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UNDERSTANDING LEVERS by Mitch Ricketts Math Toolbox Is Designed to Help Readers Apply STEM Principles to Everyday Safety Issues

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UNDERSTANDING LEVERS by Mitch Ricketts Math Toolbox Is Designed to Help Readers Apply STEM Principles to Everyday Safety Issues MATH TOOLBOX UNDERSTANDING LEVERS By Mitch Ricketts Math Toolbox is designed to help readers apply STEM principles to everyday safety issues. Many readers may feel apprehensive about math and science. This series employs various communication strategies to make the learning process easier and more accessible. This Math Toolbox article is the is a push or pull in a particular direction class levers are commonplace in anatomy, second of two installments on the role of (e.g., up, down, left, right, rotational). however, as illustrated by the human el- levers in occupational safety. In the first The magnitude (or amount) of force is bow, where the bicep exerts upward effort, installment (“The Case of the Tipping often specified in units of weight, such the elbow serves as the fulcrum, and the Forklift,” PSJ September 2020, pp. 45-51), as ounces, pounds or newtons. In a tendon attachment denotes the line of we considered the hazards associated first-class lever, the effort arm and load applied effort on the forearm. with out-of-control levers in workplaces. arm are located on opposite sides of the Recall from the first installment on This article will delve more deeply to un- fulcrum, and effort is applied in the same levers that the rotational force of a pivot- derstand the three main classes of levers, direction as the force exerted by the load. ing lever is known as torque (τ). Torque the concept of mechanical advantage and In a second-class lever (Figure 2), the is stated in units of distance times some associated calculations. effort arm and load arm are located on the weight, such as foot-pounds (ft-lb) and same side of the fulcrum. Effort is applied newton-meters (N-m). The direction of Classes of Levers in the opposite direction, compared with torque may be denoted with a positive Figure 1 depicts two examples of the force exerted by the load. Furthermore, sign for clockwise motion and a negative first-class levers: a stylized lever at top the load lies between the point of effort sign for counterclockwise rotation. As in and a pry bar as an everyday example at and the fulcrum. A wheelbarrow is an ev- the first installment, we will simplify by bottom. Remember from the first install- eryday example of a second-class lever. using the directionless, absolute value of ment that a lever is a rigid device that In a third-class lever (Figure 3), the ef- torque, designated as |τ|. pivots on a fulcrum. The fulcrum is the fort arm and load arm are located on the pivot point, or the point about which the same side of the fulcrum. The direction of Calculating Torque lever rotates. The lever has two “arms”: effort is opposite the direction of the force for Each Lever Class The load arm (or output arm) is the por- exerted by the load. Uniquely for third- In the first installment, we applied the tion of the lever directly connected to the class levers, the point of effort lies between formula for torque (absolute value) to load. The effort arm (or arm of applied the load and the fulcrum. Since these first-class levers as follows: force) is the portion of the lever to which levers tend to be inefficient, they are less we apply the effort, or input force. Force common in everyday technology. Third- || = ∙ FIGURE 1 FIGURE 2 FIGURE 3 FIRST-CLASS SECOND-CLASS THIRD-CLASS LEVER EXAMPLES LEVER EXAMPLES LEVER EXAMPLES 52 PSJ PROFESSIONAL SAFETY OCTOBER 2020 assp.org FIGURE 4 OPPOSING TORQUES OF EFFORT & LOAD IN A FIRST-CLASS LEVER FIGURE 5 EXAMPLE: BALANCED FIRST-CLASS LEVER Example problem. Equivalent torques result in a balanced first-class lever. In this case, a 30-lb force applied to an effort arm of 1.5 ft bal- ances the torque created by a 45-lb force applied to a load arm of 1 ft. where: consider this 30-lb force to be applied at a •The weight applies a downward force |τ| = torque (absolute value); a turning point that represents the average location of 45 lb perpendicularly to the load arm. or twisting force where your hand presses on the lever, which This is the value of f2 in the formula. f = force applied perpendicularly (at a in this case is a distance of 1.5 ft to the left •The weight applies this force at an right angle) to the lever arm of the fulcrum. We will assume the weights average distance of 1 ft from the fulcrum. d = distance at which the force is ap- of the lever arms are negligible (or, alterna- This is the value of d2 in the formula. plied; the distance from the fulcrum to tively, that the weights of the lever arms are Based on these data, we calculate the line of applied force (measured per- balanced on both sides of the fulcrum). We torque generated by the load arm (|τload|) pendicularly to the line of applied force) will also assume friction is negligible. Under in ft-lb as follows: Note: When force is not applied per- these conditions, how many foot-pounds of Step 1: Start with the equation for pendicularly to the lever arm, a more torque are created by the effort arm to bal- torque of the load arm: general formula is used to account for the ance the torque created by the load arm? angle of applied force: τ = f ∙ d ∙ sin θ. We calculate torque created by the effort Continuing our review, we can calculate arm (|τeffort|) based on the following data: #$%& ) ) torque for both sides of a first-class lever •Your hand applies a downward force Step 2: Insert| the| known= ∙ values for as shown in Figure 4. For the effort arm, of 30 lb perpendicularly to the effort arm. weight of the load (f2 = 45 lb) and distance torque may be designated as |τeffort|. Force This is the value of f1 in the formula. (d2 = 1 ft). Then solve forτ | load| in ft-lb: and distance in the effort arm may be des- •Your hand applies this force at an av- ignated as f1 and d1, respectively. For the erage distance of 1.5 ft from the fulcrum. load arm, torque may be designated as |τload|, This is the value of d1 in the formula. |#$%&| = ) ∙ ) = 45 ∙ 1 = 45 − with force and distance in the load arm Based on these data, we calculate Step 3: Our calculation indicates denoted as f2 and d2. If the opposing torques torque generated by the effort arm that the load arm is generating a torque are equal, then |τeffort| = |τload| and the lever (|τeffort|) in ft-lb as follows: of 45 ft-lb, which is equal to the torque will be balanced. On the other hand, if Step 1: Start with the equation for of the effort arm, meaning the torques torque created by the effort arm is greater torque of the effort arm: are balanced. In contrast, if the torques than torque created by the load arm, then had been unbalanced, the load would |τeffort| > |τload| and the load will rise. Finally, either rise (if |τeffort| > |τload|) or descend if torque created by the effort arm is less #$$%&' * * (if |τeffort| < |τload|). than torque created by the load arm, then Step 2: Insert! the! known= ∙ values for Second-class lever calculated example: |τeffort| < |τload| and the load will descend. applied force (f1 = 30 lb) and distance (d1 We calculate torque for a second-class First-class lever calculated example: = 1.5 ft). Then solve forτ | effort| in ft-lb: lever using the same formula. However, Imagine you have used a first-class lever note that the effort and load arms over- to raise a load consisting of a 45-lb weight, lap because they are located on the same as illustrated in Figure 5. The load’s center #$$%&' * * side of the fulcrum, as shown in Figure of gravity is located 1 ft to the right of the !Step 3:! Our= ∙calculation = 30 ∙ 1. 5indicates= 45 −that 6 (p. 54). Also note that in second-class fulcrum. (Center of gravity is the average the effort arm is generating a torque of levers the effort is applied in a direction location of a body’s weight, or the location 45 ft-lb to balance the torque created by opposite to the direction of the force of where a body’s total weight is assumed to the load arm. the load. The value of d1 will be the entire be concentrated.) We will now calculate torque created distance from the fulcrum to the average To keep the load balanced (so it moves by the load arm (|τload|) to confirm that point of applied effort. The value of d2 neither up nor down), you exert a down- the torques are balanced, based on the will be the distance from the load’s center ward force of 30 lb with your hand. We will following data: of gravity to the fulcrum. Once again, assp.org OCTOBER 2020 PROFESSIONAL SAFETY PSJ 53 MATH TOOLBOX the opposing torques are designated as force of 15 lb with your hand. Your Based on these data, we calculate |τeffort| and |τload|. Similarly, the effort and hand exerts this force at an average dis- torque generated by the effort arm load forces designated as f1 and f2, and tance of 2 ft to the left of the fulcrum. (|τeffort|) in ft-lb as follows: the distances are designated as d1 and d2 Assuming friction and the weights of Step 1: Start with the equation for for the effort and load arms respectively.
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