2. What are the conditions for equilibrium of a rigid body? the page? Is the torque clockwise, or is it counterclockwise? Show your work.

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1 Name Section Date Read carefilly the entire description of the oratory and answer questions based on the material contained in reading assignme completed prelaboratory assignment at the beginning of the 1 oratory period prior to the performance of the laboratory. 1. State a definition of torque and give an equation for torque. Define the terms in the equation. 2. What are the conditions for equilibrium of a rigid body? 3. For the meter stick shown in Figure 10.1, the force I;; = 10.0 M acts at 18.0 cm. What is the magnitude of the torque due to F1 about an axis through point A perpendicular to the page? Is it clockwise, or is it counterclockwise? Show your work. Figure 10. I Meter stick with tvvo forces FI and F2 acting at points shown. 4. In Figure 10.1, the force Fz = 15 N acts at the point 70.0 cm. at is the magnitude of the torque due to Fz about an axis through point B and perpendicular to the page? Is the torque clockwise, or is it counterclockwise? Show your work. 5. For the meter stick in Figure 1.I, what is the m itude of the net torque due to both forces PI and F2 about an axis perpendicular to the page through point A? Is it ckochise or counterclockwise? Show your work.

2 6. For the meter stick in Figure 10.1, what is the magnitude of the net torque due to both forces F1 and F2 about an axis perpendicular to the page through point B? Is it clockwise or counterclockwise? Show your work. 7. In Figure 10.2, if mass rnl = kg acts at 2 em, what is the value of mass m2 that must be placed at the position 70.0 em shown to put the system in equilibrium? Write the equation for ZT,, = ZaCw with the rnass m2 as unknown and solve for m2. Assume the meter stick is uniform and symmetric. Show your work. Figure 10.2 Meter stick with forces applied by hanging two masses mi and m2. 8. The meter stick in Figure 10.3 has ml kg at P.0 em, ma = kg at 20.0 cm, and mg = kg at 70.0 cm. The meter stick mass is kg, and it is symmetric and uniform. At what position should a kg mass be placed to put the system in equilibrium? Write the equation for ZT,, = Z+rCwith the lever arm of the kg mass as the unknown and solve for the lever arm and thus the position. Show your work. Figure 10.3 A mass of kg is to be applied to place the system in equilibrium. 118 Laboratory 8 0

3 Forces acting on a body of finite size ten to both translate and rotate the body. is to be in equilibrium, it must be in equilibrium both with respect on and to rotation. In this laboratory, a meter stick pivoted on a support whose position is adjustable will be subjected to various forces by hanging weights on the meter stick, Measurements of the magnitude and position of e meter stick will be used to accomplish the following objectives: 1. Application of the complete conditions for equilibrium of a rigid body to a meter stick pivoted on a knife edge 2. Experimental determination of the center of gravity of the meter stick 3. Determination of the mass of the meter stick by the application of known torques to the meter stick 4, Comparison of the experimentally determined location for a given applied force to produce rotational equilibrium with the location predicted theoretically 5. Determination of the mass of an unknown object EQUIPMENT LIST 1. Meter stick with adjustable knife-edge clamp and support stand 2. Laboratory balance and calibrated hooked masses 3. Thin nylon thread and unknown mass with hook THEORY If a foree F acts on a rigid body that is pivoted about some axis, the body tends to rotate about that axis. e tendency of a force to cause a body to rotate about some axis is measured a quantity called torque r. e torque T is defined by 7 = FdI (11 where F is the m tude of the force, and d, stan r the lever arm of the force that is the perpe lar distance from the force t location of the axis of rotation. The units of torque are Nm. mplication is that the torque due to a given force must be defined relative to ific axis of rotation. Figure 10.4 shows two forces PI and F2 acting on an arb Laboratory

4 igure Lever arms about the point 0 for o forces acting on a body. The axis of rotation of the body is labele is along a line through 0 perpendicular to the page. For each force, t section of the line of action of the force is shown by a dotted line extended in either directio force vector. The lever arm for each force is shown as the perpendicular rn O to the line of action of the force. In this case there are two torques, acting on the body, where Torques tend to rotate the about the axis. The conventi torques be positive agld clo net torque due to forces F1 ise or a counterclockwise direction ory will be to let counterclockwise gative. Using that convention, the rough 0 is given by The net torque T, could he either eounterelockwise or eloekvvise depending on whether F2 d2 or F, d, is greater in magnitu e. If the magnitu es of F2 d2 and F1 dl e same, the net torque is zero. In this laboratory, a meter stick y to which forces will be applied to produce mechanical equili re are two conditions that must be satisfied for corn equilibrium of a rigid body. Th are as follows: (1) The vector sum of e forces acting on the body mus e zero. This ensures slational equilibrium. net torque about any axis of the e zero. This means nitude of CT~, (the sum of the counterclockwise toqu must be equal to the magnitude of ZT,, (the sum of the clockwise torques). is ensures rotational equilibrium. he center of gravity which the sum of all the torques due to all ody is zero. If it is true that the gravitation then the center of t. Efieetively, the entire mass of the niform and sym- that i.s uniform an

5 Figure Meter stick alanced at point away from the center of gravity..the stick has torques from the three applied masses an from the meter stick mass at the center of gravity. Consider a meter stick with a port stand as shown in ass of the meter stick e other masses mg? m2, hung from the meter s s places. The masses ml, m2, and m3 produce forces e an2 g, and rn3 g at the point where they are pl irected upward at the point of the support. The weight of the meter stick mog is a force exerted at the tenter of gravity of the meter stick, labeled as xg. Assume that the meter stick in gure 10.5 is in corn lete eguilibx-iurn. Forces in the upward direction are consider positive, and force n the downward direction are considered negative. Counterclockwis orques are considered positive, and clockwise torques are considered negative. ques are taken about an axis perpendicular to the page through the point of support, which is labeled as the position xo. The lever arms for each mass di are calculated from the equation di = ]xi- x, 1 where xi is the position of the ith mass. e application of the two conditions for cornplete equilibrium of a rigid body to this example leads to the following equations: Consider the following numerical example of the situation described in Figure The mass of the meter stick is mo =.I200 kg, and masses mp = kg and m2 = kg n in the figure. The point of FB; is at the m mrark. s rn3 must be placed at the I. rn mark in order to put the syste at is the resulting force Fs with which the support pushes r stick? Solution: ZF = Ois Note that since the aceelerati to gravity was a common factor in the torque equation, it was cancelle term and was in the calculation of the mass m3. On the 0th factor g m.ust in the calculation of the force Fs that the support exerts.

6 e numerical exampl torques about an axis throu complete equilibrium have any axis is then ensure ove, consider the torque end of the meter stick. Fo values gives mined for which the sum of ar to the page through the left 2, mot and m3 exert clockwise ise torque. Calculating t Thus, the net torque about an axis per left end of the meter stick is equal to clockwise and clockwise torques are torque will be zero about any axis con In the experimental arrangements satisfying the conditions of rotational act through the support position. If not contribute to th rotational equilibriu force Fs will be such to ensure trans In all of the experimental proced thread to hang the hooked masses a very small piece of tape to hold the t the mass. support force Fs will always mall loop sf nylon e helpful to use a s desired to place EXPERIMENTAL PROCEDURE L Remove the knife-edge clamp from t e meter stick. Determine the mass of the meter stick using the laboratory recvrd the value in the Data Table. 2. Place the knife-edge cl p on the meter sti place it on the support. Adjust the position of the cla ntil the best bal e is achieved. Be sure the clamp is tight at balance. The ion of the knife e clamp is now the center of gravity of the meter stick. ecord this position as xg in Data Table.. With the meter stick s osition xg, place a the m mark. Ex x2 at which a mass rn2 = stick. Record the value of xz in Data Table I. 4. Calculate the lever a m the position of the ith mass. Not center of gravity xg, th tribute to the tor di for ea or this case d, = I xg - xi f, where xi is rt is at the position of the d thus will not conations Table I.. Laboratory 10

7 5. Calculate an e sum of the counterclockwise torques XT,. In ly eountercloekwise torque is due to ml, and C T - ~ nil&. ~ ~ C in Calculations Table 1 the value sum of the clockwise only clockwise torque is m2, and C T = ~ m2 ~ According Lo theory, I;he magnitn be equal since the riu.m. Calculate the percentage difference between and result in Calcdations Table 1. eter stick at xg, the cen ty of the meter stick. Place ml = 000 kg at the rn mark. the position x3 at which a mass balance the system. Record t 2. For this experimental arrangement, the meter-stick mass mo again makes no contribution to the torque. Calculate the lever arm for each of the other masses the values in Calculations Ta Is 2 (di = lxg - xi I ). 3. Calculate the ue of ZT~, and Calculations Table 2. Cal the record it in Gal ble 2. Note that you must for each mass whether it contributes to the counterclockwise or clockwise torque. Use a value of ds2 for g. 4. Calculate the excentage dif'ference between CT~,, and Cr,, and record it in Calculations Table Place ta mass ml m position. Loosen the knifeedge until the torque exerted by the weight of the meter stick acting when the best balw supported is xo. 2. The values of the lever arms are give y dl = Ixl - xo 1 and do - Ixg - xo 1. Cab culate and record the values of dl an in Calculations Table For these con itions Z T = ~ ml& ~ ~ and 2,7,, = rnogdo, where ano stands for the mass of the meter stick, w to be unknown. cpating the two torques gives ml& = mogd hat equation gives mo = rn3 (d,lce,). Calculate the value of the m equation and record it in Caleulations Table 3 as meter-stick mass (mo)exp. 4. Calculate andl re in Calculations ercentage error in t mental value (m to the meter-stick mass recorded in Dat y balance. ass Laboratory 10

8 The location of mass m4 needed to place the system into eqlsili 0th experimen~aliy and theoretically and then compare cdatioss Table 4 as (d4)exp. ace provided in Data n which the counter ilzckude the eontrkbukion from $he me ity of the meter stick xg. 4. Using this value of x4 caqrd this value of d4 inn Cal- ymbols for the appropriate ever arms. Be sure to also ctkng at the center ofgrau- treat the lever arm dq of own. Solve the equation to obtain a value result in Calculations Table 4 as (d4)theo. percentage error in (d4 compare to (4)thee and recor n experimental arr 2. Determine the unknown mass centage error in your measure c~ratsry balance. n an equation that own mass as the imelntal your

9 Name Section - Date s tational Equili Table 1 Table 2 Table 3 Support position xo = M Laboratory

10 Table 4 Support position ro = m - rn3 = x3 = / ds = I Solving equation above for d4 gives Table 5 SAMPLE CALCU 126 Laboratory 10

11 QUESTIONS 1. For the data for the first two parts of the laboratory when known forces are balanced, discuss the agreement between Zro and Xr,,. Are the experimental results sufficiently close to consider that the data verifies the theory? 2. In part 3 of the laboratory, consider the percentage difference between the two determinations of the meter-stick mass. Do the data agree suffnciently well to be considered to verify the theory? 3. In part 4 of the laboratory, agreement between experimental and theoretical, values of the lever of rn4 sufficiently goo be considered to verify the theory?

12 4* In all of the experimental arrangements the ass of the knife-e-edge elamp is ignored. Is this an approximation because its m s is small, or is there some reason it makes no contribution at all to th torque? If you think there is some reason that it does not have to be eonside, state the reason. 5. Suppose an experimental arrangement like the one in part 2 has a mass ml = kg at the m mark and a mass n2 = kg at the m mark. Codd the system be put into equilibrium by a kg mass? If so, state where it would be placed. If it cannot be one, state why not. 6. In part 1 of the laboratory, ue of the force Fs with which the support pushes upward on the meter stick? 7. For the equilibrium conditions established 4 of the laboratory, calculate the counterclockwise and clockwise torques an axis perpendicular to the page through a point at the left end of the m ck. Calculate the percentage difference between the net counterclockwise d the net clockwise torque.