A body rotating with uniform angular acceleration covers `100 pi` (radian) in the first 5 s after the start. Its angular speed at the end of 5 s (in rad/s) is

Options

A. `40 pi`
B. `30 pi`
C. `20 pi`
D. `10 pi`

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Identify the given values
- Angular displacement (θ) = \(100\pi\) radians
- Initial angular speed (\(ω_0\)) = 0 rad/s (the body starts from rest)
- Time (t) = 5 seconds

### Step 2: Use the second equation of motion for rotational motion
The second equation of motion for rotational motion is:
\[ \theta = ω_0 t + \frac{1}{2} \alpha t^2 \]
Since the initial angular speed \(ω_0\) is 0, the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

### Step 3: Rearrange the equation to solve for angular acceleration (α)
Rearranging the equation gives:
\[ \alpha = \frac{2\theta}{t^2} \]

### Step 4: Substitute the known values into the equation
Substituting the values we have:
\[ \alpha = \frac{2 \times 100\pi}{5^2} \]
Calculating \(5^2\):
\[ 5^2 = 25 \]
Now substituting this back into the equation:
\[ \alpha = \frac{200\pi}{25} = 8\pi \, \text{rad/s}^2 \]

### Step 5: Use the angular acceleration to find the final angular speed (ω)
The formula to find the final angular speed is:
\[ ω = ω_0 + \alpha t \]
Substituting the known values:
\[ ω = 0 + (8\pi)(5) \]
Calculating this gives:
\[ ω = 40\pi \, \text{rad/s} \]

### Final Answer
The angular speed at the end of 5 seconds is:
\[ ω = 40\pi \, \text{rad/s} \]

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