A particle is kept fixed on as turntable rotating uniformly. As seen from the ground the particle goes in a circle,its speed is `20 cm//s` and acceleration is `20 cm//s^2` The particle is now shifted to a new positon to make the radius half of the original value. The new values of the speed and acceleration will be

Options

A. `10 cm//s, 10 cm//s^2`
B. `10 cm//s, 80 cm//s^2`
C. `40 cm//s, 10 cm//s^2`
D. `40 cm//s, 40 cm//s^2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the initial conditions
We are given:
- Initial speed \( V_1 = 20 \, \text{cm/s} \)
- Initial acceleration \( A_1 = 20 \, \text{cm/s}^2 \)

### Step 2: Relate speed and acceleration to radius
The centripetal acceleration \( A \) is related to speed \( V \) and radius \( R \) by the formula:
\[ A = \frac{V^2}{R} \]
We can rearrange this to find the radius \( R_1 \):
\[ R_1 = \frac{V^2}{A} \]

### Step 3: Calculate the initial radius
Substituting the known values:
\[ R_1 = \frac{(20 \, \text{cm/s})^2}{20 \, \text{cm/s}^2} = \frac{400 \, \text{cm}^2/\text{s}^2}{20 \, \text{cm/s}^2} = 20 \, \text{cm} \]

### Step 4: Determine the new radius
The problem states that the new radius \( R_2 \) is half of the original radius:
\[ R_2 = \frac{R_1}{2} = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Step 5: Find the angular velocity
The angular velocity \( \omega \) is constant as the turntable rotates uniformly. We can find \( \omega \) using the initial conditions:
\[ \omega = \frac{V_1}{R_1} = \frac{20 \, \text{cm/s}}{20 \, \text{cm}} = 1 \, \text{rad/s} \]

### Step 6: Calculate the new speed
Using the new radius \( R_2 \) to find the new speed \( V_2 \):
\[ V_2 = \omega \times R_2 = 1 \, \text{rad/s} \times 10 \, \text{cm} = 10 \, \text{cm/s} \]

### Step 7: Calculate the new acceleration
Now we can find the new acceleration \( A_2 \) using the new speed \( V_2 \) and the new radius \( R_2 \):
\[ A_2 = \frac{V_2^2}{R_2} = \frac{(10 \, \text{cm/s})^2}{10 \, \text{cm}} = \frac{100 \, \text{cm}^2/\text{s}^2}{10 \, \text{cm}} = 10 \, \text{cm/s}^2 \]

### Final Results
- New speed \( V_2 = 10 \, \text{cm/s} \)
- New acceleration \( A_2 = 10 \, \text{cm/s}^2 \)

### Conclusion
The new values of speed and acceleration are both \( 10 \, \text{cm/s} \) and \( 10 \, \text{cm/s}^2 \), respectively.

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