Rotation in a Collision

We can answer the first question using conservation of angular momentum as noted. Because angular momentum is $\mathrm{I\omega }$, we can solve for angular velocity.

Conservation of angular momentum states

where primed quantities stand for conditions after the collision and both momenta are calculated relative to the pivot point. The initial angular momentum of the system of stick-disk is that of the disk just before it strikes the stick. That is,

where $I$ is the moment of inertia of the disk and $\omega$ is its angular velocity around the pivot point. Now, $I={\text{mr}}^{\mathrm{\text{2}}}$ (taking the disk to be approximately a point mass) and $\omega =v/r$, so that

After the collision,

It is $\omega \prime$ that we wish to find. Conservation of angular momentum gives

Rearranging the equation yields

where $I\prime$ is the moment of inertia of the stick and disk stuck together, which is the sum of their individual moments of inertia about the nail. Figure 10.12 gives the formula for a rod rotating around one end to be $I={\text{Mr}}^2/3$. Thus,

Entering known values in this equation yields,

The value of $I\prime$ is now entered into the expression for $\omega \prime$, which yields

The kinetic energy before the collision is the incoming disk’s translational kinetic energy, and after the collision, it is the rotational kinetic energy of the two stuck together.

First, we calculate the translational kinetic energy by entering given values for the mass and speed of the incoming disk.

After the collision, the rotational kinetic energy can be found because we now know the final angular velocity and the final moment of inertia. Thus, entering the values into the rotational kinetic energy equation gives

The linear momentum before the collision is that of the disk. After the collision, it is the sum of the disk’s momentum and that of the center of mass of the stick.

Before the collision, then, linear momentum is

After the collision, the disk and the stick’s center of mass move in the same direction. The total linear momentum is that of the disk moving at a new velocity $v\prime =\mathrm{r\omega }\prime$ plus that of the stick’s center of mass,

which moves at half this speed because $v_{\text{CM}}=\left(\frac{r}{2}\right)\omega \prime =\frac{v\prime }{2}$. Thus,

Gathering similar terms in the equation yields,

so that

Substituting known values into the equation,

First note that the kinetic energy is less after the collision, as predicted, because the collision is inelastic. More surprising is that the momentum after the collision is actually greater than before the collision. This result can be understood if you consider how the nail affects the stick and vice versa. Apparently, the stick pushes backward on the nail when first struck by the disk. The nail’s reaction (consistent with Newton’s third law) is to push forward on the stick, imparting momentum to it in the same direction in which the disk was initially moving, thereby increasing the momentum of the system.