Twirling Skater

After using one of the methods described in Part A to determine the acceleration, we can use kinematics to find the speed. The most direct approach is to use:

where we choose x_i = 0 m. Solving gives:

we choose the plus sign, since the ball is translating in the + x direction. Since the y-velocity is zero, the total speed of the ball's center of mass is:

Another way to solve the problem is to use energy. It turns out that in this problem, the mechanical energy of the ball will be constant. This assertion requires justification. The forces present in the system are gravity, normal force and friction. Gravity is a conservative force. The normal force is non-conservative, but it does no work because it is perpendicular to the motion of the object. Friction, however, is both non-conservative and directed anti-parallel to the motion of the ball, and so it should clearly do work. The reason we can assume the energy is constant is the problem's statement that the ball rolls without slipping. This means that the friction is static rather than kinetic. Kinetic friction converts mechanical energy into thermal energy and so it is not appropriate to use conservation of mechanical energy when kinetic friction is present. The work done by static friction, however, does not convert mechanical energy into thermal energy. Instead, the static friction acts to divert some of the lost potential energy into rotational kinetic energy (rather than simply translational kinetic energy). Thus, all of the energy remains in a mechanical form.

With this realization, we can write the equation of mechanical energy conservation in the form:

If we select h = 0 at the point of release of the ball, then by the time the ball has moved a distance x along the ramp, it has reached a height:

Substituting zeros and appropriate expressions into the conservation of energy formula gives:

Finally, the assumption that the ball is rolling without slipping implies the relationship:

so: