A 10-kg disk-shaped flywheel of radius 9.0 cm rotates with a rotational speed of 320 rad/s.
Determine the rotational momentum of the flywheel.
With what magnitude rotational speed must a 10-kg solid sphere of 9.0 cm radius rotate to have the same rotational momentum as the flywheel?
To determine the rotational momentum of the flywheel, we need to use the formula:
Rotational momentum = Moment of inertia × Angular velocity
The moment of inertia of a disk-shaped flywheel can be calculated using the equation:
Moment of inertia (I) = (1/2) × mass × radius^2
Given:
Mass of the flywheel (m) = 10 kg
Radius of the flywheel (r) = 9.0 cm = 0.09 m
Angular velocity of the flywheel (ω) = 320 rad/s
First, let's calculate the moment of inertia of the flywheel:
I = (1/2) × m × r^2
I = (1/2) × 10 kg × (0.09 m)^2
I = 0.405 kg·m^2
Now we can calculate the rotational momentum:
Rotational momentum = I × ω
Rotational momentum = 0.405 kg·m^2 × 320 rad/s
Rotational momentum = 129.6 kg·m^2/s
To determine the magnitude of the rotational speed that the solid sphere must rotate with to have the same rotational momentum, we can rearrange the formula:
Rotational momentum = Moment of inertia × Angular velocity
Angular velocity = Rotational momentum / Moment of inertia
Given:
Rotational momentum = 129.6 kg·m^2/s
Mass of the sphere (m) = 10 kg
Radius of the sphere (r) = 9.0 cm = 0.09 m
We can apply the same formula to calculate the rotational speed for the sphere:
Moment of inertia (I) = (2/5) × m × r^2 (for a solid sphere)
I = (2/5) × 10 kg × (0.09 m)^2
I = 0.081 kg·m^2
Angular velocity = Rotational momentum / Moment of inertia
Angular velocity = 129.6 kg·m^2/s / 0.081 kg·m^2
Angular velocity = 1600 rad/s
Therefore, the magnitude of the rotational speed that the solid sphere must rotate with to have the same rotational momentum as the flywheel is 1600 rad/s.