Rolling straight?
Intuitively, it is fairly clear that the sphere cannot change direction, but it is a little tricky to prove. Qualitatively, we can reason as follows. Assume that there is a nonzero friction force at the contact point. (The normal force is irrelevant here, because it doesn't provide a torque about the center, because the contact point is always directly below the center. This is what is special about a sphere.) Then the sphere accelerates in the direction of this force. However, you can show with the right-hand rule that this force produces a torque that causes the angular momentum to change in a way that corresponds to the sphere accelerating in the direction opposite to the friction force (assuming that the sphere is rolling without slipping). There is thus a contradiction, unless the friction force is equal to zero.
Let's now be rigorous. Let the angular velocity of the sphere be ω, which may point diagonally if the contact points don't form a vertical great circle on the sphere. The nonslipping condition says that the sphere's CM velocity equals
v = ω× (Rzˆ). (9.199)
The sphere's angular momentum is
L = Iω. (9.200)
The friction force (if it exists) from the ground at the contact point changes both the momentum and the angular momentum. F = dp/dt gives
F = m dv/dt. (9.201)
And τ = dL/dt (relative to the center) gives
(−Rzˆ) × F = dL/dt, (9.202)
because the force is applied at position −Rzˆ relative to the center. We will now show that the preceding four equations imply that ω˙ = 0. From Eq. (9.199), this then implies that v˙ = 0, as desired.
Equations (9.199) and (9.201) give F = mω˙ × (Rzˆ). Plugging this F, along with the L from Eq. (9.200), into Eq. (9.202) then gives
(−Rzˆ) × [mω˙ × (Rzˆ)] = Iω˙. (9.203)
9.10 Solutions 453
The vector ω˙ must lie in the horizontal plane, because it equals the cross product of two vectors, one of which is the vertical zˆ. This implies that (as you can verify) zˆ × (ω˙ × zˆ) = ω˙. Therefore, Eq. (9.203) gives
−mR²ω˙ = 2/5 mR²ω˙, (9.204)
and so ω˙ = 0, as we wanted to show.