A turn table which is rotating uniformly has a particle placed on it. As seen from the ground, the particle goes in a circle with speed 20 `cm//s` and acceleration 20 `cm//s^2`. The particle is now shifted to a new position where radius is half of the original value . the new values of speed and acceleration will be

Options

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

Answer

To solve the problem step by step, we will analyze the situation using the formulas for circular motion.

### Step 1: Understand the initial conditions
The particle is initially moving in a circle with:
- Speed (V1) = 20 cm/s
- Acceleration (A1) = 20 cm/s²

The relationship between speed, radius (r), and angular velocity (ω) in circular motion is given by:
\[ V = r \cdot \omega \]
And the centripetal acceleration is given by:
\[ A = r \cdot \omega^2 \]

### Step 2: Relate speed and acceleration to angular velocity
From the given data:
1. \( V1 = r \cdot \omega \)
2. \( A1 = r \cdot \omega^2 \)

### Step 3: Find the angular velocity (ω)
We can express ω in terms of V1 and A1:
From the speed equation:
\[ \omega = \frac{V1}{r} \]
From the acceleration equation:
\[ \omega^2 = \frac{A1}{r} \]

### Step 4: Substitute values to find ω
Using the values:
1. From \( V1 = 20 \, \text{cm/s} \) and \( A1 = 20 \, \text{cm/s}^2 \):
- We can express \( \omega \) as:
\[ \omega = \frac{20}{r} \]

2. Substitute ω into the acceleration equation:
\[ A1 = r \cdot \left(\frac{20}{r}\right)^2 \]
\[ 20 = r \cdot \frac{400}{r^2} \]
\[ 20r = 400 \]
\[ r = 20 \, \text{cm} \]

### Step 5: New radius
The new radius (r2) is half of the original radius:
\[ r2 = \frac{r}{2} = \frac{20}{2} = 10 \, \text{cm} \]

### Step 6: Calculate new speed (V2)
Using the relationship \( V = r \cdot \omega \):
\[ V2 = r2 \cdot \omega \]
Since ω remains constant:
\[ V2 = \frac{r}{2} \cdot \omega = \frac{20}{2} \cdot \omega = \frac{20}{2} \cdot \frac{20}{20} = 10 \, \text{cm/s} \]

### Step 7: Calculate new acceleration (A2)
Using the relationship \( A = r \cdot \omega^2 \):
\[ A2 = r2 \cdot \omega^2 \]
\[ A2 = \frac{r}{2} \cdot \omega^2 = \frac{20}{2} \cdot \frac{20}{20} = 10 \, \text{cm/s}^2 \]

### Final Results
- New speed (V2) = 10 cm/s
- New acceleration (A2) = 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