Question

Figure 12-17 shows four overhead views of rotating uniform disks that are sliding across a frictionless floor. Three forces, of magnitude F, 2F, or 3F, act on each disk, either at the rim, at the center, or halfway between rim and center. The force vectors rotate along with the disks, and, in the “snapshots” of Fig. 12-17, point left or right. Which disks are in equilibrium?

Short answer

Disks a and c are in equilibrium.

Step-by-step solution

The given data

Forces $\overrightarrow{F}.2\overrightarrow{F}.3\overrightarrow{F}$ are acting at various points on the disks, either on the rim, at the center, or halfway between the rime and the center.

Understanding the concept of equilibrium

We can use the conditions for equilibrium for the net force and the net torque to determine which disk is in equilibrium.

Formulae:

At equilibrium, the net torque of the system is given by, $\overrightarrow{{\tau }_{net}}=0$ (i)

At equilibrium, the net force of the system is given by, $\overrightarrow{F_{net}}=0$ (ii)

The torque acting on a point, $\tau =r\times F$ (iii)

Calculation of finding the disk in equilibrium

a)

To check whether disk a is at equilibrium:

All the forces arein ahorizontal direction. Thus, the value of the net force can be given as:

$$\begin{gathered} \overrightarrow{F_{net}}=-F+3F-2F \\ =0 \end{gathered}$$

Thus, the net force on disk a is zero, which satisfies the equilibrium condition of equation (i).

Consider the pivot point at the center of the disk, so the angle between the moment arm and the forces is $90°$. The force 2F is acting at the center, so its moment arm and hence the torque is zero.

The net torque acting can be given using equation (iii) as follows:

$$\begin{gathered} {\overrightarrow{\tau }}_{net}=\left(r\right)\left(F\right)-\left(\frac{r}{2}\right)2F \\ =rF-rF \\ =0 \end{gathered}$$

Thus, net torque on disk a is zero, which satisfies the equilibrium condition of equation (ii).

As the forces, as well as torques, are balanced, disk a is in equilibrium.

b)

To check whether disk b is at equilibrium:

All the forces arein ahorizontal direction. Thus, the value of the net forces can be given as follows:

$$\begin{gathered} F_{net}=F+2F+2F \\ =4F \end{gathered}$$

Thus, the net force on disk b is not zero. Hence forces are not balanced, and this does not satisfy the value of equation (i).

Consider the pivot point at the center of the disk, so the angle between the moment arm and the forces is. The force is acting at the center, so its moment arm and hence the torque is zero.

The net torque by forces F at the top and F at the bottom will be given using equation (iii) as follows:

$$\begin{gathered} {\overrightarrow{\tau }}_{net}=\left(r\right)\left(F\right)-\left(r\right)\left(F\right) \\ =0 \end{gathered}$$

Thus, net torque on disk b is zero, which satisfies the condition of equation (ii).

Though torques are balanced, forces are not balanced; hence disk b is not in equilibrium.

c)

To check whether disk c is at equilibrium:

All the forces arein ahorizontal direction. Thus, the value of the net forces is given as follows:

$$\begin{gathered} F_{net}=F+F-2F \\ =0 \end{gathered}$$

Thus, the net force on disk c is zero, which satisfies the value of equation (i).

Consider the pivot point at the center of the disk, so the angle between the moment arm and the forces is. The force is acting at the center, so its moment arm and hence the torque is zero.

The net torque by forces F and 2F will be given using equation (iii) as follows:

$$\begin{gathered} {\overrightarrow{\tau }}_{net}=\left(r\right)\left(F\right)-\left(r\right)\left(F\right) \\ =0 \end{gathered}$$

Thus, net torque on disk c is zero, which satisfies the condition of equation (ii).

As the forces, as well as torques, are balanced, disk c is in equilibrium.

$$\begin{gathered} F_{net}=-F-F+2F \\ =0 \end{gathered}$$

Thus, the net force on disk d is zero. Hence forces are balanced, which satisfies the condition of equation (i).

Consider the pivot point at the center of the disk, so the angle between the moment arm and the forces is $90°$. The force directed towards the left acts at the center, so its moment arm and hence the torque is zero.

The net torque by forces F at the top and F at the bottom will be given using equation (iii) as follows:

$$\begin{gathered} {\overrightarrow{\tau }}_{net}=\left(r\right)\left(F\right)-\left(r\right)\left(2F\right) \\ =-rF \end{gathered}$$

Thus, net torque on disk d is not zero, which does not satisfy the condition of equation (ii).

Though forces are balanced, torques are not balanced, and hence disk d is not in equilibrium.

Hence, disks a and c are in equilibrium.