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

52584533.33joules

a) Use the formula E = (1/2) I w^2, where w is the angular speed in rad/s and I is the moment of inertia.

(b) Time = (Full speed energy)/Power
You will have to convert horsepower to Watts when using the formula

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

Kinetic Energy (KE) = 0.5 * moment of inertia * angular velocity^2

(a) First, we need to 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 and r is the radius.

Given:
Mass of the flywheel (m) = 600 kg
Radius of the flywheel (r) = 1.00 m

Using these values, we can calculate the moment of inertia:

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

Now, we can find the angular velocity. The initial rotational speed of the flywheel is given as 4,000 rev/min. To convert this to radians per second, we use the following conversion:

1 rev = 2π radians

Angular velocity = (4000 rev/min) * (2π radians/1 rev) * (1 min/60 s)
Angular velocity = (4000 * 2π) / 60 radians/sec
Angular velocity = (800π) / 60 radians/sec
Angular velocity ≈ 41.8879 radians/sec

Using these values in the formula for kinetic energy, we get:

KE = 0.5 * 300 kg * m^2 * (41.8879 radians/sec)^2
KE ≈ 0.5 * 300 kg * m^2 * (1750.8302 radians^2/sec^2)
KE ≈ 263524155 J

So, the kinetic energy stored in the flywheel is approximately 263,524,155 J.

(b) To find the length of time the car could run before the flywheel would have to be brought back up to speed, we need to know the power requirement of the car.

Given that the flywheel is to supply energy to the car as a 20.0-hp motor would, we need to convert horsepower to watts since power is usually measured in watts. 1 horsepower (hp) is equal to 746 watts.

Power requirement = 20.0 hp * 746 watts/hp
Power requirement ≈ 14,920 watts

Now, to find the length of time the car could run, we can use the formula:

Time = KE / Power requirement

Using the value of KE calculated in part (a) and the power requirement, we have:

Time = 263524155 J / 14920 watts
Time ≈ 17,663 seconds

To convert this to hours, we divide by 3600 (since there are 3600 seconds in an hour):

Time ≈ 17,663 seconds / 3600 seconds/hour
Time ≈ 4.905 hours

So, the car could run for approximately 4.905 hours before the flywheel would have to be brought back up to speed.