In a city with an air-pollution problem, a bus has no combustion engine. It runs on energy drawn from a large, rapidly rotating flywheel under the floor of the bus. The flywheel is spun up to its maximum rotation rate of 5150 rev/min by an electric motor at the bus terminal. Every time the bus speeds up, the flywheel slows down slightly. The bus is equipped with regenerative braking so that the flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1600 kg and radius 0.650 m. The bus body does work against air resistance and rolling resistance at the average rate of 18.0 hp as it travels with an average speed of 40.0 km/h. How far can the bus travel before the flywheel has to be spun up to speed again?

It will travel until the initial kinetic energy of the flywheel equals the work that has to be done to keep the bus moving for a time t.

(1/2)Iw^2 = (Power)*time

Power = 18.0 hp = 13,428 J/second
You don't need to know or use the average speed.

Sngular velocity w = 5150 rev/min*(2 pi rad/rev)/(60 s/min)
= 539.3 rad/s

I = (1/2)M*R^2 = 338 kg*m^2

Solve for time

To find how far the bus can travel before the flywheel needs to be spun up again, we need to calculate the energy lost by the flywheel due to the bus's work against air resistance and rolling resistance.

Step 1: Convert the average power to watts.
1 horsepower (hp) = 745.7 watts
18.0 hp * 745.7 W/hp = 13423.6 watts

Step 2: Calculate the total energy lost by the flywheel.
The total energy lost by the flywheel is given by the product of power and time.
Energy lost = Power * time

We need to find the time taken for the flywheel to slow down to a speed requiring recharging.

Step 3: Calculate the initial kinetic energy of the flywheel.
The initial kinetic energy of the flywheel is given by the formula:
KE = (1/2) * I * ω^2
where I is the moment of inertia and ω is the angular velocity in radians per second.

Given: Mass (m) = 1600 kg, Radius (r) = 0.650 m, Angular velocity (ω) = 5150 rev/min

Let's convert ω from rev/min to rad/s.
1 rev = 2π radians, and 1 min = 60 seconds
ω = (5150 rev/min) * (2π radians/rev) * (1 min/60 s)

Step 4: Calculate the moment of inertia (I) of the flywheel.
The moment of inertia of a solid cylinder is given by the formula:
I = (1/2) * m * r^2

Substituting the given values:
I = (1/2) * (1600 kg) * (0.650 m)^2

Step 5: Calculate the initial kinetic energy of the flywheel.
KE = (1/2) * I * ω^2

Substituting the values of I and ω calculated above, we can find the initial kinetic energy.

Step 6: Calculate the time taken for the flywheel to slow down.
The time taken can be expressed as the ratio of the energy lost to the average power.
Time taken = (Initial kinetic energy) / (Power)

Step 7: Calculate the distance traveled by the bus before the flywheel needs to be spun up again.
The distance traveled is given by the formula:
Distance = Average speed * Time taken

By substituting the given average speed and the time taken calculated previously, we can find the distance the bus can travel.

Following these steps, you can calculate the distance the bus can travel before the flywheel has to be spun up again.

To find out how far the bus can travel before the flywheel needs to be spun up again, we need to calculate the energy loss of the flywheel due to the bus's energy consumption.

First, let's calculate the initial kinetic energy of the flywheel. The formula for kinetic energy (KE) of a rotating object is given by:

KE = (1/2) * I * ω^2

Where:
I is the moment of inertia of the flywheel
ω is the angular velocity of the flywheel in radians per second

To find the moment of inertia of the flywheel (I), we can use the formula for a solid cylinder:

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

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

Plugging in the values:
m = 1600 kg
r = 0.650 m

I = (1/2) * 1600 kg * (0.650 m)^2

Next, we convert the maximum rotation rate of the flywheel from revolutions per minute to radians per second. Recall that there are 2π radians in one revolution:

ω = (5150 rev/min) * (2π radians/1 rev) * (1 min/60 s)

Now we can use the moment of inertia and angular velocity to calculate the initial kinetic energy (KE) of the flywheel:

KE = (1/2) * I * ω^2

Next, let's calculate the power loss of the flywheel. Power is the rate of energy consumption per unit time. In this case, the power loss is given as 18.0 horsepower (hp). We need to convert that into watts (W):

1 horsepower (hp) = 745.7 watts (W)

So, the power loss is:

P_loss = 18.0 hp * 745.7 W/hp

Now we can calculate the time it takes for the flywheel to slow down to a stop. We know the power loss and the initial kinetic energy of the flywheel. The relationship between time (t) and power (P) is given by:

P = ΔKE / t

Rearranging the formula, we can solve for time (t):

t = ΔKE / P_loss

Now we need to calculate the change in kinetic energy (ΔKE) of the flywheel while it slows down. This is given by:

ΔKE = 1/2 * I * (ω_final^2 - ω_initial^2)

Since the flywheel slows down until it stops, ω_final is equal to 0. Therefore, ΔKE simplifies to:

ΔKE = 1/2 * I * (-ω_initial^2)

Finally, we can calculate the time (t) it takes for the flywheel to slow down to a stop using the values we have:

t = (1/2 * I * (-ω_initial^2)) / P_loss

Now we know the time it takes for the flywheel to slow down to a stop. To find out how far the bus can travel during this time, we need to know the average speed of the bus. Given that the average speed of the bus is 40.0 km/h, we can calculate the distance (d) using the formula:

d = v * t

Where:
v is the average speed of the bus (40.0 km/h)
t is the time it takes for the flywheel to slow down to a stop

Plugging in the values, we can calculate the distance (d) the bus can travel before the flywheel needs to be spun up again.