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 575 kg. Before a trip, the flywheel is attached to an electric motor, which brings the flywheel's rotational speed up to 5,000 rev/min.
(a) Find the kinetic energy stored in the flywheel.
Answer in J
(b) If the flywheel is to supply energy to the car as a 10.0-hp motor would, find the length of time the car could run before the flywheel would have to be brought back up to speed.
Answer in h
To find the kinetic energy stored in the flywheel, we can use the formula:
Kinetic Energy (KE) = 1/2 * moment of inertia * angular speed^2
(a) First, let's calculate the moment of inertia of the flywheel. The moment of inertia of a solid disk rotating about its axis is given by the formula:
I = (1/2) * mass * radius^2
Given:
mass of the flywheel (m) = 575 kg
radius of the flywheel (r) = 1.00 m
I = (1/2) * 575 kg * (1.00 m)^2
Now, let's calculate the kinetic energy (KE) using the formula above:
KE = 1/2 * I * (angular speed)^2
Given:
angular speed = 5000 rev/min
To convert revolutions per minute (rpm) to radians per second (rad/s), we use the conversion factor 1 rev = 2π rad.
angular speed = 5000 rev/min * (2π rad/1 rev) * (1 min/60 s) = (5000 * 2π) rad/s
Now we can calculate the kinetic energy:
KE = 1/2 * (1/2 * 575 kg * (1.00 m)^2) * ((5000 * 2π) rad/s)^2
Simplifying this expression, we get:
KE ≈ 2.75 × 10^7 J
Therefore, the kinetic energy stored in the flywheel is approximately 2.75 × 10^7 Joules (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 can use the formula for power:
Power (P) = Energy (E) / Time (t)
Given:
Power of a 10.0-hp motor = 10 hp = 10 hp * (745.7 W/1 hp)
= 7457 W
We need to convert this power value to the same unit as the kinetic energy (Joules), so we can use the equation:
1 J = 1 W * 1 s
Therefore, the power in Joules per second (Watts) is equivalent to Joules.
Now, rearranging the formula:
t = E / P
t = (2.75 × 10^7 J) / (7457 W)
Simplifying this expression, we get:
t ≈ 3685.31 s
Now, let's convert the time from seconds to hours:
t = 3685.31 s * (1 min/60 s) * (1 h/60 min)
Simplifying this expression, we get:
t ≈ 1.02 h
Therefore, the car could run for approximately 1.02 hours before the flywheel would have to be brought back up to speed.
To solve part (a), we need to calculate the kinetic energy stored in the flywheel.
The formula for kinetic energy (KE) of a rotating object is:
KE = 0.5 * I * ω^2,
where I is the moment of inertia and ω is the angular velocity.
First, let's calculate the moment of inertia using the formula:
I = 0.5 * m * r^2,
where m is the mass of the flywheel and r is its radius.
Given:
m = 575 kg,
r = 1.00 m.
Substituting the values, we get:
I = 0.5 * (575 kg) * (1.00 m)^2.
Now, we need to convert the angular velocity from rev/min to rad/s. The conversion factor is 2π rad/rev and 60 s/min. Therefore:
angular velocity (ω) = (5000 rev/min) * (2π rad/rev) * (1 min/60 s).
Now, we have all the necessary information to calculate the kinetic energy (KE) using the formula mentioned earlier:
KE = 0.5 * I * ω^2.
Solve the equation to find the answer in Joules (J).
To solve part (b), we need to calculate the length of time the car could run before the flywheel would need to be brought back up to speed.
Given that the flywheel is supplying energy to the car as a 10.0-hp motor would, we can use the following relationship:
Power (P) = Energy (E) / Time (t).
We know that 1 horsepower (hp) is equal to 746 Watts (W). So, we can express the power in terms of Watts.
First, calculate the kinetic energy (E) stored in the flywheel using the formula derived in part (a).
Next, convert the power from 10.0 hp to Watts.
Now, use the power formula mentioned earlier to find the length of time (t) in seconds that the car can run before the flywheel needs to be brought back up to speed.
Finally, convert the time from seconds to hours to get the answer in hours.
Given: r=1m, m=575kg, V = 5000rev/min.
KE = ?
a. C = pi*D=3.14 * 2=6.28m= Circumference.
V=5000rev/min * 6.28m/rev *(1/60)min/s
= 523.3m/s.
KE=0.5mV^2=0.5 * 575 * (523.3)^2 =
78,729,831J.
b. Po = 10hp * 746W/hp = 7460 Watts =
7460 Joules/s.
Po = 7460J/s * 3600s/h = 26,856,000J/h.
T = KE / Po = 78,729,831 / 26,856,000 =
2.93h.