Torque and Rotational Equilibrium Name Section Theory Torque is the rotational analog of force. If you want something to move (translate), you apply a force; if you want something to rotate, you apply a torque. Torque is defined as τ = F d (1) where F is a force and d a distance. This distance is known as the moment (or lever) arm of the torque and is defined as the perpendicular distance from the line of action of the force to the axis of rotation. The SI unit of torque is a Nm. By convention, torques which would produce a clockwise rotation are considered negative; torques which would produce a counterclockwise rotation are considered positive. If a system is in equilibrium, then two conditions must be met: 1. The sum of the forces (the net force) on the system is zero. 2. The sum of the torques (the net torque) about any point is zero. Figure 1: A System in Equilibrium Consider the system shown in Figure 1. A meter stick is supported at its center of gravity (CG) with a mass suspended from each end. The first condition states that the sum of the forces on the system is zero. Here, the sum of the weights of m 1 and m 2 (both acting downward) is equal to the force of the support on the meter stick (acting upward); i.e., F support m 1 g m 2 g = 0. The second condition states that the sum of the torques about any point must be zero. If we sum torques about the support point, then the counterclockwise torque due to m 1 is equal to the clockwise torque due 1
to m 2 ; i.e., F 1 d 1 F 2 d 2 = 0 m 1 gd 1 m 2 gd 2 = 0. The upward force on the meter stick at the support contributes no torque about this point because d = 0! Figure 2: The Same System However, notice that this condition states that the sum of the torques about any point is zero. If we instead sum torques about the left end of the meter stick (Figure 2), then m 1 gd 1 + F support d support m 2 gd 2 = 0. The force of the support on the meter stick will contribute a torque about the left end of the meter stick (d 0). Look at it this way - if F support were the only force acting on the meter stick, it would rotate the meter stick counterclockwise about the left end. Keep this in mind. The moment arm for a torque changes when you change the point about which you are summing torques (the axis of rotation). Also, it is possible that you will have a different number of torques about different points in a system. Finally, a torque that would produce a clockwise rotation about one point may produce a counterclockwise rotation about another point. You may be wondering why the weight of the meter stick was not used in any of the calculations above. Certainly it is another force acting downward, and it would certainly contribute a torque about the left end of the meter stick. While both are true, it does not matter. The force is trivial; the total force acting downward would increase, but the upward force of the support on the meter stick would increase by the same amount; i.e., the sum would still be zero. As for torque, we have an analogous situation because the meter stick is supported at its CG. The clockwise torque about the left end of the meter stick due to the weight of the meter stick itself would be exactly offset by the larger counterclockwise torque about this point due to the increase in the upward force of the support on the meter stick. Apparatus Meter stick, Knife-edge clamp, Support, Hooked Masses, String, Scissors, Triple-beam balance. Procedure The system you assemble in each of the following procedures starts with the meter stick balanced on the support by itself. The hooked masses are suspended from the meter stick with loops of string. 2
Two Masses in Equilibrium 1. Suspend a from the meter stick on one side of the support and a on the other side. Adjust the locations of these masses until equilibrium is achieved (the meter stick is again balanced). Place the data in Table 1, then calculate the torques about the support point. Location of support (cm) Location of (cm) Location of (cm) Table 1: Two Mass System; Torques about Support Three Masses in Equilibrium 1. Suspend a and a 50g mass on one side of the support (different locations), and a on the other side. Adjust the locations until the system is again balanced. Place the data in Table 2, then calculate the torques about the support point. 2. In Table 3, calculate the torques about the left end (0cm) of the meter stick. 3. In Table 4, calculate the torques about a third point of your choosing in the system (other than the previous two). Note: if one of your moment arms (hence torque) is 0, record it as such. Location of support (cm) Location of 50g mass (cm) Location of (cm) Location of (cm) 3
50g mass Table 2: Three Mass System; Torques about Support 50g mass Support Table 3: Three Mass System; Torques about Left End of Meterstick Location of point about which you are summing torques (cm) 50g mass Support Table 4: Three Mass System; Torques about a Third Point Determining an Unknown Mass 1. Suspend a on one side of the support and the unknown mass on the other side. Adjust the locations until the system is again balanced. Place the data in Table 5. Using that fact that the system is in equilibrium, calculate the mass of the unknown. 4
2. Determine the mass of the unknown with the beam balance for comparative purposes. Location of support (cm) Location of (cm) Location of unknown mass (cm) Unknown mass Mass of unknown from torques (kg) Mass of unknown from triple-beam (kg) Percent error in masses Table 5: Determination of Unknown Mass Analysis 1. For the two masses in equilibrium procedure, what was the net torque about the support point? What should be the net torque about this point? 2. For the three masses in equilibrium, what was the net torque about the support point? What should the net torque be about this point? 5
3. How did the net torque about the other two points you determined compare with the net torque about the support point in the three mass system? Is this surprising? Explain. 4. How close were you in determining the mass of the unknown from the summation of torques? Are you confident that a mass could be determined via this method? 6
Pre-Lab: Torque and Rotational Equilibrium Name Section Answer the questions at the bottom of this sheet, below the line - continue on the back if you need more room. Any calculations should be shown in full. 1. What is the clockwise torque about the support point in the system shown above? 2. What is the counterclockwise torque about the support point in the system shown above? 3. Is the system shown above in equilibrium? How do you know? 4. How many torques are there about the left end of the meter stick (0cm) in the system shown above? 7