Question

(8%) Problem 16:   A flywheel is a mechanical device used to store rotational kinetic energy for later use. Consider a flywheel in the form of a uniform solid cylinder rotating around its axis, with moment of inertia I = 1/2 mr2.

33% Part (a) If such a flywheel of radius r1 = 1.5 m and mass m1 = 22 kg can spin at a maximum speed of v = 55 m/s at its rim, calculate the maximum amount of energy, in joules, that this flywheel can store. Emax = 16637 Emax = 16640     ✔ Correct!   33% Part (b) Consider a scenario in which the flywheel described in part (a) (r1 = 1.5 m, mass m1 = 22 kg, v = 55 m/s at the rim) is spinning freely at its maximum speed, when a second flywheel of radius r2 = 1.1 m and mass m2 = 13 kg is coaxially dropped from rest onto it and sticks to it, so that they then rotate together as a single body. Calculate the energy, in joules, that is now stored in the wheel. PART C Return now to the flywheel of part (a), with mass m1, radius r1, and speed v at its rim. Imagine the flywheel delivers one third of its stored kinetic energy to car, initially at rest, leaving it with a speed vcar. Enter an expression for the mass of the car, in terms of the quantities defined here.

Answer 1

part a)

moment of inertia of the flywheel, I1 = (1/2)*m1*r1^2

= (1/2)*22*1.5^2

= 24.75 kg.m^2

given speed at rim, v = 55 m/s

angular speed of the flywheel, w = v/r1

= 55/1.5

= 36.67 rad/s

The maximum amount of energy stored by the fly wheel,

E_max = (1/2)*I1*w^2

= (1/2)*24.75*36.67^2

= 16640 J

b) moment of inertia of the second fly wheel, I2 = (1/2)*m2*r2^2

= (1/2)*13*1.1^2

= 7.865 kg.m^2

let w' is the combined angular speed of both flywheels

apply conservation of angular momentum

(I1 + I2)*w' = I1*w

w' = I1*w/(I1 + I2)

= 24.75*36.67/(24.75 + 7.865)

= 27.827 rad/s

kinetic energy stored in the wheels = (1/2)*(I1 + I2)*w'^2

= (1/2)*(24.75 + 7.865)*27.827^2

= 12628 J <<<<<<<<<<----------------------------------Answer

part C)

(1/2)*m_car*v_car^2 = (1/3)*(1/2)*I1*w^2

(1/2)*m_car*v_car^2 = (1/3)*(1/2)*(1/2)*m1*r1^2*w^2

(1/2)*m_car*v_car^2 = (1/3)*(1/2)*(1/2)*m1*(r1*w)^2

(1/2)*m_car*v_car^2 = (1/3)*(1/2)*(1/2)*m1*v^2 (since v = r1*w)

m_car = m1*v^2/(6*v_car^2) <<<<<<<<<<----------------------------------Answer