You set out to design a car that uses the energy stored in a flywheel consisting of a uniform 101-kg cylinder of radius r that has a maximum angular speed of 470 rev/s. the flywheel must deliver an average of 2.40 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.

Ok, assuming "mj" in the question is Megajoules MJ) you need a total amount of rotational kinetic energy in the fly wheel at the beginning of the trip that equals

(2.4e6 J/km) x (300 km) = 7.2e8 J

The expression for rotational kinetic energy is

E = (1/2) Iω²

where I is the moment of inertia of the fly wheel and ω is the angular velocity.

So this comes down to finding the value of I that gives the required energy. We know the mass is 101kg. The formula for a solid cylinder's moment of inertia is

I = (1/2) mR²

We want (1/2) Iω² = 7.2e8 J and we know ω is limited to 470 revs/sec. However, ω must be in radians per second so multiply it by 2π to get

ω = 2953.1 rad/s

Now let's use this to solve the energy equation, E = (1/2) Iω², for I:

I = 2 (7.2e8 J) / (2953.1 rad/s) ² = 165.12 kg·m²

Now find the radius R,

165.12 kg·m² = (1/2) (101) R²,

√ (2·165/101) = 1.807m

R = 1.807m