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Equilibrium LAB Rev. 8/16/22 DLD 1 LAB Exploring Equilibrium Objectives: Examine the calculation of torque Examine the conditions for an object to be in static equilibrium Use the condition of static equilibrium to determine an unknown mass Equipment: Computer with internet access Rigid, straight, uniform solid rod or tube at least 40cm long 18 pennies Masking tape Fulcrum (cardboard, wood, binder clip, etc.) Thin string, thread, fishing line, or similar (optional) Plastic baggies (2-optional) A small unknown mass (ex: an eraser, a small toy, a battery, etc.) Lab balance Cell phone Theory: Recall that the conditions for the static equilibrium of a rigid body are  F = 0 (Equation 1a)  = 0 (Equation 1b) Object is at rest The first condition,  F = 0, is concerned with translational equilibrium and ensures that the object is at a particular location (not moving linearly) or that it is moving with a uniform linear velocity (Newton’s first law of motion). The second condition,  = 0, indicates that the object is in rotational equilibrium, and either it does not rotate (static case) or it rotates with a uniform angular velocity. The third condition that the object is at rest is required for the object to be in static equilibrium. A torque results from the application of a force acting at a distance from an axis of rotation (Fig. 1). The magnitude of the torque is equal to the product of the force’s magnitude F and the perpendicular distance r from the axis of rotation to the force’s line of action (a straight line through the force vector arrow). That is,  = rF (Equation 2) Figure 1: The moment arm is the perpendicular distance from the axis of rotation to the line of action of the force.

Equilibrium LAB Rev. 8/16/22 DLD 2 LAB The perpendicular distance r is called the lever arm or moment arm. The unit of torque can be seen to be the meter-Newton (m-N), to emphasize the difference from energy units of Joules. Relative to an axis of rotation, a rigid body can rotate only clockwise (CW) or counterclockwise (CC) due to torques in one of those senses. For example, taking the axis of rotation at the 50 cm position, F 1 and F 2 produce counterclockwise torques and F 3 and F 4 produce clockwise torques, but no rotation takes place if the torques are balanced and the system is in rotational equilibrium. The condition for rotational equilibrium is  =  CC +  CW = 0 (Equation 3a) where  CC and  CW are counterclockwise and clockwise torques, respectively. Designating the directions arbitrarily by plus and minus signs, Equation 3 can be written  CC -  CW = 0 (Equation 3b) Figure 2. Rotational equilibrium requires balanced torques in opposite directions. The forces are due to weights suspended from the rod, and with F = mg, m 1 gr 1 + m 2 gr 2 = m 3 gr 3 + m 4 gr 4 (Equation 5). Procedure 1. First, you need a uniform rod from which to suspend your masses. Something straight, solid and more than 40cm long will work the best. The most important thing is that the item has a uniform mass distribution along its length. If you brought an object from home, identify it here and have it approved by your instructor. Object brought from home_____________________________________________________ If you did not bring something, you will need to construct your rod from the cardboard and masking tape provided in class. This is not difficult, but please be VERY careful when you cut

Equilibrium LAB Rev. 8/16/22 DLD 3 LAB and assemb le the cardboard so you don’t cut yourself and so that the mass distribution of your rod remains uniform. The photo shows a cardboard “rod” made by one of your instructors. It is made from 6 layers of cardboard. It was cut into 3 strips and scored down the middle and bent in half to make a doubled thickness. The three 1.5inch wide strips were then taped together at 4 spots using masking tape that was measured to exactly 6” in length. The position of the tape was also measured so that the mass distribution remained uniform . 2. Using a ruler or tape measure, determine the length of your uniform rod and record it. Length = m. 3. Now you need a fulcrum. The fulcrum MUST be pointed on the top (or round) so that your uniform rod balances, but does not simply sit flat on top. It should also be high enough so that if you choose to suspend your masses, they don’t touch the table. You can make your fulcrum taller by putting it block or book, but it must be stable. You can make the fulcrum out of cardboard as shown in the photo on the left. Other alternatives may include a binder clip with the prongs removed or a triangular block of wood if available. Alternatively, you may suspend your rod from a string and hang it up. 4. Balance your rod with nothing on it on the fulcrum so that it is in static equilibrium. You want no rotational or translational motion! You might want to tape a piece of graph paper or lined paper to the wall behind your system so that you have a something with which to compare. 5. Place a mark on your rod with something so you know where the fulcrum was located in case you bump the system. Now you are ready to start adding masses. You will be using pennies as your masses.

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Equilibrium LAB Rev. 8/16/22 DLD 4 LAB Please note: all of the measurements being used are from the zero position on your rod. Zero will be considered the LEFT end of the rod. Part 1: Two known masses. 6. With the uniform rod balanced, suspend by a thin loop of thread or place on top of your rod, a mass m 1 = 7 pennies (7 M) at a position equal to 25% of the total length of your rod from the zero position. For example: if your rod is 40cm long, you would place 7 pennies at a location 10cm from the left end of your rod. NOTE: If you are completing this lab off campus (and ONLY if you are off campus) the US Treasury has stated that the standard mass of any penny minted after 1982 is 2.500g. You may use this value as the weight of your individual pennies, but check the dates and use only pennies dated after 1982. If you don’t have enough pennies, you may use any other object (M &Ms, Skittles, etc.) for which you can find a standard mass on the internet and that you can either easily stack or hang from your rod. You must be able to measure to the center of mass of the pennies. It is acceptable to stack them and use the center of symmetry of the stack as the center of mass for the objects. Alternatively, you may elect to suspend them from a string under your rod. Cut the corner from a sandwich or snack bag and use it as a small “baggie” to hold the pennies. In the photos, dental floss was used as the “string.” N either the bag nor the string adds significant mass to the system. 7. Set up the conditions for static equilibrium by adjusting the position, and thus the moment arm, of a mass m 2 = 4 pennies (4M) on the side of the rod opposite m 1 . Record the masses of each stack of pennies from the balance and the measured moment arms below. Remember the moment arms are the distances from the fulcrum to the masses.

Equilibrium LAB Rev. 8/16/22 DLD 5 LAB Completing the example: this means that you need to suspend 4 pennies on the right side of the rod and move them around until the rod is perfectly balanced again. Find the distances from the fulcrum (these are your moment arms) and complete the calculations requested. m 1 = 7M = r 1 = m 2 = 4M = r 2 = Compute the magnitudes of the torques. Show ALL your work below. (the choice of CW and CC is arbitrary-it is just what you observe from your perspective) (equation, substitution with units, answer)  cw =  cc = Find the percent difference in the computed values of torque (i.e., compare the clockwise torque with the counterclockwise torque). Show ALL your work below. (equation, substitution with units, answer) Percent difference = 8. Take a photo of your system and insert here.

Equilibrium LAB Rev. 8/16/22 DLD 6 LAB Part 2: Three known masses-Case (a) Measuring r 3 9. Again, with the rod balanced with no masses , suspend, m 1 = 5M at the 30% mark from your zero position and m 2 = 10M at the 70% mark from your zero position. Add a third mass, m 3 = 3M, and adjust its position so that the rod is in static equilibrium again. Measure all of the moment arms and record them. m 1 = 5M = r 1 = m 2 = 10M = r 2 = m 3 = 3M = r 3 = Compute the magnitudes of the torques. Show ALL your work below. (equation, substitution with units, answer)  cw  cc Find the percent difference in the computed values of torque. Show ALL your work below. (equation, substitution with units, answer) Percent difference =

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Equilibrium LAB Rev. 8/16/22 DLD 7 LAB Part 2: Three known masses-Case (b) Confirming r 3 10. Calculate, theoretically, the lever arm (r 3 ) for the mass m 3 = 3M for the system to be in equilibrium if m 1 = 5M is suspended at the 30% mark from your zero position and m 2 = 10M is suspended at the 70% mark from your zero position. Show ALL your work below. (equation, substitution with units, answer) Record your information from #9 again here m 1 = 5M = r 1 = m 2 = 10M = r 2 = m 3 = 3M = Compute the percent difference of the experimental value of r 3 . Show ALL your work below. (equation, substitution with units, answer) r 3 (calculated) = r 3 (measured) = Percent difference =

Equilibrium LAB Rev. 8/16/22 DLD 8 LAB Part 3: An unknown mass. 11. Choose a small object of unknown mass from the items provided by your instructor. If you have something personal you really want to try, have it approved by your instructor. Place your unknown mass at a location close to the zero point of your rod. Set up the conditions for static equilibrium by adjusting the position and the mass, if necessary, of your known small objects . We suggest starting with 10M as your known mass, but adjust according to the object you chose. Record the masses and moment arms below. Identity of the unknown object______________________________________ m 1 =unknown r 1 = m 2 = r 2 = Use the methods from the lab to determine the unknown mass (in appropriate mks units), showing all your work here (equation, substitution with units, answer). 12. Measure the mass of your unknown on the classroom balance and compare it to the mass you determined from your equilibrium calculations. Show all your work here (equation, substitution with units, answer). Mass of the unknown from the balance ____________________

Equilibrium LAB Rev. 8/16/22 DLD 9 LAB Part 4: Finding the mass of the rod-using a pivot that is not at the center of mass 13. Take as many of your pennies (small objects) as possible and place them at a single location 10% of the distance from the end of your rod. You can use a single stack, or two or three stacks side- by-side, as long as the center of mass of each stack is the same distance from the end of the rod. Now adjust the position of the pivot point so that the rod balances again. Clearly you will have one very long end without any additional masses on it, and one short end where your pennies are located. Record the mass of the pennies and the moment arm for the pennies. The moment arm for the rod itself should be determined by measuring the distance from the pivot point to the center of mass of the rod. Record that value as well. m 1 =pennies=_______________ r 1 = m 2 = rod, unknown r 2 = Use the methods from the lab to determine the unknown mass of the rod (in appropriate mks units), showing all your work here (equation, substitution with units, answer). Measure the mass of your rod on the classroom balance and compare it to the mass you determined from your equilibrium calculations. Show all your work here (equation, substitution with units, answer). Mass of the rod from the balance ____________________

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Equilibrium LAB Rev. 8/16/22 DLD 10 LAB 14. Think carefully about this experiment. Identify the 3 errors that you feel most influenced your results for confirming static equilibrium in each case. Keep in mind the analysis of error you have been doing in previous labs. Complete the table for those 3 errors. If your complete answers do not fit in the space provided, enlarge the height of the rows or create a new table. Complete Description of Error Type of Error Influence on Data and/or Results Suggestion for Mitigation