Calculating the Work and Energy for Spinning a Grindstone

To find the work, we can use the equation $\text{net}W=\left(\text{net τ}\right)\theta$. We have enough information to calculate the torque and are given the rotation angle. In the second part, we can find the final angular velocity using one of the kinematic relationships. In the last part, we can calculate the rotational kinetic energy from its expression in ${\text{KE}}_{\text{rot}}=\frac{1}{2}{\mathrm{I\omega }}^2$.

The net work is expressed in the equation

where net $\tau$ is the applied force multiplied by the radius $(\text{rF})$ because there is no retarding friction, and the force is perpendicular to $r$. The angle $\theta$ is given. Substituting the given values in the equation above yields

Noting that $\mathrm{1\; N}·\text{m}=\mathrm{1\; J}$,

Figure 10.17 A large grindstone is given a spin by a person grasping its outer edge.

To find $\omega$ from the given information requires more than one step. We start with the kinematic relationship in the equation

Note that ${\omega }_0=0$ because we start from rest. Taking the square root of the resulting equation gives

Now we need to find $\alpha$. One possibility is

where the torque is

The formula for the moment of inertia for a disk is found in Figure 10.12:

Substituting the values of torque and moment of inertia into the expression for $\alpha$, we obtain

Now, substitute this value and the given value for $\theta$ into the above expression for $\omega$:

The final rotational kinetic energy is

Both $I$ and $\omega$ were found above. Thus,

The final rotational kinetic energy equals the work done by the torque, which confirms that the work done went into rotational kinetic energy. We could, in fact, have used an expression for energy instead of a kinematic relation to solve part (b). We will do this in later examples.