Lever Problems

16LESSON 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. 172 Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use. Where must Mary.LESSON 16 Solving Lever Problems 173 Fig. 16-1. who weighs 96 pounds. . assume the fulcrum is in the center of the lever. Given any of the three variables. Unless otherwise specified. w ¼ 96. 16-2. STRATEGY: Use the formula WL ¼ wl where W ¼ 120. you can set up an equation and solve for the fourth one. IMPLEMENTATION: Solve the equation: 120ð3Þ ¼ 96ðxÞ 360 ¼ 96x Fig. L ¼ 3. EXAMPLE: Bill weighs 120 pounds and sits on a seesaw 3 feet from the fulcrum. l ¼ x. WL ¼ wl 120ð3Þ ¼ 96ðxÞ See Figure 16-2. 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. 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. On the short end rests an 84-pound weight. Fig. 16-3.75 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. WL ¼ wl 84ð3Þ ¼ xð7Þ See Figure 16-3. STRATEGY: Let x ¼ the weight of the object needed. .174 360 96 x ¼ 1 96 96 3:75 ¼ x 1 LESSON 16 Solving Lever Problems Hence she must sit 3. The equation is WL ¼ wl 36x ¼ 64ð18 À xÞ 175 Fig. .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. 16-4. 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. See Figure 16-4. STRATEGY: Let x ¼ the length of the lever from the fulcrum to the 36-pound weight and (18 À x) ¼ the length of the lever from the fulcrum to the 64-pound weight. the equation is W1 L1 þ W2 L2 ¼ w1 l1 þ w2 l2 EXAMPLE: On a 16-foot seesaw Fred. Where should Sam. weighing 84 pounds. two on each side. Next to Fred sits Bill. On the other side at the end sits Pete.176 LESSON 16 Solving Lever Problems STRATEGY: Solve the equation: 36x ¼ 64ð18 À xÞ 36x ¼ 1152 À 64x 36x þ 64x ¼ 1152 À 64x þ 64x 100x ¼ 1152 1001 x 1152 ¼ 100 1001 x ¼ 11:52 Hence the fulcrum must be placed 11. weighing 80 pounds. If 4 weights are used. See Figure 16-5. . sits on one end. weighing 75 pounds. Bill is 4 feet from the fulcrum. STRATEGY: Let x ¼ the distance from the fulcrum where Sam needs to sit. EVALUATION: Check the equation: WL ¼ wl 36ð11:52Þ ¼ 64ð18 À 11:52Þ 414:72 ¼ 414:72 You can place 3 or more weights on a lever and it still can be balanced. weighing 95 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.52 feet from the 36-pound weight. 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 À 760 ¼ 760 À 760 þ 75x 216 ¼ 75x 216 75 x ¼ 75 75 2:88 ¼ x Sam needs to sit 2. 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 1 .LESSON 16 Solving Lever Problems 177 Fig.88 feet from the fulcrum.