In the design of a passenger bus, it is proposed to derive the motive power
from the energy stored in a flywheel. The flywheel, which has a moment of
inertia of 4.0 x 102 kgm2, is accelerated to its maximum rate of rotation of 3.0 x
103 revolutions per minute by electric motors at stations along the bus route.
a) Calculate the maximum kinetic energy that can be stored in the flywheel.
b) If, at an average speed of 36 kilometres per hour, the power required by the
bus is 20 kW, what will be the maximum possible distance between stations
on the level?
To solve this problem, we need to use the concept of rotational kinetic energy and power. Let's break it down step by step:
a) To calculate the maximum kinetic energy stored in the flywheel, we need to use the formula for rotational kinetic energy:
K.E. = (1/2) I ω^2
Where:
- K.E. is the kinetic energy in Joules
- I is the moment of inertia in kgm^2
- ω is the angular velocity in radians per second
First, let's convert the maximum rate of rotation from revolutions per minute to radians per second. To do this, we need to know that 1 revolution is equivalent to 2π radians.
Given:
- Maximum rate of rotation = 3.0 x 10^3 revolutions per minute
To convert revolutions per minute to radians per second:
ω = (2π x maximum rate of rotation) / 60
Now we can calculate ω:
ω = (2π x 3.0 x 10^3) / 60
Having obtained ω, we can calculate the maximum kinetic energy stored in the flywheel:
K.E. = (1/2) I ω^2
Substitute the given values and solve for K.E.
b) To calculate the maximum possible distance between stations on the level, we need to use the concept of power. Power is defined as the rate at which work is done or energy is transferred.
Given:
- Average speed of bus = 36 km/h
- Power required by the bus = 20 kW
To convert the average speed from km/h to m/s:
Average speed = Average speed (in km/h) x (1000 m/1 km) x (1 h/3600 s)
Now, let's use the power formula to calculate the maximum possible distance between stations:
Power = Work / Time
Since the work done is equal to the change in kinetic energy, we can rewrite the formula as:
Power = ΔK.E. / Time
Rearranging the formula, we get:
ΔK.E. = Power x Time
To find the time, we can use the formula:
Time = Distance / Speed
Rearranging again, we get:
ΔK.E. = Power x (Distance / Speed)
Now we can substitute the given values and solve for the maximum possible distance.
a) To calculate the maximum kinetic energy stored in the flywheel, we can use the formula:
Kinetic Energy (KE) = 0.5 * moment of inertia * (angular velocity)^2
First, we need to convert the angular velocity from revolutions per minute to radians per second. There are 2π radians in one revolution.
Angular velocity (ω) = (3.0 x 10^3 rev/min) * (2π rad/1 rev) * (1 min/60 s)
= (3.0 x 10^3 * 2π) / 60 rad/s
= 314.16 rad/s (approx)
Now, we can calculate the maximum kinetic energy:
KE = 0.5 * (4.0 x 10^2 kgm^2) * (314.16 rad/s)^2
KE = 0.5 * 4.0 x 10^2 kgm^2 * (314.16 rad/s)^2
= 0.5 * 4.0 x 10^2 kgm^2 * 98632.8256 rad^2/s^2
≈ 98,632.83 J
Therefore, the maximum kinetic energy stored in the flywheel is approximately 98,632.83 Joules.
b) To calculate the maximum possible distance between stations on level ground, we can use the formula:
Distance = Power / (Force * Speed)
First, we need to convert the average speed from kilometers per hour to meters per second:
Speed = (36 km/h) * (1000 m/1 km) * (1 h/3600 s)
= (36 * 1000) / 3600 m/s
≈ 10 m/s
The force required by the bus can be calculated using the equation:
Force = Power / Speed
= 20,000 W / 10 m/s
= 2,000 N
Finally, we can calculate the maximum possible distance:
Distance = Power / (Force * Speed)
= 20,000 W / (2,000 N * 10 m/s)
= 1 meter
Therefore, the maximum possible distance between stations on level ground is 1 meter.