Smallest radius a flywheel can be and still provide enough energy

[mitch_1211]

I am on my second year of study for a bachelor of medical and radiation physics and one of my friends who is studying mechanical engineering ran this question by me, I haven't really had time to go into it in any detail, i thought some of you guys might like to give it a crack:

You set out to design a car that uses the energy stored in a flywheel
consisting of a uniform 100-kg cylinder of radius R. The flywheel
must deliver an average of 2 MJ of mechanical energy per kilometer,
with a maximum angular velocity of 400 rev/s. Find the least value of
R such that the car can travel 300 km without the flywheel having to
be recharged.

Mitch

 

[tiny-tim]

Hi Mitch!

… i thought some of you guys might like to give it a crack …

No, you go first, and then we'll comment.

 

[jschmidt]

I don't believe this forum exists for *doing other people's homework for them*. If your friend is working on the problem, and is running into an error, and wants to post his work thus far, I bet people would be willing to help him find where he's going wrong. If he needs help getting started, if he can explain what he doesn't understand, I bet they would be willing to provide some help that way too, but they aren't going to cheat for him/her.

That said, I can't help because the problem's beyond me. *grin*

 

[timthereaper]

I crunched some numbers and got something like R=1.378 m or something like that. I'm not sure if my process was entirely thorough and I'm not going to post it. I'm just trying to give you something to compare your result with.

As good as your intentions were posting this problem, this isn't like a one-hour cleaners where you can drop off your unsolved problems and pick up the ones that got solved later. There are some really smart people on here, but they want to see a bit of effort first and then they'll teach you how to do the problem.

 

[PJC01]

ell, thank you for your question. I can provide some insights into the smallest radius a flywheel can be and still provide enough energy for a car to travel 300 km without recharging.

To begin, let's first define some key terms and concepts. A flywheel is a mechanical device that stores rotational energy and can release it as needed. In this case, the flywheel is being used in a car to provide mechanical energy for movement. The energy stored in the flywheel is dependent on its mass and angular velocity, with a higher mass and angular velocity resulting in a greater amount of stored energy.

Based on the given information, we can calculate the total energy needed for the car to travel 300 km without recharging. 2 MJ of mechanical energy per kilometer for 300 km would require a total of 600 MJ of energy. Now, let's consider the maximum angular velocity of 400 rev/s. This means that the flywheel can rotate 400 times in one second, or 24,000 times in one minute.

To determine the smallest radius that the flywheel can be, we can use the formula for rotational kinetic energy: E = 1/2 * I * ω^2, where E is the energy, I is the moment of inertia, and ω is the angular velocity. Solving for I, we get I = 2E/ω^2.

Substituting the values we have, I = 2 * 600 MJ / (400 rev/s)^2 = 3.75 x 10^5 kg·m^2. This is the moment of inertia required for the flywheel to store 600 MJ of energy at an angular velocity of 400 rev/s.

Now, we can use the formula for moment of inertia for a cylindrical object: I = 1/2 * m * R^2, where m is the mass and R is the radius. Rearranging the formula, we get R = √(2I/m).

Substituting the values we have, R = √(2 * 3.75 x 10^5 kg·m^2 / 100 kg) = 86.6 meters.

Therefore, the smallest radius the flywheel can be to provide enough energy for the car to travel 300 km without recharging is approximately 86.6 meters. This is a very large radius and may not be practical for a