Question

A machine part rotates at an angular speed of ωi. Its speed is then increased to ωf at an angular acceleration of α. If both the initial and final angular speeds are now doubled and the angular acceleration remains the same, by what factor is the angular displacement changed?

Answer 1

Given data

Initial angular speed is ωi

Final angular speed is ωf

Angular acceleration is α.

The expression for the angular displacement of machine is

$\Delta\theta=\frac{((\omega_f)^2-(\omega_i)^2)}{(2\alpha)}$ ........(1)

When the both the initial and final angular speeds are now doubled that is ( ω'_i =2ω_i) and ( ω'_f=2ω_f), then the new angular displacement is

$\Delta\theta '=\frac{((\omega'_ f)^2-(\omega'_i)^2)}{(2\alpha)}$

$=(\frac{(2\omega_f)^2-(2\omega_i)^2)}{(2\alpha) } =\frac{4((\omega _f)^2-(\omega_i)^2))}{(2\alpha)}$ ........ (2)

From equations (1) and (2), we get

$\Delta\theta '=4 \Delta\theta$

Hence , the angular displacement is increased by the factor 4.