A wheel starts from rest and attains an angular velocity of 20 radian/s after being uniformly accelerated for 10 s. The total angle in radian through which it has turned in 10 second is

Options

A. `20 pi`
B. `40 pi`
C. 100
D. `100 pi`

Answer

To solve the problem of finding the total angle in radians through which the wheel has turned in 10 seconds, we can follow these steps:

### Step 1: Identify the given information
- Initial angular velocity (\( \omega_0 \)) = 0 rad/s (starts from rest)
- Final angular velocity (\( \omega \)) = 20 rad/s
- Time (\( t \)) = 10 s

### Step 2: Use the first equation of rotational motion to find angular acceleration (\( \alpha \))
The first equation of motion for rotational motion is:
\[ \omega = \omega_0 + \alpha t \]
Substituting the known values:
\[ 20 = 0 + \alpha \cdot 10 \]
Solving for \( \alpha \):
\[ \alpha = \frac{20}{10} = 2 \text{ rad/s}^2 \]

### Step 3: Use the second equation of rotational motion to find the angle (\( \theta \))
The second equation of motion for rotational motion is:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
Substituting the known values:
\[ 20^2 = 0^2 + 2 \cdot 2 \cdot \theta \]
This simplifies to:
\[ 400 = 4\theta \]
Solving for \( \theta \):
\[ \theta = \frac{400}{4} = 100 \text{ radians} \]

### Final Answer
The total angle through which the wheel has turned in 10 seconds is \( \theta = 100 \) radians.

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