Question

A particle moves with a constant velocity of \(5\ \text{m/s}\) in a circular path of radius \(2\ \text{m}\); calculate its centripetal acceleration.

• A. \(5\ \text{m/s}^2\)
• B. \(10\ \text{m/s}^2\)
• C. \(12.5\ \text{m/s}^2\)
• D. \(25\ \text{m/s}^2\)

Hint:

Centripetal acceleration can be found directly from speed and radius, or by first finding the angular velocity $\omega = v/r$ and then using $a_c = \omega^2 r$. Both routes must give the same result, so use whichever set of values you are confident squaring correctly.

Answer 1

The Correct Option is C

Concept: For circular motion, the centripetal acceleration is given by \[ a_c = \frac{v^2}{r} \] where \(v\) = velocity of the particle \(r\) = radius of the circular path.

Step 1: Substitute the given values. \[ v = 5\ \text{m/s}, \qquad r = 2\ \text{m} \] \[ a_c = \frac{5^2}{2} \]

Step 2: Calculate the acceleration. \[ a_c = \frac{25}{2} = 12.5\ \text{m/s}^2 \] Thus, the centripetal acceleration is \[ \boxed{12.5\ \text{m/s}^2} \]

Answer 2

Concept:

• Centripetal acceleration can also be written in terms of angular velocity: $a_c = \omega^2 r$, where $\omega$ measures how fast the angle swept by the particle changes with time.
• Linear speed and angular velocity are connected by $v = \omega r$, so this route must give the same numerical result as the direct formula.

Step 1: Find the angular velocity of the particle.
$\omega = \dfrac{v}{r} = \dfrac{5}{2} = 2.5\ \text{rad/s}$

Step 2: Apply the angular form of the centripetal acceleration formula.
$a_c = \omega^2 r$

Step 3: Substitute the values and calculate.
$a_c = (2.5)^2\times 2 = 6.25\times 2 = 12.5\ \text{m/s}^2$

Final Answer: $12.5\ \text{m/s}^2$