A body is moving in a circular path with acceleration a. If its speed gets doubled, find the ratio of centripetal acceleration after and before the speed is changed

Options

A. `1:4`
B. `1:2`
C. `2:1`
D. `4:1`

Explanation

To solve the problem step by step, we need to determine the ratio of centripetal acceleration before and after the speed of the body is doubled.

### Step 1: Understand the formula for centripetal acceleration
Centripetal acceleration (\( a_c \)) is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \( v \) is the speed of the body and \( r \) is the radius of the circular path.

### Step 2: Write the expression for initial centripetal acceleration
Let the initial speed of the body be \( v_i \). The initial centripetal acceleration (\( a_{c_{initial}} \)) can be expressed as:
\[ a_{c_{initial}} = \frac{v_i^2}{r} \]

### Step 3: Write the expression for final centripetal acceleration
According to the problem, the speed of the body is doubled. Therefore, the final speed (\( v_f \)) is:
\[ v_f = 2v_i \]
Now, we can express the final centripetal acceleration (\( a_{c_{final}} \)):
\[ a_{c_{final}} = \frac{v_f^2}{r} = \frac{(2v_i)^2}{r} = \frac{4v_i^2}{r} \]

### Step 4: Find the ratio of final to initial centripetal acceleration
Now we can find the ratio of the final centripetal acceleration to the initial centripetal acceleration:
\[ \text{Ratio} = \frac{a_{c_{final}}}{a_{c_{initial}}} = \frac{\frac{4v_i^2}{r}}{\frac{v_i^2}{r}} \]

### Step 5: Simplify the ratio
Since \( r \) and \( v_i^2 \) are common in both the numerator and the denominator, they cancel out:
\[ \text{Ratio} = \frac{4v_i^2}{v_i^2} = 4 \]

### Conclusion
Thus, the ratio of centripetal acceleration after the speed is changed to that before the speed is changed is:
\[ \text{Ratio} = 4:1 \]