Question

Two identical wheels are moving on horizontal surfaces. The center of mass of each has the same linear speed. However, one wheel is rolling, while the other is sliding on a frictionless surface without rolling. Each wheel then encounters an incline plane. One continues to roll up the incline, while the other continues to slide up. Eventually they come to a momentary halt, because the gravitational force slows them down. Each wheel is a disk of mass 3.73 kg. On the horizontal surfaces the center of mass of each wheel moves with a linear speed of 5.84 m/s. (a),(b) What is the total kinetic energy of each wheel? (c), (d) Determine the maximum height reached by each wheel as it moves up the incline.

Answer 1

Mass of the disk, m = 3.73 kg

Linear speed of the disc, v = 5.84 m/s

(a) The wheel that is sliding on the horizontal surface will only have translational kinetic energy.

So, its total kinetic energy = (1/2)mv²

= (1/2)(3.73 kg)(5.84 m/s)²

= 63.6 J

(b) The wheel which is rotating also has rotational kinetic energy.

So, its total kinetic energy = 63.6 J + (1/2)Iω²

= 63.6 J + (1/2)[(1/2)mr²](v/r)²

= 63.6 J + (1/4)mv²

= 63.6 J + (1/2)(63.6 J)

= 95.4 J

(c) Maximum height reached by the sliding disc.

Potential energy attained by the disc = mgh

So,

63.6 J = mgh

63.6 J = (3.73 kg)(9.81 m/s²)h

=> h = 63.6 / (3.73 x 9.81) = 1.74 m

(d) Maximum height reached by the rotating wheel.

95.4 J = mgh

=> 95.4 J = (3.73 kg)(9.81 m/s²)h

=> h = 95.4 / (3.73 x 9.81) = 2.61 m (Answer)