A solid sphere rolls without slipping or sliding along a horizontal surface towa

Question

A solid sphere rolls without slipping or sliding along a horizontal surface toward an incline. If the sphere's speed is 4.12 m/s at the base of the incline and the angle of inclination is 30 degrees.

A.) Calculate the kinetic energy of sphere as it rolls along the horizontal surface.

B.) Calculate the maximum vertical height of the sphere above the horizontal sphere when it reaches its maximum point on the incline.

C.) Calculate how far along the incline the sphere traveled before coming to a stop.

Explanation / Answer

Part A)

Since you left out the mass, I can only solve part A in terms of mass. Here it is...

We need to add up the rotational and translational KE's

KE = .5mv2 + .5Iw2

I for a solid sphere is 2/5mr2

w = v/r, so

KE = .5(2/5)mr2v2/r2 which simplifies to .2mv2

Thus the total KE = .5mv2 + .2mv2 = .7mv2

KE = (.7)(m)(4.12)2 = 11.9m Joules

Multiply 11.9 by the mass to get the energy in Joules.

Part B)
PE = KE

mgh = 11.9m (m cancels)

(9.8)(h) = 11.9

h = 1.21 m

Part C)

The distance along the plane is found from the sin function

sin(30) = h/d

d = h/sin(30)

d = (1.21)/sin 30

d = 2.42 m

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