# S-2 PROBLEMS IN EQUILIBRIUM

1. PURPOSE

To investigate several static equilibrium situations with nonconcurrent forces, and to analyze these with forces and torques.

2. APPARATUS

Loaded meter stick, support frame made from lab hardware, two ball-bearing pulleys, string, two weight hangers, slotted weight set, beam balance.

In this experiment you will balance a meter stick under the action of several applied forces. One force will be the stick's own weight. The stick has been deliberately "loaded" with lead at one end so that its center of mass will not be at its geometric center.

The other forces will be measured. Two methods are available for measuring them.

(1) Spring balances. Advantage: The setup is stable and easy to construct. Disadvantages: The spring balances' accuracy and precision are not very good. The balances only read correctly when hanging upright. A ball bearing pulley can be used to redirect a non-vertical string to a vertical spring balance.

(2) Strings over ball bearing pulleys (or other low friction pulleys) to weight hangers. Advantage: Very sensitive and accurate. Disadvantage: Very sensitive, making the system a bit tricky to balance. However, this is clearly the superior method, and should be used if such quality pulleys are available.

3. BACKGROUND

Reread the theory section of experiment S-1.

4. THEORY

(1) Torque. Torque is a physical quantity which plays an important role in systems which can rotate, and in the conditions which determine whether rotation occurs. Torque involves a force and an axis, and the geometric relation of the force to the axis. The axis can be any line in space. The torque about a particular axis is the turning effect about that axis due to an applied force.

[The axis in the definition of torque does not depend on whether there is actual rotation.]

An important class of torque problems consists of physical situations in which all of the forces lie in one plane. In that case it is usual to choose a torque axis perpendicular to that plane. The point where it crosses the plane is called the "center of torques." The "line of action" of a force is a line extended infinitely far along the direction of the force. The perpendicular distance from the center of torques to the line of action of the force is called the lever arm, l. The size of the torque about that center of torques is then given by Torque = f l , where F is the applied force and l is its lever arm about the chosen axis.

The rotational effect of a torque on a body in the earth's gravitational field is often associated with the center of gravity of the body, which is defined to be that point at which a single upward force can balance the gravitational attraction on all parts of the body for any position of the body. The line of action of the resultant of all gravitational forces on the body will pass through the center of gravity, no matter what position the body is in. The center of gravity may also be defined as the point about which the algebraic sum of all the gravitational toques is equal to zero for any orientation of the body.

(2) A couple is a pair of oppositely directed, but equal size, forces which do not lie along the same line. The size of a couple is defined to be the size of its torque, which is the same value no matter what center of torques is chosen. The torque of a couple may be expressed in terms of the size of each force and the length of the perpendicular between the lines of action of the forces:

The size of a couple does not depend on the choice of center of torques, and for a given F and D, the couple's effect on the system does not depend on where the couple is applied to the system. The size of the Torque of this couple has size FD.

The resultant force of a couple is obviously zero, but its torque is non-zero.

5. PROCEDURE

A. Suspended Meter Stick

In this part, and part B, you are presented with the problem of balancing a meter stick under certain given conditions. The stick will be considered balanced when it is stationary and horizontal. The forces will be supplied by strings suspending the stick, and by weights hanging from the stick. In the balanced condition we will insist the strings be perpendicular to the stick (to keep calculations simpler).

(1) FINDING THE CENTER OF MASS. The meter stick has been loaded with lead in one end, so you can not assume that its center of mass is at the 50 cm mark. The center of mass and the mass of the stick may be determined by a simple experiment. Attach two cords over two pulleys to weight hangers as shown in Fig. 1. Now determine experimentally what size weights placed on the hangers will bring the stick into balance. The three torques on the stick (what is the third?) now add to zero, so the torque equation may easily be solved for the two desired quantities.

The position of the center of mass found from the above calculation may be roughly checked by finding the point at which the stick will balance when suspended from a single string. The weight may be checked by weighing the stick on a beam balance.

Does the accuracy of the above method depend on the choice of the two suspension points? If so, what points are best. You must do the error analysis to answer this.

Using the data obtained above, solve the following problems mathematically, then check your answers by balancing the stick experimentally.

(2) PROBLEM: Consider the stick suspended from strings at 30 and 90 cm. If 500 grams were placed on each of these hangers, the stick would, of course, be unbalanced. What single additional force applied to the stick would bring it into balance, and where must that force be applied?

(3) Do some careful experimentation to determine, as well as you can, the size of the frictional torque in each pulley when the stick is balanced under the conditions of the problem. Use this information to do a complete error analysis on this situation.

Obtain a third pulley, and try to devise a situation where the stick balances in a horizontal position, but two of the applied forces are not perpendicular to the stick. Analyze this mathematically.

B. Couples

Read part A for a general description of meter stick balancing problems.

In all meter stick balancing problems, the stick's own weight provides one downward force, -W. Suppose that a force +W (equal to the stick's weight) were applied upward at the 40 cm mark. These two forces, +W and -W, are a couple, and though their total force is zero, their total torque is not zero. If now an equal and opposite couple is now also applied, the stick will balance. Calculate the required couple, then apply it and see if the stick balances.

This couple will balance the stick no matter where it is applied to the stick. Check this assertion for several different locations. [Relocate the two forces, but keep the horizontal displacement of their lines of action the same.]

C. Non-Parallel Forces

(1) Relocate one pulley to one of the upright rods. Run a string, H, horizontally from one end of the meter stick and over that pulley. Adjust the weights to duplicate Fig. 4, with the meter stick and the string, H, both exactly horizontal.

(2) Measure the angle θ that string S makes with the horizontal. There is a goniometer on the back cover of this lab. manual.

(3) Use force and torque analysis to calculate W and a, using only the experimental values of S, V, H and θ. Explicitly show the equation(s) you develop for this, and their error equation(s).

(4) Calculate the tension, H, in the horizontal string. Check this by replacing this string with a sufficiently sensitive spring balance, or with another pulley and weight-hanger. Be certain the stick remains in its previous position when doing this. It is especially important that the angle θ not be changed.

6. QUESTIONS

In all questions asking for a proof, or using the language "show that", you must supply a general mathematical (geometric) proof of the proposition which follows from the given assumptions. Do not include experimental data in these.

(1) Does a spring balance correctly measure forces in all positions; hanging down; upside down; and suspended horizontally?

(2) A student doing a lab problem with a balanced meter stick as in parts A and B notes that all of the weight hangers have equal mass. The student concludes that the hanger masses may safely be ignored, since their effects would "balance out." Is this correct? Explain.

(3) Prove that the torque of a force is equal to the sum of the torques of the components of that force.

(4) If only two forces, forming a couple, act on a body, show that that body cannot be brought into equilibrium by adding just one more force.

(5) Prove that the size of a couple is independent of the choice of the center of torques. You may limit the discussion to a two-dimensional situation.

(6) Prove that if a body is in equilibrium under the action of several forces, among which is a couple, that the body will still balance if the couple is moved to any other location, provided the couple's size is kept unchanged. You may limit the discussion to a two-dimensional situation.

(7) [Bernard and Epp] If, in any part of the procedure, the meter stick were balanced resting at an angle, rather than in a horizontal position, would the meter stick be in equilibrium?

(8) [Bernard and Epp] Suppose the stick made a 10° angle with the horizontal in procedure A, but this fact was not taken into account in the calculations. When the oversight is discovered the situation is recalculated by explicitly including the 10° angle and correctly calculating the true lever arms. How do the "right" and "wrong" results compare?

Text and diagrams © 1997, 2004 by Donald E. Simanek.