A solid sphere of with a mass of 2.0 kg starts at rest at the top of a ramp. It

Question

A solid sphere of with a mass of 2.0 kg starts at rest at the top of a ramp. It then begins to roll, without slipping, to the bottom of the ramp. The sphere has a radius of 10 cm. The ramp has a height of 1.0 m and an angle of 25*. What is the velocity of the sphere when it is at the bottom of the ramp? What angle has the sphere rotated through to get to the bottom of the ramp?

The picture shows: on the left 1.0 m up top is the sphere and in the middle is 25*. I need to find where I wrote the question mark, which is the bottom of the ramp.

Explanation / Answer

Given data : Height of the ramp, h = 1 m Mass of the solid sphere, m = 2 kg Radius of the solid sphere, r = 10 cm                                            = 0.1 m Angle of inclined plane, = 25 deg Solution: Length of the inclined plane, L = h / sin                                                = 1 / sin 25                                                = 2.37 m Moment of inertia of solid sphere, I = (2/5) m r^2 (a) Velocity, v = { 2 g L sin / [ 1 + ( I / m r^2 ) ] }                  = { 2 * 9.8 * 2.37 * sin 25 / [ 1 + (2/5) ] }                  = ( 19.6 / 1.4 )                  = 3.74 m/s      v = 3.74 m/s (b) No. of rev = L / 2r                  = 2.37 / ( 2 * 0.1 )                  = 3.77 Angle rotated = 3.77 * 2 rad                       = 23.7 rad       = 23.7 rad       = 23.7 rad

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