Two cars of mass ` m_(1) and m_(2) ` are moving in circle of radii `r_(1) and r_(2) ` , respectively . Their speeds are such that they make complete circles in the same time `t` . The ratio of their centripetal acceleration is :

Options

A. (a) ` m _ 1 r _ 1 : m _ 2 r _ 2 `
B. (b)` m_1 : m_2 `
C. (c)`r_1 : r _ 2 `
D. (d)` 1: 1 `

Explanation

To find the ratio of the centripetal accelerations of two cars moving in circles of different radii but completing their circles in the same time, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Centripetal Acceleration**:
The formula for centripetal acceleration \( a \) is given by:
\[ a = \frac{v^2}{r} \]
where \( v \) is the linear speed and \( r \) is the radius of the circular path.

2. **Relating Speed and Time**:
Since both cars complete their circles in the same time \( t \), we can express their speeds in terms of the radius and the time taken to complete the circle. The speed \( v \) for a circular motion can be expressed as:
\[ v = \frac{2\pi r}{t} \]
where \( r \) is the radius of the circle.

3. **Calculating Speeds for Both Cars**:
For car 1 (with radius \( r_1 \)):
\[ v_1 = \frac{2\pi r_1}{t} \]
For car 2 (with radius \( r_2 \)):
\[ v_2 = \frac{2\pi r_2}{t} \]

4. **Finding Centripetal Accelerations**:
Now, we can find the centripetal accelerations for both cars using the formula:
- For car 1:
\[ a_1 = \frac{v_1^2}{r_1} = \frac{\left(\frac{2\pi r_1}{t}\right)^2}{r_1} = \frac{4\pi^2 r_1}{t^2} \]
- For car 2:
\[ a_2 = \frac{v_2^2}{r_2} = \frac{\left(\frac{2\pi r_2}{t}\right)^2}{r_2} = \frac{4\pi^2 r_2}{t^2} \]

5. **Finding the Ratio of Centripetal Accelerations**:
Now, we can find the ratio of the centripetal accelerations \( a_1 \) and \( a_2 \):
\[ \frac{a_1}{a_2} = \frac{\frac{4\pi^2 r_1}{t^2}}{\frac{4\pi^2 r_2}{t^2}} = \frac{r_1}{r_2} \]

6. **Final Result**:
Thus, the ratio of the centripetal accelerations of the two cars is:
\[ \frac{a_1}{a_2} = \frac{r_1}{r_2} \]

### Conclusion:
The ratio of the centripetal acceleration of the two cars is \( \frac{r_1}{r_2} \).