The rotational kinetic energy of two bodies of moment of inertia `9 kg m^(2)` and `1kg m^(2)` are same . The ratio of their angular momenta is

Options

A. (a)`3:1`
B. (b)`1:3`
C. (c)`9:1`
D. (d)`1:9`

Answer

To solve the problem, we need to find the ratio of the angular momenta of two bodies given that their rotational kinetic energies are the same. Here’s a step-by-step solution:

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

### Step 2: Set up the equation for both bodies
Let the moment of inertia of the first body be \(I_1 = 9 \, \text{kg m}^2\) and for the second body \(I_2 = 1 \, \text{kg m}^2\). Since the kinetic energies are the same, we can write:
\[ \frac{1}{2} I_1 \omega_1^2 = \frac{1}{2} I_2 \omega_2^2 \]
The \(\frac{1}{2}\) cancels out from both sides, so we have:
\[ I_1 \omega_1^2 = I_2 \omega_2^2 \]

### Step 3: Substitute the values of \(I_1\) and \(I_2\)
Substituting the values of the moments of inertia:
\[ 9 \omega_1^2 = 1 \omega_2^2 \]

### Step 4: Rearranging to find the ratio of angular velocities
From the equation above, we can express the ratio of angular velocities:
\[ \frac{\omega_2^2}{\omega_1^2} = 9 \implies \frac{\omega_2}{\omega_1} = 3 \]

### Step 5: Relate angular momentum to angular velocity
The angular momentum \(L\) of a body is given by:
\[ L = I \omega \]
For the first body:
\[ L_1 = I_1 \omega_1 = 9 \omega_1 \]
For the second body:
\[ L_2 = I_2 \omega_2 = 1 \omega_2 \]

### Step 6: Find the ratio of angular momenta
Now, we can find the ratio of the angular momenta:
\[ \frac{L_1}{L_2} = \frac{9 \omega_1}{1 \omega_2} \]
Substituting \(\omega_2 = 3 \omega_1\) into the equation:
\[ \frac{L_1}{L_2} = \frac{9 \omega_1}{1 \cdot 3 \omega_1} = \frac{9}{3} = 3 \]

### Final Answer
Thus, the ratio of their angular momenta is:
\[ \frac{L_1}{L_2} = 3:1 \]

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