Question

A particle moves in a circle of radius $5 \,cm$ with constant speed and time period $0.2 \,\pi\, s$. The acceleration of the particle is

• A. $15 \,m/s^2 $
• B. $25 \,m/s^2 $
• C. $36 \,m/s^2 $
• D. $5 \, m/s^2 $

Answer 1

The Correct Option is D

To determine the acceleration of a particle moving in a circle with constant speed, we need to use the formula for centripetal acceleration. The centripetal acceleration \(a_c\) for a particle moving in a circle of radius \(r\) with speed \(v\) is given by:

a_c = \frac{v^2}{r}

Alternatively, since the particle repeats its motion every time period \(T\), we can also express centripetal acceleration in terms of the radius \(r\) and the time period \(T\) as:

a_c = \frac{4\pi^2 r}{T^2}

Given:

• Radius of the circle, \(r = 5 \, \text{cm} = 0.05 \, \text{m}\)
• Time period, \(T = 0.2\pi \, \text{s}\)

Substitute the given values into the formula:

a_c = \frac{4\pi^2 \times 0.05}{(0.2\pi)^2}

Calculate the denominator:

(0.2\pi)^2 = 0.04\pi^2

Now substitute back:

a_c = \frac{4\pi^2 \times 0.05}{0.04\pi^2}

Cancel \(\pi^2\) and simplify:

a_c = \frac{4 \times 0.05}{0.04} = \frac{0.2}{0.04} = 5 \, \text{m/s}^2

Thus, the acceleration of the particle is 5 \, \text{m/s}^2.

Therefore, the correct answer is 5 \, m/s^2, which matches with one of the provided answer choices.