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A spherical boulder of mass 139 kg and radius 24 cm rolls without slipping down a hill 17 m high from rest.

(a) What is its angular momentum about its center when it is halfway down the hill? (Enter the magnitude in kg•m^2/s.)

__________ kg•m^2/s

(b) What is its angular momentum about its center when it is at the bottom? (Enter the magnitude in kg•m^2/s.)

_________ kg•m^2/s

Answer :

Final answer:

The angular momentum of the rolling sphere when it is halfway down the hill is approximately 65.38 kg·m^2/s. The angular momentum of the sphere remains the same when it reaches the bottom of the hill.

Explanation:

To find the angular momentum of the rolling sphere when it is halfway down the hill, we need to calculate its angular velocity at that point.

First, let's calculate the moment of inertia of the sphere:

Given mass of the sphere = 139 kg

Given radius of the sphere = 24 cm = 0.24 m

Moment of inertia of a solid sphere = (2/5) * mass * radius^2

Moment of inertia = (2/5) * 139 kg * (0.24 m)^2

Moment of inertia = 6.2976 kg·m^2

Next, let's calculate the sphere's angular velocity when it is halfway down the hill:

Given height of the hill = 17 m

At the halfway point, the potential energy of the sphere is converted to kinetic energy.

Using the conservation of energy principle:

Potential energy at the top = Kinetic energy at the halfway point

m * g * h = (1/2) * I * ω^2

139 kg * 9.8 m/s^2 * 17 m = (1/2) * 6.2976 kg·m^2 * ω^2

ω^2 = (2 * 139 kg * 9.8 m/s^2 * 17 m) / (6.2976 kg·m^2)

ω^2 = 107.876

ω = √107.876

ω ≈ 10.386 rad/s

Finally, we can calculate the angular momentum of the sphere when it is halfway down the hill:

Angular momentum = Moment of inertia * Angular velocity

Angular momentum = 6.2976 kg·m^2 * 10.386 rad/s

Angular momentum ≈ 65.38 kg·m^2/s

For part (b), when the sphere reaches the bottom of the hill, its angular momentum remains the same since there are no external torques acting on it. Therefore, the angular momentum at the bottom is also approximately 65.38 kg·m^2/s.

Learn more about angular momentum of a rolling sphere here:

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