Question

Two racing cars of masses ${{m}_{1}},{{m}_{2}}$ are moving in circles of radii ${{r}_{1}}$, ${{r}_{2}}$​ respectively. Their speeds are such that each makes a complete circle in the same duration of time t. The ratio of the angular speed of the first to the second car is
$\begin{align}
  & a){{m}_{1}}:{{m}_{2}} \\
 & b){{r}_{1}}:{{r}_{2}} \\
 & c)1:1 \\
 & d)none \\
\end{align}$

Answer 1

Hint: Let us find out the angular speed of the cars. As the time taken by the cars to complete one circle is the same, we must write the angular speed in terms of time taken by the cars. Next, find the ratio between the angular speed and we can get the ratio.
Formula used:
$\omega =\dfrac{2\pi }{t}$

Complete answer:
Let us find the relation between the angular speed and the time taken by the cars to complete one complete circle. Speed depends on the distance travelled and the time taken. The angular speed depends on only the time taken because the distance is equal to $2\pi $.
\[\begin{align}
  & {{\omega }_{1}}=\dfrac{2\pi }{t} \\
 & {{\omega }_{2}}=\dfrac{2\pi }{t} \\
\end{align}\]
As the time taken is same to complete one circle,
Ratio between the singular velocities,
${{\omega }_{1}}:{{\omega }_{2}}=1:1$

So, the correct answer is “Option C”.

Additional Information:
Centripetal acceleration, property of the motion of a body transferred in a circular path is always directed radially towards the centre of the circle and has a magnitude equal to the square of the body speed along the curve divided by the distance from the centre of the circle. The force causing this acceleration is directed towards the centre of the circle and named centripetal force. Acceleration is a change in velocity either in its magnitude or in his direction or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. The sharper the curve and the greater the airspeed, the more noticeable this acceleration will be. The direction of centripetal acceleration is towards the centre of curvature always.

Note:
The angular velocity in this case only depends on the time taken by the cars to complete one circle because the radius will get cancelled while solving the angular velocity using linear velocity. As the radius gets cancelled, only time will be present in the formula of angular velocity.