Question 20.12: Two wheels have the same mass M. One can be modelled by a la...
Two wheels have the same mass M. One can be modelled by a large hollow cylinder of radius R and the other by a smaller solid cylinder of radius r. When they are rotating, they are each subject to a frictional couple of constant magnitude C. While the wheels are being driven, there is a break in power which lasts for a time t. Assuming all units are compatible, find expressions for
i) the angular retardation of each wheel while the power is off
ii) the reduction in the angular velocity of each wheel during this time.
The wheels have the same initial angular velocity, and the radius of the larger is twice that of the smaller.
iii) Show that the percentage reduction in the angular velocity during the time the power is off is eight times greater for the smaller wheel.
iv) When the wheels are rotating with the same angular speed, show that the kinetic energy of the larger is eight times that of the smaller.
i) When a wheel has an angular acceleration \ddot{\theta } as a result of the action of a couple, -C, the equation of motion gives
– C = I\ddot{\theta }
⇒ \ddot{\theta } = \frac{- C}{I}
The moment of inertia of the hollow cylinder about its axis is MR².
Its acceleration is \frac{- C}{MR²}.
The moment of inertia of the solid cylinder about its axis is \frac{1}{2}Mr².
Its acceleration is \frac{- 2C}{Mr²}.
ii) For constant angular acceleration α, the new angular velocity, ω, of a wheel after t seconds is
ω = ω_{0} + αt.
So the reduction in the angular speed is
\begin{matrix} \omega _{0} – \omega = \frac{Ct}{I}. & \boxed{\alpha = \frac{-C}{I}} \end{matrix}
The reductions in the angular speeds of the wheels are
\frac{Ct}{Mr^{2}} and \frac{2Ct}{Mr^{2}}.
iii) When the initial angular speeds are the same, the percentage reductions in the
angular speeds are proportional to the actual reductions.
These are in the ratio (larger : smaller) of
\frac{Ct}{Mr^{2}} : \frac{2Ct}{Mr^{2}}.
= r² : 2R²
= r² : 2 × 4r² (R = 2r)
= 1 : 8.
The angular speed is reduced for both wheels, but the percentage reduction is 8
times greater for the smaller wheel.
iv) When it rotates with angular speed ω, the kinetic energy of a wheel is \frac{1}{2}Iω². The ratio of the kinetic energy of the larger wheel to that of the smaller is
\frac{1}{2}(MR^{2})ω² : \frac{1}{2}(\frac{1}{2}Mr^{2})ω²
= R^{2} : \frac{1}{2}r^{2}
= 4r^{2} : \frac{1}{2}r^{2}
= 8 : 1.
