Two bodies have their moments of inertia `I` and `2 I` respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio.

Options

A. `1/2`
B. `sqrt 2`
C. `1 : sqrt 2`
D. `2/1`

Explanation

To solve the problem, we need to find the ratio of the angular momenta of two bodies given their moments of inertia and equal kinetic energies of rotation.

### Step 1: Write the formula for kinetic energy
The kinetic energy (K) of a rotating body is given by the formula:
\[ K = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

### Step 2: Set up the equations for both bodies
Let the moments of inertia of the two bodies be:
- Body 1: \( I_1 = I \)
- Body 2: \( I_2 = 2I \)

Since the kinetic energies are equal, we can write:
\[ K_1 = K_2 \]
Thus,
\[ \frac{1}{2} I_1 \omega_1^2 = \frac{1}{2} I_2 \omega_2^2 \]

### Step 3: Simplify the equation
Substituting the values of \( I_1 \) and \( I_2 \):
\[ \frac{1}{2} I \omega_1^2 = \frac{1}{2} (2I) \omega_2^2 \]
Cancelling \( \frac{1}{2} \) and \( I \) (assuming \( I \neq 0 \)):
\[ \omega_1^2 = 2 \omega_2^2 \]

### Step 4: Find the ratio of angular velocities
Taking the square root of both sides gives:
\[ \frac{\omega_1}{\omega_2} = \sqrt{2} \]

### Step 5: Write the formula for angular momentum
The angular momentum (L) of a rotating body is given by:
\[ L = I \omega \]

### Step 6: Set up the equations for angular momentum
For both bodies, we have:
- Angular momentum of Body 1: \( L_1 = I_1 \omega_1 = I \omega_1 \)
- Angular momentum of Body 2: \( L_2 = I_2 \omega_2 = 2I \omega_2 \)

### Step 7: Find the ratio of angular momenta
Now, we can find the ratio:
\[ \frac{L_1}{L_2} = \frac{I \omega_1}{2I \omega_2} \]
Cancelling \( I \) (assuming \( I \neq 0 \)):
\[ \frac{L_1}{L_2} = \frac{\omega_1}{2 \omega_2} \]

### Step 8: Substitute the ratio of angular velocities
Substituting \( \frac{\omega_1}{\omega_2} = \sqrt{2} \):
\[ \frac{L_1}{L_2} = \frac{\sqrt{2}}{2} \]

### Step 9: Simplify the ratio
This simplifies to:
\[ \frac{L_1}{L_2} = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the ratio of the angular momenta \( L_1 : L_2 \) is:
\[ 1 : \sqrt{2} \]