What is relationship between linear and angular momentum? I'm confused about imparting momentum to a sphere, which causes it to rotate without slipping. Does the momentum applied go:


*

*only to the rotational motion and the linear is a consequence of the rolling, 

*or does the momentum split between the two.
In essence: can you hit a ball so it starts out only turning but not moving forward. Is the equation $$p = I\omega$$ or do you add $$mv~?$$
 A: If a ball of say radius $R$ rolls without slipping it has both linear ($p$) and rotational momentum ($L$):
$$p=mv$$
$$L=I\omega$$
Where $m$ is mass of ball, $I$ is inertial moment of ball, $v$ is translational (linear) speed and $\omega$ is angular speed.
For rolling without slipping the following condition also holds:
$$v=\omega R$$
The ball will have a total kinetic energy $K$:
$$K=\frac12 mv^2+ \frac12 I\omega^2$$

So in essence can you hit a ball so it starts out only turning but not moving forward.

Yes, it's possible to make a ball spin on a surface without it translating ('moving forward'), provided there's no friction between ball and surface or if the axis of rotation is perpendicular to the surface (like a spinning top). If the axis of rotation is not perpendicular to the surface and there is friction, the friction force will exert a force on the ball that will cause it to start translating ('move linearly'). The angular speed $\omega$ will then decrease and the linear speed $v$ increase, until the condition $v=\omega R$ is met (rolling without slipping).
A: To hit a rigid body such that it rotates about a specified point, you need to hit it at the instant center of percussion. If the pivot point is a distance $c$ from the center of mass, then the percussion center is located a distance $$\ell = \frac{I_{cm}}{m c} = \frac{\kappa^2}{c} $$ away from the center of mass. Here, $I_{cm}$ is the mass moment of inertia about the center of mass, $m$ is the mass and $\kappa$ is the radius of gyration.
As a result, when the spin point is coincident with the center of mass, the percussion center is located at infinity. The only way to impart rotation about the center of mass would be to apply a pure torque on the body, with zero net force.
In reverse, an impact (or force) with moment arm $\ell$ from the center of mass would make the body rotate about a point away from the center of mass at a distance $$c = \frac{\kappa^2}{\ell}$$
Again, if the impact goes through the center of mass ($\ell=0$ for a sphere) then the rotation point is at infinity, or the motion is a pure translation.
