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 step by step, we'll use the equations of rotational motion.

### Step 1: Identify the given values
- Initial angular velocity (\( \omega_i \)) = 0 rad/s (since the wheel starts from rest)
- Final angular velocity (\( \omega_f \)) = 20 rad/s
- Time (\( t \)) = 10 s

### Step 2: Calculate the angular acceleration (\( \alpha \))
We can use the formula relating final angular velocity, initial angular velocity, angular acceleration, and time:
\[ \omega_f = \omega_i + \alpha t \]
Substituting the known values:
\[ 20 = 0 + \alpha \cdot 10 \]
This simplifies to:
\[ \alpha = \frac{20}{10} = 2 \text{ rad/s}^2 \]

### Step 3: Calculate the total angle (\( \theta \)) turned in 10 seconds
We can use the equation for angular displacement:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ \theta = 0 \cdot 10 + \frac{1}{2} \cdot 2 \cdot (10)^2 \]
This simplifies to:
\[ \theta = 0 + \frac{1}{2} \cdot 2 \cdot 100 = 100 \text{ radians} \]

### Final Answer
The total angle through which the wheel has turned in 10 seconds is **100 radians**.

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