Uniform solid sphere of mass m and radius r rotates with an angular speed w about an axis through is center a uniform solid cylinder of mass m radius r, and length 2r rotates through an axis running through the central axis of the cylinder what must be the angular speed of the cylinder so it will have the same rotational kinetic energy the sphere?

a. 2 omega/5
b. squareroot 2/5 omega
c. 4 omega/5
d. omega/5
e. 2omega/squareroot 5

e. 2ωs / √5

Explanation:

The rotational kinetic energy of any rigid body, like an extension of the translational kinetic energy, is defined as follows:

Krot = 1/2 *I * ω²

For a solid sphere of mass M and radius R, the moment of inertia regarding any axis through its center, is as follows:

I =2/5 M*R²⇒ Krot(sp) = 1/2 (2/5 M*R²)*ωs² (1)

For a solid cylinder, rotating through an axis running through the central axis of the cylinder, the moment of inertia can be calculated as follows:

I = 1/2 M*R² ⇒ Krot(c) = 1/2 (1/2*M*R²)*ωc² (2)

As both rotational kinetic energies must be equal each other, we can equate (1) and (2), as follows:

1/2 (2/5 M*R²)*ωs² = 1/2 (1/2*M*R²)*ωc²

Simplifying common terms, and solving for ωc, we have:

ωc = 2*ωs /√5

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