSON 16 Solving Lever Problems

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STRATEGY: Solve the equation: 36x ¼ 64ð18
xÞ 36x ¼ 1152
64x 36x þ 64x ¼ 1152
64x þ 64x 100x ¼ 1152 1001 x 1152 ¼ 100 1001 x ¼ 11:52 Hence the fulcrum must be placed 11.52 feet from the 36-pound weight. EVALUATION: Check the equation: WL ¼ wl 36ð11:52Þ ¼ 64ð18
11:52Þ 414:72 ¼ 414:72 You can place 3 or more weights on a lever and it still can be balanced. If 4 weights are used, two on each side, the equation is W1 L1 þ W2 L2 ¼ w1 l1 þ w2 l2 EXAMPLE: On a 16-foot seesaw Fred, weighing 80 pounds, sits on one end. Next to Fred sits Bill, weighing 84 pounds. Bill is 4 feet from the fulcrum. On the other side at the end sits Pete, weighing 95 pounds. Where should Sam, weighing 75 pounds, sit in order to balance the seesaw? SOLUTION: GOAL: You are being asked to find the distance from the fulcrum where Sam should sit in order to balance the seesaw. STRATEGY: Let x ¼ the distance from the fulcrum where Sam needs to sit. See Figure 16-5.

LESSON 16 Solving Lever Problems

Fig. 16-5.

The equation is W1 L1 þ W2 L2 ¼ w1 l1 þ w2 l2 80ð8Þ þ 84ð4Þ ¼ 95ð8Þ þ 75ðxÞ 640 þ 336 ¼ 760 þ 75x 976 ¼ 760 þ 75x 976
760 ¼ 760
760 þ 75x 216 ¼ 75x 1

216 75 x ¼ 75 75 2:88 ¼ x Sam needs to sit 2.88 feet from the fulcrum. EVALUATION: Check the equation: W1 L1 þ W2 L2 ¼ w1 l1 þ w2 l2 80ð8Þ þ 84ð4Þ ¼ 95ð8Þ þ 75ð2:88Þ 640 þ 336 ¼ 760 þ 216 976 ¼ 976

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