You set out to design a car that uses the energy stored in a flywheel consisting of a uniform 97-kg cylinder of radius R that has a maximum angular speed of 330 rev/s. The flywheel must deliver an average of 1.70 MJ of energy for each kilometer of distance. Find the smallest value of R for which the car can travel 300 km without the flywheel needing to be recharged.
To find the smallest value of R for which the car can travel 300 km without the flywheel needing to be recharged, we need to determine the energy stored in the flywheel and calculate the amount of energy required for the car to travel 300 km.
First, let's find the energy stored in the flywheel using the formula for rotational kinetic energy:
E = (1/2)Iω^2
Where E is the energy stored, I is the moment of inertia of the flywheel, and ω is the angular speed of the flywheel.
The moment of inertia of a solid cylinder can be calculated using the formula:
I = (1/2)mR^2
Where m is the mass of the flywheel and R is the radius of the flywheel.
Therefore, the energy stored in the flywheel is:
E = (1/2)(1/2)mR^2ω^2
Now, let's calculate the energy required for the car to travel 300 km. Given that the average energy required per kilometer is 1.70 MJ:
Total energy required = 1.70 MJ/km * 300 km
Next, let's set the energy stored in the flywheel equal to the total energy required to find the value of R:
(1/2)(1/2)mR^2ω^2 = 1.70 MJ/km * 300 km
We can cancel out the units and simplify the equation further:
(mR^2ω^2)/4 = 1.70 * 10^6 J/km * 300 km
(mR^2ω^2)/4 = 5.10 * 10^8 J
Now, let's plug in the given values and solve for R:
m = 97 kg
ω = 330 rev/s (convert to radians per second by multiplying with 2π)
1 J = 1 kg·m^2/s^2 (unit conversion)
(97 kg * R^2 * (330 rev/s * 2π rad/rev)^2) / 4 = 5.10 * 10^8 J
Simplifying the equation:
R^2 = (4 * 5.10 * 10^8 J * 4) / (97 kg * (330 rev/s * 2π rad/rev)^2)
Now, calculate the value of R:
R = √[(4 * 5.10 * 10^8 J * 4) / (97 kg * (330 rev/s * 2π rad/rev)^2)]
After substituting the values and solving the equation, you will find the smallest value of R for which the car can travel 300 km without the flywheel needing to be recharged.