Question

A particle moves in a circular path of radius 4 cm. It completes 5 revolutions in 2 minutes. Find its linear velocity.

Hint:

Always convert minutes into seconds before calculating frequency to ensure your final velocity unit is consistently in $\text{cm/s}$ or $\text{m/s}$.

Answer 1

Step 1: Understanding the Concept:
Linear velocity in uniform circular motion is the continuous distance traveled along the circumference of the circle divided by the elapsed time.
It directly relates to the angular frequency, which measures how rapidly the particle cycles through revolutions.
Step 2: Key Formula or Approach:
The fundamental relationship between linear velocity $v$, radius $r$, and angular velocity $\omega$ is given by $v = r \omega$.
The angular velocity $\omega$ can be calculated from the frequency of revolutions $f$ using the formula $\omega = 2\pi f$.
Step 3: Detailed Explanation:
First, carefully identify the given physical values from the problem statement.
Radius $r = 4\text{ cm}$.
The particle effectively completes $5\text{ revolutions}$ in a total time of $2\text{ minutes}$.
Convert the given time entirely into standard SI base units (seconds): $2\text{ minutes} = 120\text{ seconds}$.
Now, calculate the fundamental frequency $f$, which is the number of revolutions per second:
\[ f = \frac{5\text{ revolutions}}{120\text{ seconds}} = \frac{1}{24}\text{ Hz} \] Next, calculate the corresponding angular velocity $\omega$:
\[ \omega = 2\pi \left( \frac{1}{24} \right) = \frac{\pi}{12}\text{ rad/s} \] Finally, apply the linear velocity formula using the radius and angular velocity:
\[ v = 4 \times \left( \frac{\pi}{12} \right) \] \[ v = \frac{4\pi}{12} = \frac{\pi}{3}\text{ cm/s} \] Step 4: Final Answer:
The linear velocity of the particle is exactly $\frac{\pi}{3}\text{ cm/s}$.