A chain couples and rotates two wheels in a bicycle . The radii of bigger and smaller wheels are 0.5 m and 0.1 m respectively . The bigger wheel rotates at the rate of 200 rotations per minute , then the rate of rotation of smaller wheel will be :-

Options

A. 1000 rpm
B. 50/3 rpm
C. 200 rpm
D. 40 rpm

Explanation

To solve the problem, we need to relate the rotational speeds of the two wheels using their radii. Here's a step-by-step solution:

### Step 1: Understand the relationship between the two wheels
The two wheels are connected by a chain, which means that the linear velocities at the points where the chain contacts each wheel must be equal. This can be expressed as:
\[ v_1 = v_2 \]
where \( v_1 \) is the linear velocity of the bigger wheel and \( v_2 \) is the linear velocity of the smaller wheel.

### Step 2: Express linear velocity in terms of angular velocity
The linear velocity \( v \) of a wheel can be expressed in terms of its angular velocity \( \omega \) and radius \( r \):
\[ v = \omega \cdot r \]
Thus, for the two wheels, we have:
\[ \omega_1 \cdot r_1 = \omega_2 \cdot r_2 \]

### Step 3: Substitute the known values
From the problem, we know:
- Radius of the bigger wheel, \( r_1 = 0.5 \) m
- Radius of the smaller wheel, \( r_2 = 0.1 \) m
- Angular velocity of the bigger wheel, \( \omega_1 = 200 \) revolutions per minute (rpm)

Now, we need to convert \( \omega_1 \) into radians per second for consistency in units:
\[ \omega_1 = 200 \, \text{rpm} = 200 \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} \]
\[ \omega_1 = \frac{200 \times 2\pi}{60} \, \text{radians/second} \]
\[ \omega_1 \approx 20.94 \, \text{radians/second} \]

### Step 4: Set up the equation
Now substitute the values into the equation:
\[ \omega_1 \cdot r_1 = \omega_2 \cdot r_2 \]
\[ (20.94) \cdot (0.5) = \omega_2 \cdot (0.1) \]

### Step 5: Solve for \( \omega_2 \)
Now, we can solve for \( \omega_2 \):
\[ \omega_2 = \frac{(20.94) \cdot (0.5)}{0.1} \]
\[ \omega_2 = \frac{10.47}{0.1} \]
\[ \omega_2 = 104.7 \, \text{radians/second} \]

### Step 6: Convert \( \omega_2 \) back to rpm
To convert \( \omega_2 \) back to revolutions per minute:
\[ \omega_2 = 104.7 \, \text{radians/second} \times \frac{1 \, \text{revolution}}{2\pi \, \text{radians}} \times 60 \, \text{seconds} \]
\[ \omega_2 \approx 1000 \, \text{rpm} \]

### Final Answer
The rate of rotation of the smaller wheel is approximately **1000 revolutions per minute (rpm)**.

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