Equilibrium of Non-­‐Concurrent Forces Objectives: •
•
Experimental objective – Students will verify the law of moments. Learning objectives (students should learn…) – The requirements to balance the moments of unequal masses. – The principles of parallel forces Equipment list: Meter stick, fulcrum with knife-­‐edge, scale, balance, rigid support, known mass set with a pair of looped threads, two unknown masses. Apparatus: Theory: In this experiment, we will focus on a concept called “Moment of force” or “torque”. This is closely related to torque, which you will encounter later in the semester. Here, we are concerned with balancing these moments to establish equilibrium. The law of moments allows us to determine when an object is balanced. It has important applications in aviation because pilots need to know if their aircraft will fly strait and level. The law of moments says that an object such as a scale will be in equilibrium, and will not tip in either direction, when the moments of force in either direction are equal. We will consider a situation (pictured above) where the horizontal object can either rotate clockwise or counterclockwise. These movements are due to a clockwise moment and a counterclockwise moment. Each moment is defined as follows: 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑡𝑜𝑟𝑞𝑢𝑒 =
𝑀𝑐! ×𝑅𝑙𝑎! 𝑔 !
And 𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒, 𝑡𝑜𝑟𝑞𝑢𝑒 =
𝑀𝑐𝑐! ×𝐿𝑙𝑎! 𝑔 !
Accordingly, torque is expressed in the units Nm (be sure to convert your measured values to SI units so that your result has the appropriate units). Thus, at equilibrium, the torques can be equated (condition for rotational equilibrium). 𝑀𝑐𝑐! ×𝐿𝑙𝑎! =
!
𝑀𝑐! ×𝑅𝑙𝑎! !
Note the acceleration of gravity, g, can be removed from the equation at equilibrium by dividing on both sides. For the situation above, with only two masses, the equation above reduces to 𝑀𝑐𝑐! ×𝐿𝑙𝑎! = 𝑀𝑐! ×𝑅𝑙𝑎! Because of the definition of torque, some force with a given lever or moment arm is the same as half the force with twice the lever arm, and this gives a mechanical advantage in the use of a lever. Think of trying to pry a nail out of a board… you can apply more force to the nail by either pushing harder on the lever or using a lever with a longer lever arm. This is why crowbars have a long side and a short side. The same principle can be used to break free a stubborn frozen bolt using a cheater bar. The relationship between moment of force and the lever arm gives rise to another interesting property. If the balanced object is shifted so that the fulcrum is no longer at the center of mass of the object, the system behaves the same as if a mass equal to the mass of the object is placed at its center of mass. While we will not concern ourselves with a mathematical proof of this concept, you will confirm it through experimentation. In a system with a meter stick set so that its center of mass is away from the fulcrum, and different masses are placed on each side so as to balance the meter stick, there is an effective mass at the point of the meter stick’s center of mass. You will find the value of that mass. Equations: For this experiment, you will need to calculate the %difference between torques. First, you will need to calculate the mean moment with the equation 𝑚𝑒𝑎𝑛 𝑚𝑜𝑚𝑒𝑛𝑡 =
𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑡𝑜𝑟𝑞𝑢𝑒 + 𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑡𝑜𝑟𝑞𝑢𝑒
2
Then, the %difference can be calculated as follows: %𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =
|𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑡𝑜𝑟𝑞𝑢𝑒 − 𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑡𝑜𝑟𝑞𝑢𝑒|
𝑚𝑒𝑎𝑛 𝑡𝑜𝑟𝑞𝑢𝑒
Procedure: Part 1 – conditions of equilibrium 1.
2.
3.
4.
5.
Place the meter stick in the sliding knife-­‐edge support so that the (unloaded) meter stick balances horizontally. And note the position of the knife-­‐edge on the stick. Caution: when adjusting the position of the knife-­‐edge, gently secure the meter stick with the setscrew. Do not over-­‐tighten or drive the screw into the wood! Place a 100g mass at the 10cm mark and place a 200g mass on the opposite side of the fulcrum in a position that so that the meter stick remains balanced horizontally. Record the masses and lever arm in table 1. Repeat steps 2 and 3 with different masses at different positions. Calculate the clockwise and counterclockwise torques and the percent difference between them Part 2 – using torques to calculate the mass of the meter stick 6.
7.
Adjust the knife edge so that it is at the 30cm mark on the meter stick. Place a 300g mass at the 10cm mark and place a 100g mass on the other side of the fulcrum in a position so as to balance the system. 8.
9.
10.
11.
Record the mass and lever arm in table 2. Repeat steps 7 and 8 with different masses at different locations. Calculate the moment from the masses. Calculate the moment of the effective mass at the meter stick’s center of mass. This is simply the difference of the two torques calculated from the masses. 12. Calculate the mass of the meter stick from the moment calculated in step 11, located at the meter stick’s center of mass. 13. Weigh the meter stick on a balance to measure its mass. 14. Calculate the %error of the mass from step 12 compared to the actual mass measured in step 13. Part 3 – determining an unknown mass 15. Adjust the knife-­‐edge so that it is at the center of mass of the meter stick. 16. Place one of the unknown masses at the 80cm mark. 17. Place a known mass on the opposite side of the fulcrum so that it balances the system (choose the known mass such that its lever arm is significantly different from the unknown mass lever arm). 18. Record the known mass and both lever arms in table 3. 19. Repeat steps 17 and 18 with a different known mass. 20. Calculate the moment from the known mass, and determine the moment from the unknown mass. 21. Calculate the unknown mass. 22. Weigh the unknown mass on a balance to measure its mass. 23. Calculate the %error of the determined mass from step 18 compared to the actual mass from step 19. Prelab: 1.
2.
3.
4.
5.
What are the two conditions for equilibrium of a rigid body? How would this experiment demonstrate their validity? Why do you not need to take into consideration the force exerted on the meter stick by the support? In part 3, you can calculate the moment of the known mass and use that to determine the unknown mass, but explain how the unknown mass can be determined without directly calculating the torque (hint: use ratios). What defines the center of gravity of a rigid body? How does it relate to center of mass? Assume a 150g meter stick with center of mass at the 48cm mark is supported by the 60cm mark. There are two masses placed on the meter stick; m1=50g and is placed at the 16cm mark and m2 is an unknown mass at an undisclosed location. The meter stick is in equilibrium. What is the torque induced by each of the masses (don’t forget about the mass of the stick itself)? Now, supposing m2 is located on the 100cm mark, what must its mass be? What if it is at the 80cm mark? Report: Table 1 – part 1 Mass on right side Lever arm on right side Mass on left side Lever arm on left side Clockwise torque Counterclockwise torque %difference Position of fulcrum = Mass on right side Lever arm on right side Mass on left side Lever arm on left side Clockwise torque Counterclockwise torque Difference (meter stick mass) Calculated mass of meter stick Measured mass of meter stick %error Position of fulcrum = Center of mass of meter stick = Table 2 – part 2 Known mass Lever arm (known mass side) Lever arm (unknown mass side) Torque (known mass side) Torque (unknown mass side) Calculated unknown mass Measured unknown mass %error Position of fulcrum = Table 3 – part 3 Questions: 1.
2.
3.
4.
How well do your experimental results compare to the expected results? Did you confirm the theory you were testing? Cite your results and errors specifically. In most cases, the balancing position of the meter stick (unloaded) is not on the 50cm mark. Why is that the case? What does that mean about the center of mass of the meter stick? What are the largest sources of error in your experiment? Explain how the old fashioned scales in doctors’ offices and gyms (with all of the sliding blocks) work.