Example

Question:

An insect trapped in a circular groove of radius \(12\,\mathrm{cm}\) moves along the groove steadily and completes 7 revolutions in \(100\,\mathrm{s}\).
(a) What is the angular speed and the linear speed of the motion?
(b) Is the acceleration vector a constant vector? What is its magnitude?

Solution:

This is uniform circular motion. Here \(R = 12\,\mathrm{cm}\).
The angular speed \(\omega\) is: \[ \omega = \frac{2\pi N}{T} = \frac{2\pi \times 7}{100} = 0.44\,\mathrm{rad/s} \] The linear speed \(v\) is: \[ v = \omega R = 0.44\,\mathrm{s^{-1}} \times 12\,\mathrm{cm} = 5.3\,\mathrm{cm/s} \] The direction of velocity \(\vec{v}\) is tangential to the circle at every point. The acceleration is directed towards the centre of the circle; since this direction changes continuously, acceleration is not a constant vector. However, its magnitude is constant: \[ a = \omega^2 R = (0.44\,\mathrm{s^{-1}})^2 \times 12\,\mathrm{cm} = 2.3\,\mathrm{cm/s^2} \]