A solid cylinder of mass 10 kg and radius 20 cm is free to rotate about its axis. It receives an angular impulse of `4 kgm^(2)rad//s` . What is the angular speed of the cylinder if the cylinder is initially at rest ?

Options

A. 20 rad/s
B. 15 rad/s
C. 10 rad/s
D. 5 rad/s

Explanation

To solve the problem step by step, we will use the concepts of angular impulse and angular momentum.

### Step 1: Understand the given data
- Mass of the cylinder (m) = 10 kg
- Radius of the cylinder (r) = 20 cm = 0.2 m
- Angular impulse (J) = 4 kg m² rad/s
- Initial angular velocity (ω₀) = 0 rad/s (since the cylinder is initially at rest)

### Step 2: Relate angular impulse to change in angular momentum
Angular impulse (J) is defined as the change in angular momentum (ΔL):
\[ J = L_f - L_i \]
Where:
- \( L_f \) = final angular momentum
- \( L_i \) = initial angular momentum

Since the cylinder starts from rest, the initial angular momentum \( L_i = 0 \).
Thus, we have:
\[ J = L_f - 0 \]
\[ J = L_f \]

### Step 3: Express angular momentum in terms of moment of inertia and angular velocity
Angular momentum (L) can be expressed as:
\[ L = I \cdot \omega \]
Where:
- I = moment of inertia
- ω = angular velocity

Thus, we can rewrite the equation for angular impulse as:
\[ J = I \cdot \omega \]

### Step 4: Calculate the moment of inertia (I) for a solid cylinder
The moment of inertia for a solid cylinder rotating about its axis is given by:
\[ I = \frac{1}{2} m r^2 \]
Substituting the values:
\[ I = \frac{1}{2} \cdot 10 \, \text{kg} \cdot (0.2 \, \text{m})^2 \]
\[ I = \frac{1}{2} \cdot 10 \cdot 0.04 \]
\[ I = \frac{1}{2} \cdot 0.4 \]
\[ I = 0.2 \, \text{kg m}^2 \]

### Step 5: Substitute the values into the angular impulse equation
Now, substituting \( I \) and \( J \) into the equation:
\[ J = I \cdot \omega \]
\[ 4 = 0.2 \cdot \omega \]

### Step 6: Solve for angular velocity (ω)
To find the angular velocity, rearrange the equation:
\[ \omega = \frac{J}{I} \]
Substituting the values:
\[ \omega = \frac{4}{0.2} \]
\[ \omega = 20 \, \text{rad/s} \]

### Final Answer
The angular speed of the cylinder is \( \omega = 20 \, \text{rad/s} \).

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