Structures Activity 2 Levers Object: The objects of this activity are to explore the concepts of torque and rotational equilibrium, study the three classes of lever, and apply the concepts of torque and rotational equilibrium to a model of the human biceps muscle and elbow joint. Equipment: This manual Meterstick Mass-hanger Hanger clamps (3) Masses Fulcrum stand Spring scale Theory: Equilibrium Mechanics, the study of motion, is concerned with forces and how those forces act on different objects. Objects that are at rest or in motion with constant velocity are in equilibrium. Knowledge of the conditions that prevail when an object is in equilibrium is important in a variety of fields. For example, mechanical engineers benefit from an understanding of the forces that act on buildings or large machines, while some biologists benefit from an understanding of the forces at work in muscles and bones. Newton's second law is the basis for an attempt to understand how these forces act on objects. In order to understand objects in equilibrium fully, it is necessary to understand the concept of torque and its effect on objects. We have previously encountered Newton's second law, which states that a force, F r, acting on a body of mass, m, produces an acceleration, a r, in the direction of the applied force: r r F = ma (Eq. 1) For an object to be in equilibrium, two conditions must be met. The first condition of equilibrium is that the vector sum of the forces acting on the object be zero: F r = 0 (Eq. 2) However, this condition alone does not suffice to ensure that an object at rest (or in motion with constant velocity) will remain at rest (or in motion with constant velocity). To see this consider the following example, illustrated in Figure 1, in which a block is acted on by two forces, F 1 and F 2, of equal magnitude. Applying the first condition of equilibrium one sees that 1
the vector sum of the forces does add to zero. However, anyone who has played with blocks can see the block is still able to rotate clockwise about its center, C, because the forces do not act through a common point. Torque is the tendency of a force acting on an object to rotate it about some axis, as illustrated in Figure 2. The torque, r τ, due to a force, F r, acting at a position r, away from the center of rotation is perpendicular to both r and F r and is given by r r r τ = F. (Eq. 3) The magnitude of τ, as usual for the cross product, is τ = rf sin(θ ), (Eq. 4) where θ is the angle between the position vector to the application point and the direction of the force, as shown in Figure 2. The direction of the torque can be found from the right-hand rule for the cross product. If you curl the fingers of your right hand in the direction of the tendency to rotate, your thumb points in the direction of the torque. Likewise, knowing the direction of the torque, you can determine the rotation direction by pointing your right thumb in the direction of the torque and examining your curled fingers. The second and final condition for equilibrium is that the sum of all torques acting on the object must also be zero. τ r = 0 (Eq. 5) In equilibrium, then, an object will continue in whatever state it started motionless, translating, rotating, or both translating and rotating. Levers The lever, a member of the class of simple machines, is a rigid bar free to rotate about a fixed point called the fulcrum. The position of the fulcrum is fixed so that it is not free to move with respect to the bar. There are Figure 1: A solid block viewed from above. The forces F 1 and F 2 do not act through the center of the block and cause the block to rotate. The block is not in equilibrium. Figure 2: Torque, as represented by the ability of a wrench to turn a nut. The torque, τ r, is directed out of the page (and centered on the nut), indicating, by the righthand rule, a tendency for the nut to turn in the counterclockwise direction. 2
three classes of levers as shown in Figure 3. With a class 1 lever, the fulcrum is located between the applied force and the load. A crowbar is an example of a class 1 lever. With a class 2 lever, the fulcrum is at one end of the bar with the load situated between the fulcrum and the applied force. A wheelbarrow is an example of a class 2 lever. Show from the condition of rotational equilibrium that for all three types of levers the force, F r, required to balance a load of weight W r is given by r r W F = W, (Eq. 6) F a) Class 1 where r W and r F are the lever arms of W and F, respectively, as shown in Figure 3. The mechanical advantage, M, of a simple machine, such as a lever, is defined as the ratio of the output force (load) to the applied force (effort): W r M = F r F =. (Eq. 7) W b) Class 2 Figure 3: Types of levers. See text for further description. It is a measure of how easy it is to use the lever. The larger M, the more you can lift or move with the same amount of force. Note that a simple machine does not create energy from nothing, and the work you get out of it is the work you put into it. Class 1 Levers: Setup: Set up a class 1 type lever as in Figure 3a. Attach three knife-edge mass-hanger clamps to your meterstick. You should have one mass hanger that is without the wire loop dangling from the knife edges. This one is for your fulcrum stand (the wire loop would otherwise be in your way). Should you not have one, make one. Take care that the hanger for the fulcrum stand is inverted so the knife edges may rest on the fulcrum stand. Place the fulcrum at the 50 cm mark of the meterstick. Choose one hanger to take the Load and place a mass hanger and masses on it. 3
Hook the spring scale to the other end of your lever to provide the Applied Force. Note: Use the 20 N spring scale for the Class 1 lever measurements. Shift your fulcrum stand to the corner of the table to allow your spring scale to dangle over the floor while you balance it by pulling on the scale. Procedure: 1) Changing the Force location: a) Measure r W. b) Place mass totaling 400 g on your mass hanger (Load). c) Pull down on your scale to maintain the lever setup level with the floor. Be sure the spring scale is perpendicular to the lever arm. d) Measure the force registered on the spring scale for five well-separated values of r F, the distance from the fulcrum to the spring scale. Analysis: Now let s analyze what we have done and try to draw some conclusions. The questions and results of this section should be addressed in the concluding sections of your report. Changing the Force location: o Calculate the torque at each of the r F locations. o Does the torque remain constant? What does this tell you? Discuss. Class 2 Levers Setup: Set up a class 2 type lever as in Figure 3b. Place the fulcrum so that it is at the end of the meterstick, with the spring scale at the other end. Use the 5 N spring scale for the class 2 lever measurements. Place a 200 g mass at the center of the meter stick. Pull up on the spring scale to balance the Load. Note that you will likely need a partner to keep a hand on the knife edges of the fulcrum stand, to keep the stand from tilting over. Alternately, there may be clamps available to keep the fulcrum stand stationary. Procedure: 1) Changing the angle: a) Measure r F. 4
b) Vary the angle between the meterstick and the spring scale from 30 to 150 in 30 increments, measuring F at each angle. (See Figure 5.) 2) Moving the Load: a) Hold r F fixed and θ at 90 b) Vary r W, measuring F for seven different positions. Analysis: Now let s analyze what we have done and try to draw some conclusions. Changing the Angle: o As the angle increases, does the force required to balance the load increase or decrease? Why? Moving the Load: o Compare the torque caused by the spring scale and the torque caused the the load. They should be equal, i.e. τ scale = τ Load. Are they? If not, why? (Hint: Are there any other weights that you should consider?) Remove the fulcrum hanger and mass the meterstick and remaining hangers, being careful not to move the remaining hangers. Find the center of mass of the meterstick/hangers system. 1 What is the distance from the center of mass to the point where the fulcrum hanger was attached? What is the torque caused by the mass of the meterstick/hangers system? 2 Compare the torque of the meterstick/hangers system to the difference between the torque of the scale and the torque of the load, i.e. does τ τ = τ? scale Load meterstick / hangers 1 Balance it on your finger. 2 The force of gravity acts as though the mass of the system were concentrated at the center of mass, right? 5