# The tilted hemisphere.

Fig. 1. The tilted hemisphere. A solid hemisphere of radius R rests on a surface rough enough so it doesn't slip. The ramp tilt is φ.

1. Assuming that friction is sufficient that the hemisphere doesn't slip. What is the maximum angle of the tilted plane that can allow the hemisphere to rest on the plane in static equilibrium?

2. Resting at smaller angles of tilt, if the hemisphere is tweaked a bit it will vibrate for a brief while. What is the period of this small angle motion? How does this depend on the mass of the hemisphere or the angle of the plane?

3. A solid hemisphere rests on a level, flat surface, with its flat side down. It is given a forceful spin. It rises to spin on its edge a while, then settles down as it loses energy. In its final resting position is its flat face up, or is its flat face down? Before you answer, note that the hemisphere held at rest with flat side vertical will fall so it ends up at rest with flat side up. The student says, "Of course, the center of mass is not at the hemisphere's center, but is toward the curved side, so that side will fall." Does this appply to the case of the hemisphere spinning on its edge.

In all cases, show your reasoning and mathematical analysis.

### Hints.

1. The center of mass of a solid hemisphere of radius R is 3R/8 from the flat face. You can look this up, or find its derivation many places on the web.

At equilibrium on the tilted plane, the hemisphere's axis tilts from the vertical at the same angle the plane tilts to the horizontal.

2. At smaller inclination angles of the plane, the hemisphere can be in static equilibrium, its axis tilted at the same angle as the plane's inclination angle, At zero inclination the hemisphere can oscillate as a physical pendulum. It also does so on the tilted plane, but does the tilt change the oscillation period?

We can be confident that the period will not depend on the hemisphere's mass. But if you don't know why, perhaps you should start with the more fundamental question, why is the period of a simple pendulum independent of mass? Why do bodies of different mass in a vacuum fall equal distances in equal times? 