Three Ways to Calculate the Torque of a Meter Stick
Suppose you have a meter stick upon which you hang weights such that the meter stick is balanced (with a
counterbalancing weight, such as in our lab and Figure 1), but the fulcrum of the meter stick is not at its
center of mass. Here are three ways to calculate the torque.
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Figure 1
Figure 2
Method 1. Suppose the center of mass of the meter stick is a distance x from the fulcrum. The simplest
way to calculate the torque is to realize that all the mass of the meter stick can be treated as acting at the
center of mass (see Figure 2). The force is M g and the distance from the fulcrum is x so the torque is simply
τ = rF = xM g, counterclockwise. (All distances are in meters and x is a fraction of a meter.)
Figure 3
Figure 4
Method 2. In this method (see Figure 3) we treat each side of the meter stick as acting with its fraction
of the mass at its center of mass. Thus the left side of the meter stick weighs F1 = ( 12 + x)M g and acts at
its center of mass which is a distance r1 = 21 ( 12 + x) away from the fulcrum. The other side of the meter
stick has a weight of F2 = ( 12 − x)M g and acts at its center of mass r2 = 12 ( 12 − x). Since the first torque is
counterclockwise, and the second clockwise, the total torque is τ = τ1 − τ2 where τ1 = r1 F1 and τ2 = r2 F2 .
So τ1 = 21 ( 12 + x)2 M g = 12 ( 14 + x + x2 )M g and τ2 = 21 ( 12 − x)2 M g = 21 ( 41 − x + x2 )M g, and subtracting the
two, the net torque is τ = xM g.
Method 3. In the third method (see Figure 4), we recognize that a part of the meter stick on the left an
equal length to the part on the right will balance the part on the right (the lighter colored parts). So the
net torque can be considered to be entirely due to the extra part on the left which is seen to be of length
2x (so the lengths add up to 1). The distance from the fulcrum the force acts is r = 21 − x + 12 2x = 21 ,
and the weight is F = 2xM g which is the fraction of the total weight for that length. Thus the torque is
τ = rF = 21 2xM g = xM g in agreement with the previous results.
Comment: For our lab, method 1 is best, because we have experimentally determined the center of mass.
For the other methods, we do not know precisely where the center of mass of each segment is.