Two identical wheels are moving on horizontal surfaces. The center of mass of ea

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 2.69 kg. On the horizontal surfaces the center of mass of each wheel moves with a linear speed of 6.23 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

(a)
   the total kinetic energy KE of the rolling wheel isthe sum of its translational and rotational kinetic energies
   KE = (1 / 2) m v2 + (1 / 2) I2
   where w is the angular speed of the wheel
   since the wheel is a disk, its moment of inertiais
   I = (1 / 2) m R2
   where R is the radius of the disk
   furthermore, the angular speed of the rollingwheel is related to the linear speed v of its center ofmass by
   = v / R
   thus, the total kinetic energy of the rolling wheelis
   KE = (1 / 2) m v2 + (1 / 2) I2
         = (1 / 2) mv2 + (1 / 2) [(1 / 2) m R2] (v /R)2
         = 78.3 J
   for the sliding wheel
   KE = (1 / 2) m v2
         = 52.2 J
(b)
   as each wheel rolls up the incline, its totalmechanical energy is conserved
   the initial kinetic energy KE at the bottom of theincline is converted entirely into potential energy PE when thewheels
come to a momentary halt
   thus, the potential energies of the wheels are
   for rolling wheel PE = KE (in part a) = 78.3 J
   for sliding wheel PE = KE (in part a) = 52.2 J

(c)

   h = PE / m g (rolling wheel) = 2.97 m

   h = PE / m g (sliding wheel) = 1.98 m

(d)

   the required height will be
   h = 3 v2 / 4 g = 2.97 m

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