A car is designed to get its energy from a rotating flywheel (solid disk) with a radius of 1.00 m and a mass of 600 kg. Before a trip, the flywheel is attached to an electric motor, which brings the flywheel's rotational speed up to 4,000 rev/min.

(a) Find the kinetic energy stored in the flywheel.
J

(b) If the flywheel is to supply energy to the car as a 20.0-hp motor would, find the length of time the car could run before the flywheel would have to be brought back up to speed.
h

To find the kinetic energy stored in the flywheel, we can use the formula for rotational kinetic energy:

Kinetic energy (KE) = 1/2 * I * ω^2

Where:
- KE is the kinetic energy
- I is the moment of inertia of the flywheel
- ω is the angular velocity of the flywheel

(a) Let's first find the moment of inertia of the flywheel. For a solid disk, the moment of inertia is given by:

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

Where:
- m is the mass of the flywheel
- r is the radius of the flywheel

Given:
- m = 600 kg
- r = 1.00 m

Plugging these values into the formula, we get:

I = (1/2) * 600 kg * (1.00 m)^2
I = 300 kg * m^2

Now, let's convert the rotational speed from revolutions per minute (rev/min) to radians per second (rad/s). Since 1 revolution is equal to 2π radians, we have:

ω = (4000 rev/min) * (2π rad/rev) * (1 min/60 s)
ω ≈ 418.88 rad/s

Finally, let's calculate the kinetic energy:

KE = 1/2 * (300 kg * m^2) * (418.88 rad/s)^2
KE ≈ 167,768 J

So, the kinetic energy stored in the flywheel is approximately 167,768 Joules.

(b) To find the length of time the car could run before the flywheel would have to be brought back up to speed, we can use the concept of power. Power is the rate at which energy is transferred or transformed.

Given that the flywheel is to supply energy to the car as a 20.0-hp (horsepower) motor would, we need to convert the power to watts since the SI unit of power is the watt (W). 1 horsepower is equal to approximately 746 watts.

Power (P) = 20.0 hp * 746 W/hp
P ≈ 14,920 W

We also know that power is equal to the energy transferred or transformed divided by time:

P = KE / t

Rearranging this equation, we can solve for time (t):

t = KE / P

Substituting the values we calculated earlier, we get:

t = 167,768 J / 14,920 W
t ≈ 11.25 seconds

So, the car could run for approximately 11.25 seconds before the flywheel would have to be brought back up to speed.