The linear velocity of a particular moving in a circle of radius 30 mathrm(~cm) is 20 mathrm(~km) / mathrm(hr) . If the particle rotates for 3.5 mathrm(sec) , find it, (a) Angular Velocity (b) Angular acceleration (c) angular displacement (d) arc length

(a) Angular Velocity:<br />Angular velocity is defined as the rate of change of angular displacement. It can be calculated using the formula:<br /><br />Angular Velocity (\( \omega \)) = Linear Velocity (\( v \)) / Radius (\( r \))<br /><br />Given that the linear velocity is \( 20 \mathrm{~km/hr} \) and the radius is \( 30 \mathrm{~cm} \), we need to convert the units of linear velocity to \( \mathrm{cm/s} \) before calculating the angular velocity.<br /><br />\( 20 \mathrm{~km/hr} = 20 \times \frac{1000}{3600} \mathrm{~m/s} = \frac{20000}{3600} \mathrm{~m/s} = \frac{20000}{3600} \times 100 \mathrm{~cm/s} = \frac{20000}{36} \mathrm{~cm/s} \)<br /><br />Now, we can calculate the angular velocity:<br /><br />\( \omega = \frac{v}{r} = \frac{\frac{20000}{36} \mathrm{~cm/s}}{30 \mathrm{~cm}} = \frac{20000}{1080} \mathrm{~rad/s} \)<br /><br />Therefore, the angular velocity is \( \frac{20000}{1080} \mathrm{~rad/s} \).<br /><br />(b) Angular acceleration:<br />Angular acceleration is the rate of change of angular velocity. It can be calculated using the formula:<br /><br />Angular Acceleration (\( \alpha \)) = Change in Angular Velocity (\( \Delta \omega \)) / Time (\( t \))<br /><br />Since the particle is rotating for \( 3.5 \mathrm{~sec} \), we need to calculate the change in angular velocity first.<br /><br />\( \Delta \omega = \omega - \omega_0 \)<br /><br />Given that the initial angular velocity (\( \omega_0 \)) is 0 (since the particle starts from rest), the change in angular velocity is equal to the angular velocity.<br /><br />\( \Delta \omega = \omega = \frac{20000}{1080} \mathrm{~rad/s} \)<br /><br />Now, we can calculate the angular acceleration:<br /><br />\( \alpha = \frac{\Delta \omega}{t} = \frac{\frac{20000}{1080} \mathrm{~rad/s}}{3.5 \mathrm{~s}} = \frac{20000}{1080 \times 3.5} \mathrm{~rad/s^2} \)<br /><br />Therefore, the angular acceleration is \( \frac{20000}{1080 \times 3.5} \mathrm{~rad/s^2} \).<br /><br />(c) Angular displacement:<br />Angular displacement is the product of angular velocity and time. It can be calculated using the formula:<br /><br />Angular Displacement (\( \theta \)) = Angular Velocity (\( \omega \)) × Time (\( t \))<br /><br />Given that the angular velocity is \( \frac{20000}{1080} \mathrm{~rad/s} \) and the time is \( 3.5 \mathrm{~s} \), we can calculate the angular displacement:<br /><br />\( \theta = \omega \times t = \frac{20000}{1080} \mathrm{~rad/s} \times 3.5 \mathrm{~s} = \frac{70000}{1080} \mathrm{~rad} \)<br /><br />Therefore, the angular displacement is \( \frac{70000}{1080} \mathrm{~rad} \).<br /><br />(d) Arc length:<br />Arc length is the product of radius and angular displacement. It can be calculated using the formula:<br /><br />Arc Length (\( s \)) = Radius (\( r \)) × Angular Displacement (\( \theta \))<br /><br />Given that the radius is \( 30 \mathrm{~cm} \) and the angular displacement is \( \frac{70000}{1080} \mathrm{~rad} \), we can calculate the arc length:<br /><br />\( s = r \times \theta = 30 \mathrm{~cm} \times \frac{70000}{1080} \mathrm{~rad} = \frac{2100000}{1080} \mathrm{~cm} \)<br /><br />Therefore, the arc length is \( \frac{2100000}{1080} \mathrm{~cm} \).