A uniform solid sphere of radius R_0 and mass m_1 starts at rest at height H and

Question

A uniform solid sphere of radius R_0 and mass m_1 starts at rest at height H and rolls down a hill without slipping. When it reaches the bottom it continues rolling on a horizontal friction less surface. Express the rotational kinetic energy in terms of H, m_1, Ro, the gravitational acceleration g, and the velocity v of the sphere at any moment of the motion from the top of the hill to the frictionless horizontal. Not all quantities may be needed. Express the velocity at the bottom of the hill in terms of the same known quantity listed in) On the horizontal surface, the sphere hits centrally and elastically a second sphara (at rest) of radius R_0 and mass m_2=1/2 m_1. What is the velocity of the second sphere after the collision? What is the rotational kinetic energy of the second sphere after the collision In terms of a and H?

Explanation / Answer

I of sphere = 2mr^2/5

initial pe = m1 *gH
at any point on the incline, at hieght h

final pe = mgh
final ke = initial pe - final pe = m1*g(H -h) = 0.5mv^2 + 0.5(2mr^2)(v^2)/5*r^2 = 3.5mv^2
v^2 = g(H-h)/3.5

1. rotational ke = m1*g(H-h) - 0.5mv^2
2. v^2 = gH/3.5
3. m1*v = m1u + 0.5m1 u'
elastic collision, v = (u' - u)
u = u' - v
so, m1*v = m1u' - m1v + 0.5m1 u'
m1*v = 0.5m1u' - m1v
u' = 4v

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