I need help with my physics?

Question

i need to find a formula to find time, when a ball is roled down a ramp at various heights and lengths, i need to find the time it would take for that ball to reach the bottom of the ramp. So i guess the givens are height (or angle) of the ramp, length of the ramp, and i know the weight of the ball. I just need a formula to find the time it would take, given those variables if that makes any sense?

Answer 1

Hint since the ball is rolling:
I=(2/3) m r^2 if it is hollow
I=(2/5) m r^2 if it is a solid sphere

I - moment of inertia
m - mass
r - radius of the ball

Potential energy (Pe) on top equal kinetic energy (Ke) at the bottom

Pe= mgh
Ke= .5 mv^ (translational) or in our case it is also
Ke= 0.5 I w^2 (rotational)

The force F along the incline is in question.
The incline of height h, length L under an angle A is

h/L=sin(A)

then
F=mg sin(A)
If we have no friction that apposes the motion then
F=mg or
F=(1/r)T
T= I alpha
alpha= v^2/r

F=(1/r) I v^2/r
F=I (v/r)^2

a= F/m
v=at
t=v/a= v m/F
t= v m / I (v/r)^2
t=r^2 m / ( v I )
v=sqrt( 2 Pe/m)
v= sqrt (2 mgh/m)
v= sqrt(2 gh)

so finally
t= r^2 m / [ I sqrt(2 g h)]