Two identical wheels are moving on horizontal surfaces. The center of mass of ea
ID: 1697412 • Letter: T
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 2.87 kg. On the horizontal surfaces the center of mass of each wheel moves with a linear speed of 5.04 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.Explanation / Answer
The total kinetic energy KE of the rolling wheel is the sum of its translational and rotational kinetic energies
KE = 1/2mv^2 + 1/2I^2
where is the angular speed of the wheel. and where R is the radius of the disk. Furthermore, the angular speed of the rolling wheel is related to the linear speed v of its center of mass by = v/R. Thus, the total kinetic energy of the rolling wheel is
= 1/2mv^2 + 1/2 (1/2mr^2)(v/r)^2
= 3/4 mv^2
= 3/4 (2.87kg)(5.04)^2 = 54.67 J
The kinetic energy of the sliding wheel is
U = 1/2mv^2 = (0.5)(2.87)(5.04)^2
= 36.45J
As expected, the rolling wheel has the greater total kinetic energy.
b. As each wheel rolls up the incline, its total mechanical energy is conserved. The initial kinetic energy KE at the bottom of the incline is converted entirely into potential energy PE when the wheels come to a momentary halt. Thus, the potential energies of the wheels are:
Rolling
Wheel PE = K = 54.67J
Sliding PE = K = 36.45J
Wheel
As anticipated, the rolling wheel has the greater potential energy.
c. The potential energy of a wheel is given as PE = mgh, where g is the acceleration due to gravity and h is the height relative to an arbitrary zero level. The height reached by each wheel is
Rolling
Wheel h = PE/mg
= 54.67/(2.87)(9.8)
= 1.94 m
Sliding
Wheel h = PE/mg = 36.45 / (2.87)(9.8)
= 1.3m
d. The total kinetic energy of the rolling wheel is (see the results of part a). The kinetic energy KE at the bottom of the incline is converted entirely into potential energy PE when the wheels come to a monetary halt. Since the potential energy is PE = mgh, we have that
h = 3v^2 / 4g = 3(5.04)^2 / 4(9.8)
= 1.944 m
The mass has been algebraically eliminated from this expression, so the final height reached by the wheel is independent of its mass. Thus, h =1.944m , as expected.
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