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 need to analyze the relationship between the speed, acceleration, and radius of the circular motion of the particle on the turntable.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial speed (v) = 20 cm/s
- Initial acceleration (a) = 20 cm/s²
- Let the initial radius (r) be R.

2. **Understand the Relationships:**
- The speed of a particle in circular motion is given by the formula:
\[ v = \omega \cdot r \]
where \( \omega \) is the angular velocity (which is constant in this case).
- The centripetal acceleration is given by:
\[ a = \frac{v^2}{r} = \omega^2 \cdot r \]

3. **Determine the Angular Velocity:**
- Since \( v = \omega \cdot r \), we can express \( \omega \) as:
\[ \omega = \frac{v}{r} \]
- Using the initial conditions, we can substitute \( v = 20 \) cm/s and \( a = 20 \) cm/s² to find \( R \):
\[ a = \frac{v^2}{R} \implies 20 = \frac{(20)^2}{R} \implies R = \frac{400}{20} = 20 \text{ cm} \]

4. **New Radius:**
- The new radius is half of the original radius:
\[ r' = \frac{R}{2} = \frac{20}{2} = 10 \text{ cm} \]

5. **Calculate New Speed:**
- Since the angular velocity \( \omega \) is constant, the new speed \( v' \) can be calculated as:
\[ v' = \omega \cdot r' = \omega \cdot \left(\frac{R}{2}\right) = \frac{v}{2} = \frac{20}{2} = 10 \text{ cm/s} \]

6. **Calculate New Acceleration:**
- The new acceleration \( a' \) can be calculated using the formula for centripetal acceleration:
\[ a' = \frac{(v')^2}{r'} = \frac{(10)^2}{10} = \frac{100}{10} = 10 \text{ cm/s}^2 \]

### Final Results:
- New speed \( v' = 10 \) cm/s
- New acceleration \( a' = 10 \) cm/s²

### Summary:
The new values of speed and acceleration when the radius is halved are:
- Speed = 10 cm/s
- Acceleration = 10 cm/s²

---