A wheel having radius 10 cm is coupled by a belt to another wheel of radius 30 cm . 1st wheel increases its angular speed from rest at a uniform rate of `1.5 "rads"^(-2)` . The time for 2nd wheel to reach a rotational speed of 100rev/min is ... (assume that the belt does not slip )

Options

A. 20 sec
B. 10 sec
C. `1.5` sec
D. 15sec

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given values
- Radius of the first wheel, \( r_1 = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Radius of the second wheel, \( r_2 = 30 \, \text{cm} = 0.3 \, \text{m} \)
- Angular acceleration of the first wheel, \( \alpha_1 = 1.5 \, \text{rad/s}^2 \)
- Final rotational speed of the second wheel, \( \omega_2 = 100 \, \text{rev/min} \)

### Step 2: Convert the final speed of the second wheel to radians per second
To convert revolutions per minute to radians per second, we use the conversion factor:
\[ \omega_2 = 100 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = \frac{100 \times 2\pi}{60} \, \text{rad/s} = \frac{10\pi}{3} \, \text{rad/s} \]

### Step 3: Relate the angular velocities of the two wheels
Since the belt does not slip, the linear velocities at the edges of both wheels must be equal:
\[ v_1 = v_2 \]
This can be expressed in terms of angular velocities:
\[ r_1 \omega_1 = r_2 \omega_2 \]
Where \( \omega_1 \) is the angular velocity of the first wheel. Rearranging gives:
\[ \omega_1 = \frac{r_2}{r_1} \omega_2 \]
Substituting the known values:
\[ \omega_1 = \frac{0.3}{0.1} \cdot \frac{10\pi}{3} = 3 \cdot \frac{10\pi}{3} = 10\pi \, \text{rad/s} \]

### Step 4: Use the angular acceleration to find the time taken
The angular velocity of the first wheel starts from rest, so we can use the equation:
\[ \omega_1 = \omega_{0_1} + \alpha_1 t \]
Where \( \omega_{0_1} = 0 \) (initial angular velocity). Thus:
\[ 10\pi = 0 + 1.5 t \]
Solving for \( t \):
\[ t = \frac{10\pi}{1.5} = \frac{20\pi}{3} \approx 20.94 \, \text{s} \]

### Step 5: Conclusion
The time taken for the second wheel to reach a rotational speed of 100 rev/min is approximately \( 20.94 \, \text{s} \).