You set out to design a car that uses the energy stored in a flywheel consisting of a uniform 103-kg cylinder of radius R that has a maximum angular speed of 450 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.

Question

You set out to design a car that uses the energy stored in a flywheel consisting of a uniform 103-kg cylinder of radius R that has a maximum angular speed of 450 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.

My attempt:
U = 1/2 *I *w^2
w = 2 *pi *450
I = 1/12 *M*r^2
U = (1/24)103*r^2*(2pi450)^2

The total energy required to travel 300 km at 1.7 x10^6 so
U = 300 *1.7x10^6

I combined both U's together to try and solve for r

(1/24)*103*r^2*(2pi450) = 300 *1.7x10^6
r = sqrt (24*300*1.7x10^6) / 103*(2pi450)

r = sqrt (14.86)
r = 3.855 m

It told me I was wrong so any light as to why? Or am I just completely off on how to get the answer? Thanks in advance!

Answer 1

Nevermind I was able to finally figure it out xP

Answer 2

Your initial approach is correct, but it seems that there is a mistake in the calculation. Let's go through the solution step by step.

First, let's express the moment of inertia (I) in terms of mass (M) and radius (R) using the formula for the moment of inertia of a solid cylinder:

I = 1/2 * M * R^2

Given:
M = 103 kg
w (angular speed) = 450 rev/s
R (unknown) = ?

To find the energy stored in the flywheel, we can use the formula:

U = 1/2 * I * w^2

Substituting the value of I:

U = 1/2 * (1/2 * M * R^2) * w^2
U = (1/4) * M * R^2 * w^2

Next, let's find the total energy required to travel 300 km (300,000 meters) based on the given average energy per kilometer (1.7 MJ/km):

Total energy required = 300,000 m * 1.7 MJ/km

Now, we can set the energy stored in the flywheel equal to the total energy required:

(1/4) * M * R^2 * w^2 = 300,000 m * 1.7 MJ/km

Simplifying and substituting the values:

(1/4) * 103 kg * R^2 * (2pi * 450 rev/s)^2 = 300,000 m * 1.7 x 10^6 J

Now we can solve for R:

103 kg * R^2 * (2pi * 450)^2 = 300,000 * 1.7 x 10^6 * 4

Dividing both sides by 103 kg * (2pi * 450)^2, we get:

R^2 = (300,000 * 1.7 x 10^6 * 4) / (103 * (2pi * 450)^2)

R^2 = (300,000 * 1.7 x 10^6 * 4) / (103 * 2^2 * (pi * 450)^2)

Evaluating the expression:

R^2 ≈ 0.015779

Finally, taking the square root to find R:

R ≈ sqrt(0.015779) ≈ 0.1256 meters

So, the smallest value of R for which the car can travel 300 km without needing to recharge the flywheel is approximately 0.1256 meters.