A wheel having radius 10cm is coupled by a belt to another wheel of radius 30cm. `1^(st)` wheel increases its angular speed from rest at a uniform rate of `1.57 rads^(-2)`. The time for `2^(nd)` wheel to reach a rotational speed of 100 rev/min is....(assume that the belt does not slip)

Options

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

Answer

To solve the problem step by step, we will follow the given information and apply the relevant physics concepts.

### Step 1: Identify the given values
- Radius of the first wheel (r1) = 10 cm = 0.1 m
- Radius of the second wheel (r2) = 30 cm = 0.3 m
- Angular acceleration of the first wheel (α1) = 1.57 rad/s²
- Final angular speed of the second wheel (ω2) = 100 rev/min

### Step 2: Convert the final angular speed of the second wheel to radians per second
To convert revolutions per minute (rev/min) to radians per second (rad/s):
\[ \omega_2 = 100 \text{ rev/min} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{100 \times 2\pi}{60} = \frac{100\pi}{30} = \frac{10\pi}{3} \text{ rad/s} \]

### Step 3: Relate the angular accelerations of the two wheels
Since the wheels are coupled by a belt and there is no slipping, the relationship between the angular accelerations can be established as follows:
\[ \alpha_2 = \alpha_1 \cdot \frac{r_1}{r_2} \]
Substituting the known values:
\[ \alpha_2 = 1.57 \cdot \frac{0.1}{0.3} = 1.57 \cdot \frac{1}{3} = \frac{1.57}{3} \text{ rad/s}^2 \]

### Step 4: Use the kinematic equation to find the time for the second wheel to reach the desired angular speed
The kinematic equation we will use is:
\[ \omega = \omega_0 + \alpha t \]
Where:
- \(\omega\) = final angular speed (ω2)
- \(\omega_0\) = initial angular speed (0, since it starts from rest)
- \(\alpha\) = angular acceleration (α2)
- \(t\) = time

Substituting the known values:
\[ \frac{10\pi}{3} = 0 + \left(\frac{1.57}{3}\right) t \]
Solving for \(t\):
\[ t = \frac{10\pi/3}{1.57/3} = \frac{10\pi}{1.57} \]
Calculating \(t\):
\[ t \approx \frac{10 \times 3.14}{1.57} \approx \frac{31.4}{1.57} \approx 20 \text{ seconds} \]

### Final Answer
The time for the second wheel to reach a rotational speed of 100 rev/min is approximately **20 seconds**.

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