Equilibrium: Forces and Torques

In this lab exercise, you will show that equilibrium is achieved when both the sum of the forces and sum of the torques acting on a body equal zero. The "body" will be a plastic disk with many peg holes. The disk rests upon a force table with three small steel balls placed between them to reduce frictional forces.

Equipment

• Force table with hangers and masses (weights)
• Plastic disk with peg holes
• Ruler and/or straight edge, protractor

Procedure

1. Place a plain piece of paper over the top of the disk and tape it such that it cannot move. Push three pegs through the paper and into holes in the disk at random positions. The three positions should NOT lie along a line.
2. Level the force table using the adjustable feet. Place three small steel balls roughly in a triangle on the table and place the disk on top. The screw eye at the center of the disk should fit loosely into the hole at the center of the table, allowing the disk to move freely only about a centimeter in any direction.
3. Place two pulleys at arbitrary positions, but within a range of 20^o to 160^o degrees apart. Attach two strings each to two of the pegs and over the two pulleys. Add hangers and slotted masses of a few hundred grams to the end of the strings. Make sure the strings are parallel to the surface of the disk. Since we are trying to demonstrate that the principles apply for forces of arbitrary size and direction, do NOT make the weights identical.
4. Attach a string to the third peg. Pull on the string, varying the direction and strength while keeping parallel to the plane, until the disk is in equilibrium. It may be necessary to gentle nudge the disk during this procedure to insure the center screw eye on the disk is not touching the center hole of the force table. Note the angle of the third string on the force table.
5. Place a pulley at the angle just noted. Place the string over it and add a hanger and masses until equilibrium is again achieved. You may need to make fine adjustments to the third pulley's position. Estimate an uncertainty in the mass needed by adding and removing small masses to find a range of values that achieve a reasonable equilibrium.
6. With a pen or pencil, carefully make two marks directly below each string. Make the marks as far apart as possible, such that you have an accurate measure of the strings direction. Label each string pair with the total mass hanging from each string. Do not forget to include the hanger mass. In addition, make two sets of marks on either side of the center screw eye. You will use these marks to located the center of the disk.
7. Disassemble the apparatus and remove the paper. The instructor will make photocopies for each member of the group.

Report:

1. Sum of Torques
   • With a straight edge aligned with the pairs of points, use a pencil to mark the line of each of the strings. Make the lines extend completely across the page. Using the two pairs of center marks, form an "X" that marks the center.
   • Mark and then measure the perpendicular distance from the center to each line of action of the three forces. Estimate an uncertainty in the moment arm value from the precision of your ruler or the limits of your eyes, whichever is largest. Label the values and uncertainty clearly on the diagram.
   • Convert the mass values to force (weight) and calculate the torque created by each of the three forces. Choose clockwise rotations as positive. You may use any units you wish, but be clear to state those units.
   • Add the three torques and comment on the value. Is the total torque zero? If not, add or subtract the uncertainty in the weight of the third force and the uncertainties in all three moment arms as needed to push the total torque value towards zero. (The lab instructor will provide an example of how to do this.) Was the new calculation zero or of an opposite sign from the original calculation? Comment on these results.

2. Sum of Forces

   On a separate standard 8.5" by 11" sheet of paper, you will draw vectors representing the three forces and add them graphically to show that they sum to zero.

   • Identify the largest of the three forces. You will represent this force as a vector of length 20 cm. (This is slightly less than 8 " and will fit onto the sheet.) Calculate the length of the other forces by making the ratio of length to force value the same for all three.
   • Using a protractor and the original diagram, carefully measure the angles between the largest force and the other three.
   • Draw all three forces, with proper length and direction on the sheet. It should be possible to orient the forces such that all stay entirely within the page. You may wish to experiment with pencils or straws to find the best orientation.
   • Do the forces sum to zero? That is, do they form a closed triangle? If not, comment on whether or not the uncertainty in the value of the third force could account for the discrepancy.