To determine the angular velocity of a particle in circular motion that performs 30 oscillations in 6 seconds, we need to follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understanding Oscillations and Angular Velocity 1. **Definition of Angular Velocity**: Angular velocity (ω) is defined as the rate of change of angular displacement with respect to time. It is usually measured in radians per second (rad/s). 2. **Relation Between Oscillations and Angular Displacement**: One complete oscillation corresponds to one full rotation around a circle, which is equivalent to an angular displacement of \(2\pi\) radians. ### Step 2: Calculate Total Angular Displacement Given that the particle performs 30 oscillations in 6 seconds, we can calculate the total angular displacement: - **Total Angular Displacement (θ)**: \[ \theta = \text{Number of oscillations} \times \text{Angular displacement per oscillation} \] \[ \theta = 30 \times 2\pi = 60\pi \text{ radians} \] ### Step 3: Calculate Angular Velocity Now, we can calculate the angular velocity using the formula: \[ \omega = \frac{\theta}{t} \] where: - \( \theta \) is the total angular displacement, - \( t \) is the time taken. Substituting the values we have: \[ \omega = \frac{60\pi \text{ radians}}{6 \text{ seconds}} = 10\pi \text{ rad/s} \] ### Conclusion: Correct Option Thus, the angular velocity of the particle is: \[ \omega = 10\pi \text{ rad/s} \] This corresponds to option **D**. ### Step 4: Explanation of Other Options Now, let's analyze the other options to understand why they are incorrect or weaker: - **Option A: 5 rad/s** - This value is too low. It does not account for the total angular displacement of \(60\pi\) radians over 6 seconds. - **Option B: 6 rad/s** - Similar to option A, this value does not reflect the correct total angular displacement. It suggests a much slower motion than what is calculated. - **Option C: 5π rad/s** - This value is also incorrect. It suggests an angular velocity that is less than what we calculated. Specifically, \(5\pi\) rad/s would imply a total angular displacement of \(30\pi\) radians in 6 seconds, which is not consistent with the 30 oscillations performed. ### Common Pitfalls - **Misunderstanding Oscillations**: Students may confuse oscillations with rotations. Remember that each oscillation corresponds to a full rotation (2π radians). - **Forgetting to Convert**: Always ensure that you convert the number of oscillations into radians correctly. - **Time Units**: Ensure that the time is in seconds when calculating angular velocity. ### Revision Summary - Angular velocity (ω) is calculated as the total angular displacement divided by time. - Each oscillation corresponds to \(2\pi\) radians; thus, 30 oscillations equal \(60\pi\) radians. - The formula for angular velocity is \(ω = \frac{\theta}{t}\). - The correct answer for the given problem is \(10\pi\) rad/s, corresponding to option D.