A solid sphere of radius 1m and mass 10kg rolls down a hill with a height of 10m

Question

A solid sphere of radius 1m and mass 10kg rolls down a hill with a height of 10m. If friction does 100 J of work on the sphere as it rolls what is its: A.) linear velocity at the 2m mark? B.) angular velocity at the 2m mark? A solid sphere of radius 1m and mass 10kg rolls down a hill with a height of 10m. If friction does 100 J of work on the sphere as it rolls what is its: A.) linear velocity at the 2m mark? B.) angular velocity at the 2m mark? A.) linear velocity at the 2m mark? B.) angular velocity at the 2m mark?

Explanation / Answer

(A)

Apply conservation of energy to the sphere when it is at top of the inclined plane and when it is at 2 m mark

            mg(h2-h1) = 1/2 * m * v^2 + 1/2 * I * (v/r)^2

            mg(h2-h1) = 1/2 * m * v^2 + 1/2 * (2/5) mr^2 * (v/r)^2

            g(h2-h1) = 1/2 * v^2 + 1/2 * (2/5) * v^2

            2g(h2-h1) = v^2 + (2/5) * v^2

            2g(h2-h1) =   (7/5) * v^2

velociyt is v = sqrt( 10g (h1 - h2) / 7 - 100 J / m )

v = sqrt ( 10*9.8*(10 - 2) / 7 - 2*100 / 10) v = 9.6 m/s

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(B)

Angular velocity is w = v/ r = 9.6 m/s / 1 m = 9.6 rad/s

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