Question

A solid sphere and a hollow sphere have the same mass and radius. The two spheres are spun with matching angular velocities. Which statement is true? I. The hollow has the greater angular momentum II. The solid sphere has the greater angular momentum. III. The angular momentum is the same for both spheres. IV. The moment of inertia is the same for both spheres.

Answer 1

Answer:

I. The hollow sphere has the greater angular momentum.

Explanation:

Given that the mass and radius of hollow sphere and solid sphere are same. Let the mass and radius of two spheres be m and r respectively. The two spheres are rotating having same angular velocity ω .

Moment of inertia of solid sphere, I₁ = $\frac{2}{5}\times{m}r^{2}$

Moment of inertia of hollow sphere, I₂ = $\frac{2}{3}\times{m}r^{2}$

Since, I₁ and I₂ are not equal. Therefore, the statement iv is wrong.

The relation between angular momentum, moment of inertia and angular velocity is :

L = Iω

Let L₁ and L₂ be the angular momentum of solid sphere and hollow sphere respectively.

L₁ = I₁ω     and   L₂ = I₂ω

As ω is same for both spheres but I₂ is greater than I₁, hence L₂ is greater than L₁.

Therefore, statement I is correct that the hollow sphere has the greater angular momentum.

Answer 2

The angular momentum of the hollow sphere is greater than that of solid sphere.

The moment of inertia of solid sphere is given as follows;

$I_{ss} = \frac{2}{5} mr^2 = 0.4mr^2$

The moment of inertia of hollow sphere is given as follows;

$I_{hs} = \frac{2}{3} mr^2 = 0.67 mr^2$

The angular momentum of each sphere is calculated as follows;

$L = I\omega \\\\L_{ss} = 0.4mr^2 \omega \\\\L_{hs} = 0.67 mr^2 \omega$

Thus, we can conclude that the angular momentum of the hollow sphere is greater than that of solid sphere.

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