Question

A particle performing uniform circular motion of radius $\frac{\pi}{2} \text{ m}$ makes $x$ revolutions in time $t$. Its tangential velocity is

• A. $\frac{x}{\pi t}$
• B. $\frac{\pi^2}{x t}$
• C. $\frac{\pi^2 x}{t}$
• D. $\frac{\pi x}{t}$

Hint:

Distance in $x$ revolutions is $x(2\pi r)$. Velocity is total distance divided by time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Tangential velocity in circular motion relates to how fast the particle covers the circumference.
It can be calculated from angular velocity and radius.
Step 2: Key Formula or Approach:
Frequency $f = \frac{\text{Number of revolutions}}{\text{Time taken}} = \frac{x}{t}$.
Angular velocity $\omega = 2\pi f$.
Tangential velocity $v = r\omega$.
Step 3: Detailed Explanation:
Given radius $r = \frac{\pi}{2} \text{ m}$.
The particle makes $x$ revolutions in time $t$, so the frequency is $f = \frac{x}{t}$.
Calculate the angular velocity $\omega$: \[ \omega = 2\pi \left(\frac{x}{t}\right) = \frac{2\pi x}{t} \] Calculate the tangential velocity $v$: \[ v = r \cdot \omega \] Substitute the given radius and calculated angular velocity: \[ v = \left(\frac{\pi}{2}\right) \cdot \left(\frac{2\pi x}{t}\right) \] The $2$ in the numerator and denominator cancel out: \[ v = \frac{\pi \cdot \pi \cdot x}{t} \] \[ v = \frac{\pi^2 x}{t} \] Step 4: Final Answer:
The tangential velocity is $\frac{\pi^2 x}{t}$.