An ice skater spins at (4\pi )rad/s with her arms extended. If her moment of inertia with arms folded is (80\%) of that with her arms extended, what is her angular velocity when she folded her arms?
Discussion:
Conservation of angular momentum gives $$ I_1\omega_1=I_2 \omega_2$$
$$ (I_1) (4\pi)=0.8 I_1 \omega_2
$$
Hence, (\omega_2=5\pi)
A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to $$\theta(t)=\gamma t+\beta t^3$$ where (\gamma=0.4rad/s) and (\beta=0.0120 rad/s^3). What is the initial value of the angular velocity?
Discussion:
The angle through which the merry-go-round has turned varies with time according to $$\theta(t)=\gamma t+\beta t^3$$ where (\gamma=0.4rad/s) and (\beta=0.0120 rad/s^3).
$$ \omega=\frac{d\theta}{dt}$$
At (t=0) $$ \omega=\gamma=0.4 rad/s$$
Try this problem based on Angular Velocity and Acceleration, useful for Physics Olympiad.
The Problem:
A fan blade rotates with angular velocity given by $$ \omega=\gamma-\beta t^2$$ where (\gamma=5)rad/s and (\beta=0.800)rad/s. Calculate the angular acceleration as a function of time.
Solution:
The angular acceleration is given by $$\alpha=\frac{d\omega}{dt}=-2Bt=(-1.60)t $$
The unit of angular acceleration will be (rad/s^3).
Try this problem, useful for Physics Olympiad, based on the propeller's angular velocity.
The Problem:
An airplane propeller is rotating at (1900)rpm (rev/min).
(a)Compute the propeller's angular velocity in rad/s.
(b) How many seconds does it take for the propeller to run through (35^\circ)?
Solution:
An airplane propeller is rotating at (1900)rpm (rev/min).
(1)rpm = (2\pi /60)$$ \omega=(1900)(2\pi /60)=199 $$Hence, the propeller's angular velocity (\omega)=(199)rad/s.
b) (35^\circ)(\pi/180^\circ)=(0.611)rad.
Since angular velocity (\omega)=199rad/s, the time required for the propeller to run through (35^\circ)=$$ \frac{0.611}{199}=3.1\times10^{-3}s$$