Question

Figure 12-19 shows an overhead view of a uniform stick on which four forces act. Suppose we choose a rotation axis through point O, calculate the torques about that axis due to the forces, and find that these torques balance. Will the torques balance if, instead, the rotation axis is chosen to be at

(a) point A(on the stick),

(b) point B(on line with the stick), or

(c) point C(off to one side of the stick)?

(d) Suppose, instead, that we find that the torques about point Odoes not balance. Is there another point about which the torques will balance?

Short answer

a) The torques will balance about the rotation axis at point A.

b) The torques will balance about the rotation axis at point B.

c) The torques will balance about the rotation axis at point C.

d) The torque will not be balanced about any other point if it is not balanced about point O.

Step-by-step solution

The given data

The figure for the overhead view of a uniform stick on which four forces act is given.

Understanding the concept of torque

From the given condition for the net torque about axis O, we can conclude whether the stick is rotating or not. From this, we can determine whether the torques will balance or not at given points.

Formulae:

The value of the torque at equilibrium, ${\tau }_{net}=0$ (i)

a) Calculation to check whether the torques are balanced at point A

Since the torques about an axis at point O is balanced, the net torque is zero at point O. So, the stick is not rotating. It suggests that the net torque about any axis is zero according to the condition of equation (i).

Therefore the torques will balance about the rotation axis at points A, B and C.

Hence, the torques will be balanced on the stick at point A.

b) Calculation to check whether the torques are balanced at point B

From part (a), we can say that the torques will balance in line with the stick at point B.

c) Calculation to check whether the torques are balanced at point B

From part (a), we can say that the torques will balance off to one side of the stick at point C.

d) Calculation to check whether the torques will balance at any other point

Since torques about an axis at point O does not balance, the net torque is not zero at point O. So, the stick is rotating. It suggests that the net torque about any axis is not equal to zero, hence not in equilibrium according to the condition of equation (i).

Therefore the torques will not balance about the rotation axis at any point.