A particle with the constant speed in a circle of radius r and time period T.The centripetal acceleration of a particle is

Options

A. `‘r^2‘ T`
B. `‘T^2‘ r`
C. `(2π/T)^2r`
D. `(2πT)^2/r`

Answer

To find the centripetal acceleration of a particle moving in a circle with a constant speed, we can follow these steps:

### Step 1: Understand the Concept of Centripetal Acceleration
Centripetal acceleration is the acceleration that keeps a particle moving in a circular path. It is directed towards the center of the circle.

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

### Step 3: Relate Speed to Time Period
The speed \( v \) of the particle can be expressed in terms of the time period \( T \) and the radius \( r \). The distance traveled in one complete revolution (circumference of the circle) is:
\[ \text{Circumference} = 2\pi r \]
Since the particle completes this distance in one time period \( T \), we can express speed as:
\[ v = \frac{\text{Distance}}{\text{Time}} = \frac{2\pi r}{T} \]

### Step 4: Substitute Speed into the Centripetal Acceleration Formula
Now, substitute the expression for \( v \) into the centripetal acceleration formula:
\[ a_c = \frac{v^2}{r} = \frac{\left(\frac{2\pi r}{T}\right)^2}{r} \]

### Step 5: Simplify the Expression
Now simplify the expression:
\[ a_c = \frac{(2\pi r)^2}{T^2 \cdot r} = \frac{4\pi^2 r^2}{T^2 \cdot r} \]
This simplifies to:
\[ a_c = \frac{4\pi^2 r}{T^2} \]

### Final Answer
Thus, the centripetal acceleration of the particle is:
\[ a_c = \frac{4\pi^2 r}{T^2} \]

---