Question

a car is going round a circle of radius R1 with a constant speed another car is going around a circle of radius R2 with constant speed if both of them take same time to complete the circles the ratio of their angular speed and linear speed will be

Answer 1

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Answer 2

The ratio of linear velocities of two cars is$\bold{R_{1}^{2} : R_{2}^{2}}$

SOLUTION:

For car travelling in a circle of radius $R_{1}$

The angular speed ω will be $\omega=\frac{\text {Total angular distance}}{\text {time taken}}$

According to the question the time taken by both the cars are same so let the take taken by both car be “t”

So for the first car $w_{1}=\frac{2 \pi R_{1}}{t}$

For linear velocity of first car, from the relation v=r \omega

We have,

$v_{2}=R_{1} \times \frac{2 \pi R_{1}}{t}=\frac{2 \pi R_{1}^{2}}{t}$

Similarly for the second car travelling around circle of radius R_{2}.

The angular speed will be $\omega_{2}=\frac{2 \pi R_{2}}{t}$

The linear velocity of this car will be $v_{2}=R_{2} \times \frac{2 \pi R_{2}}{t}=\frac{2 \pi R_{2}^{2}}{t}.$

So, the ratio of linear velocities of these two cars is $v_{1} : v_{2}=\frac{2 \pi R_{1}^{2}}{t} : \frac{2 \pi R_{2}^{2}}{t}=R_{1}^{2} : R_{2}^{2}$