Question

In Fig.$\mathbf{\text{10-23}}$ , two forces${\overrightarrow{\mathbf{F}}}_1$ and ${\overrightarrow{\mathbf{F}}}_1$act on a disk that turns about its center like a merry-go-round. The forces maintain the indicated angles during the rotation, which is counter clockwiseand at a constant rate. However, we are to decrease the angle $\mathbf{\text{θ}}$of${\overrightarrow{\mathbf{F}}}_1$ without changing the magnitude of${\overrightarrow{\mathbf{F}}}_1$ . (a) To keep the angular speed constant, should we increase, decrease, or maintain the magnitude of ${\overrightarrow{\mathbf{F}}}_2$? Do forces (b)${\overrightarrow{\mathbf{F}}}_1$ and (c)${\overrightarrow{\mathbf{F}}}_2$ tend to rotate the disk clockwise or counter clockwise?

Short answer

1. To maintain the constant angular speed, the force${\mathrm{F}}_2$ should be decreased.
2. ${\mathrm{F}}_1$tends to rotate the disk clockwise.
3. ${\mathrm{F}}_2$tends to rotate the disk counter clockwise.

Step-by-step solution

Step 1: Given information 

Figure 10-23, with forces acting on the disk at different locations with different angles.

Understanding the concept 

In the rotational motion, the pull that causes the rotation is called torque. The torque is equal to the moment of force. The torque is equal to the cross product of the radius vector from the axis of rotation to the point of application of the force.

It is from the given figure and directions of forces that determine the magnitude of the torque caused by each force and accordingly answer the questions.

Formulae are as follows:

$\mathrm{\tau }=\mathrm{r}\times \mathrm{F}$

Where, r is radius, F is force and τ is torque.

(a) Determining if the angle of force F1  is decreased, should the value of F2  remain the same or increase or decrease in order to maintain the constant angular speed? 

If the angle is decreased, the value of the net torque would decrease. This is because a component of force${\mathrm{F}}_1$perpendicular to the vector joining the point of application of force with the axis of rotation would decrease as the angle decreases. To maintain the constant angular speed, make sure that the net torque on the disk is zero.

Therefore, the magnitude of the force ${\mathrm{F}}_2$ should be decreased to balance the net torque.

(b) Determining does force F1  tend to rotate the disk clockwise or counter clockwise?

Since the component of the force ${\mathrm{F}}_1$ perpendicular to the vector joining the point of application of force with the axis of rotation is directed in the clockwise direction, therefore, force ${\mathrm{F}}_1$ would tend to rotate the disk in the clockwise direction.

(c) Determining does force F2  tend to rotate the disk clockwise or counterclockwise?

Since the force${\mathrm{F}}_2$perpendicular to the vector joining the point of application of force with the axis of rotation is directed in the counter clockwise direction, therefore, the force${\mathrm{F}}_2$would tend to rotate the disk in the counter clockwise direction.

Therefore, using the given diagram of a rotating disk, theoretically, it can be determined how the force should change to maintain constant angular velocity and which direction the force would cause the disk to rotate.