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. `m_(1) r_(1) : m_(2) r_(2)`
B. `m_(1) : m_(2)`
C. `r_(1) : r_(2)`
D. `1 : 1`

Answer

To find the ratio of the centripetal acceleration of two cars moving in circular paths, we can follow these steps:

### Step-by-Step Solution:

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

2. **Relating Speed and Time**:
Since both cars make complete circles in the same time \( t \), we can express their speeds \( v_1 \) and \( v_2 \) in terms of the radius and time:
\[ v_1 = \frac{2\pi r_1}{t} \quad \text{and} \quad v_2 = \frac{2\pi r_2}{t} \]

3. **Calculating Centripetal Acceleration**:
Now we can calculate the centripetal accelerations \( a_1 \) and \( a_2 \) for each car:
\[ a_1 = \frac{v_1^2}{r_1} = \frac{\left(\frac{2\pi r_1}{t}\right)^2}{r_1} = \frac{4\pi^2 r_1}{t^2} \]
\[ a_2 = \frac{v_2^2}{r_2} = \frac{\left(\frac{2\pi r_2}{t}\right)^2}{r_2} = \frac{4\pi^2 r_2}{t^2} \]

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

5. **Conclusion**:
Therefore, the ratio of the centripetal accelerations of the two cars is:
\[ \frac{a_1}{a_2} = \frac{r_1}{r_2} \]

### Final Answer:
The ratio of their centripetal acceleration is \( \frac{r_1}{r_2} \).

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