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?
Look up the moment of inertia for a disk, and the sphere.
flywheel(disk)=Ifly*w
sphere=Isphere*ws
set them equal, solve for ws
put radii in meters...
To determine the rotational momentum of the flywheel, we can use the formula for rotational momentum, which is given by:
Rotational momentum (L) = mass (m) * angular velocity (ω)
Given that the mass of the flywheel is 10 kg and the rotational speed is 320 rad/s, we can calculate the rotational momentum as follows:
L = m * ω
= 10 kg * 320 rad/s
= 3200 kg·m²/s
Therefore, the rotational momentum of the flywheel is 3200 kg·m²/s.
Now, let's find the magnitude of the rotational speed that a 10-kg solid sphere must have to have the same rotational momentum as the flywheel.
The moment of inertia (I) for a solid sphere is given by the formula:
I = (2/5) * m * r²
Where m is the mass of the sphere and r is its radius.
The rotational momentum of the sphere can be calculated using the formula mentioned earlier, L = I * ω.
Setting the rotational momenta of the flywheel and the sphere equal to each other, we can solve for the magnitude of the rotational speed (ω) of the sphere:
L_flywheel = L_sphere
m_flywheel * ω_flywheel = I_sphere * ω_sphere
10 kg * 320 rad/s = (2/5) * 10 kg * (0.09 m)² * ω_sphere
3200 kg·m²/s = (2/5) * (0.09 m)² * ω_sphere
Simplifying the equation, we can solve for ω_sphere:
ω_sphere = (5 * 3200 kg·m²/s) / ((2/5) * (0.09 m)²)
ω_sphere = 71111.1 rad/s
Therefore, the magnitude of the rotational speed that the 10-kg solid sphere must have to have the same rotational momentum as the flywheel is approximately 71111.1 rad/s.