A point starts from rest and moves along a circular path with a constant tangential acceleration. After one rotation, the ratio of its radial acceleration to its tangential acceleration will be equal to

Options

A. 1
B. `2pi`
C. `(1)/(2)pi`
D. `4pi`

Explanation

To solve the problem, we need to find the ratio of radial acceleration (\(a_r\)) to tangential acceleration (\(a_t\)) after one complete rotation of a point moving along a circular path with constant tangential acceleration. Here’s how we can derive the solution step by step:

### Step 1: Understanding the Motion
The point starts from rest and moves along a circular path with a constant tangential acceleration (\(a_t\)). Since it starts from rest, its initial velocity (\(u\)) is 0.

### Step 2: Finding the Final Velocity after One Rotation
To find the final velocity (\(v\)) after one complete rotation, we can use the equation of motion:
\[ v^2 = u^2 + 2a_t s \]
Here, \(s\) is the distance traveled in one complete rotation, which is the circumference of the circle:
\[ s = 2\pi r \]
Substituting \(u = 0\) and \(s = 2\pi r\) into the equation gives:
\[ v^2 = 0 + 2a_t (2\pi r) \]
\[ v^2 = 4\pi a_t r \]

### Step 3: Calculating Radial Acceleration
Radial acceleration (\(a_r\)) is given by the formula:
\[ a_r = \frac{v^2}{r} \]
Substituting the expression for \(v^2\) from Step 2:
\[ a_r = \frac{4\pi a_t r}{r} = 4\pi a_t \]

### Step 4: Finding the Ratio of Radial to Tangential Acceleration
Now, we can find the ratio of radial acceleration to tangential acceleration:
\[ \frac{a_r}{a_t} = \frac{4\pi a_t}{a_t} = 4\pi \]

### Conclusion
Thus, the ratio of radial acceleration to tangential acceleration after one complete rotation is:
\[ \frac{a_r}{a_t} = 4\pi \]

### Final Answer
The ratio of radial acceleration to tangential acceleration is \(4\pi\).

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