A uniform sphere with know mass m, moment of inertia I, and radius Rs is traveling through free space with initial horizontal linear velocity v1 and rotational velocity w1. It then makes tangential contact with a horizontal surface and instantaneously starts rolling without slipping. What is the rotation/linear velocity the instant after it starts rolling?
Assume positive linear velocity is to the left, and positive rotation is counter-clockwise.
EDIT: assume a gravitational force sufficient enough to induce a frictional force to cause no sliding.
Fx = m*ax
ax = (v2-v1)/t
F*R = I*alpha
alpha = (w2-w1)/t
impulse = F*t
*during no slip rolling*
vx = w*Rs
The Attempt at a Solution
I would try to solve this problem by trying to apply a horizontal impulse to the sphere, changing the linear velocity and the angular velocity, until the new linear velocity matched the linear velocity induced by rolling (vx = ws*Rs) therefore:
(1) F = m*ax => (F*t)/m = v2-v1
(2) F*Rs = Irolling*alpha => (F*t)*Rs/Irolling = w2-w1
(3) v2 = w2*Rs
This is 3 equations and three unknows (v2, w2, F*t). Solving this we get:
*only here for you to check my math, actual reading not necessary*[/B]
substituting (3) into (1):
(F*t)/m = w2*Rs-v1 => (F*t) = (w2*Rs-v1)*m (equation: 4)
substituting (4) into (2):
(w2*Rs-v1)*m*Rs/Irolling = w2-w1 => and I'm not going to finish typing that because it takes a long time and the equation formatter is buggy. I realize that that I need to do a substitution of Irolling = I + m*Rs2 (parallel axis theorem because while the sphere is on the ground it is rotating about its edge, not its center of mass)
My main question is: is this the correct way to attack the problem? (apply a impulse until the linear and rotational speeds match). This concept will later be applied to a much more complex system (dynamics simulation). Is it still valid to apply this to a system with complex internal forces, but who's total mass is m, total linear velocity is v1, effective radius is R, and total moment of inertia is I?