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**1. Homework Statement**

A ball radius r mass m starts rolling without slipping up a ramp inclinced at an angle phi to the horizontal, and reaches a maximum hight, h. Derive an expression for the angular velocity, omega, that the ball has at the base of the ramp (ignore rolling friction throughout question.).

b) The ball is initially launched without rotation towards the ramp on a horizontal surface with a coefficient of sliding friction, mu. The ball slides along the surface, begins to roll, and stops slipping before it reaches the ramp. Find an expression for the time, t, raken for it to stop sliding in terms of h, g and mu.

c) by considering the initial velocity v0 and the resistence felt by the ball before it begins to roll, derive an expression relating its initial and final energies in terms of the sliding distance s.

find x, v0, and t taking values h = 0.25m, r = 2cm, m = 50g and mu = 0.3

**2. Homework Equations**

When something undergoes pure rolling motion, v = rw

**3. The Attempt at a Solution**- w = omega u = mu

a) Energy at top = mgh. mgh = 1/2 * mv^2 + 1/2 Iw^2 . I know that as it is rolling, v = rw, and solve to get omega = (10gh/7(r^2))^(0.5) (using I = 2/5 ma^2 if a is the radius)

b) here's where it gets a little dodgy - I assume that there is a rotational equation even though it is sliding (i.e. that it is rotating a little bit but not pure rotation) - is this valid?

I(dw/dt) = umgr , integrate once w.r.t time to get w = t(5ug/2r) where the moment of inertia for a sphere has been used. I then equate this to my earlier figure from a), and get (8h/35gu^2)^(0.5) - is this right?

c) I say that initial energy = 0.5m(v0)^2 and this minus the final energy of slipping (equal to 1/2 * I * w^2 where w is the omega in a) ) = Fx where F is the resistive force during the slipping, and x is slipping distance.

I can then work out t pretty easily, and am not sure how to get X and v0.

Any help is very greatly appreciated,

Cheers

Cpfoxhunt

EDIT: i initially did this using I for a disk (where I is inertia), and have corrected it a bit hurriedly - some of my multiplying fractions might be a bit out?