A flywheel in the form of a heavy circular

disk of diameter 0.83 m and mass 111 kg is
mounted on a frictionless bearing. A motor
connected to the flywheel accelerates it from
rest to 1430 rev/min.
a)What is the moment of inertia of the fly-
wheel?
Answer in units of kg · m2

b) After 1430 rev/min is achieved, the motor is
disengaged. A friction brake is used to slow
the rotational rate to 834 rev/min.
What is the magnitude of the energy dissi-
pated as heat from the friction brake?
Answer in units of J

a) To find the moment of inertia of the flywheel, we can use the formula:

I = (1/2) * m * r^2

where I is the moment of inertia, m is the mass of the flywheel, and r is the radius of the flywheel (half the diameter).

Given:
Mass of the flywheel (m) = 111 kg
Diameter of the flywheel = 0.83 m

First, we need to find the radius (r) by dividing the diameter by 2:
r = 0.83 m / 2 = 0.415 m

Now we can substitute the values into the formula to find the moment of inertia:
I = (1/2) * 111 kg * (0.415 m)^2

Calculating this equation will give us the moment of inertia of the flywheel in kg · m^2.

b) To find the magnitude of the energy dissipated as heat from the friction brake, we can use the principle of conservation of energy. The initial kinetic energy of the flywheel is equal to the energy dissipated as heat plus the final kinetic energy of the flywheel.

Given:
Initial angular velocity (ω1) = 1430 rev/min
Final angular velocity (ω2) = 834 rev/min

We need to convert the angular velocities from revolutions per minute to radians per second since the formula for kinetic energy uses angular velocity in radians per second.

Convert ω1 to radians per second:
ω1 = (1430 rev/min) * (2π rad/rev) * (1 min/60 s)

Convert ω2 to radians per second:
ω2 = (834 rev/min) * (2π rad/rev) * (1 min/60 s)

Now we can use the formula for rotational kinetic energy:

KE = (1/2) * I * ω^2

where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.

We already calculated the moment of inertia (I) in part a). Substitute the values of I, ω1, and ω2 into the formula to find the kinetic energies.

The energy dissipated as heat will be the difference between the initial and final kinetic energies.

Calculating this equation will give us the magnitude of the energy dissipated as heat in joules (J).