## S-2 PROBLEMS IN EQUILIBRIUM1. PURPOSE
To investigate several static equilibrium situations with nonconcurrent forces, and to analyze these with forces and torques.
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 The other forces will be measured. Two methods are available for measuring them. (1) (2)
Reread the theory section of experiment S-1.
(1) [The 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 that center of torques is then given by Torque = f ,
where F is the applied force and l is lits 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 (2) A
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.
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 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,
(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
(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 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
This couple will balance the stick (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
(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.
(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. |