Question

A 1000 kg car has four 10.0 kg wheels. When the car is moving, what fraction of its total kinetic energy is due to rotation of the wheels about their axles? Assume that the wheels are uniform disks of the same mass and size. Why do you not need to know the radius of the wheels?

Answer 1

Answer:

0.0196

Explanation:

$\omega$ = Angular speed

m = Mass of wheel = 10 kg

M = Mass of car = 1000 kg

v = Velocity of car

r = Radius of wheels

The wheels are assumed as disks

Rotational kinetic energy in the wheels is given by

$K_w=4\frac{1}{2}I\omega^2$

Moment of inertia of the wheels is

$I=\frac{1}{2}mr^2$

$K_w=4\frac{1}{2}mr^2\left(\frac{v}{r}\right)^2\\\Rightarrow K_w=mv^2\\\Rightarrow K=10v^2$

Kinetic energy of the car

$K_c=\frac{1}{2}Mv^2\\\Rightarrow K_c=\frac{1}{2}\times 1000v^2\\\Rightarrow K_c=500v^2$

fraction of its total kinetic energy is due to rotation of the wheels about their axles is given by

$\frac{K_w}{K_w+K_c}\\ =\frac{10v^2}{10v^2+500v^2}\\ =0.0196$

The fraction is 0.0196

As it can be seen that the radius of the wheels cancel out it is not required