Consider two cylinders that start down identical inclines from rest except that one is frictionless. Thus one cylinder rolls without slipping, while the other slides frictionlessly without rolling. They both travel a short distance at the bottom and then start up another incline. (a) Show that they both reach the same height on the other incline, and that this height is equal to their original height. (b) Find the ratio of the time the rolling cylinder takes to reach the height on the second incline to the time the sliding cylinder takes to reach the height on the second incline. (c) Explain why the time for the rolling motion is greater than that for the sliding motion.
Step-by-step solution Step 1 of 9 For a system with no frictional and other energy losses, according to conservation of energy, the initial energy of a cylinder at some height is equal to the final energy of the cylinder. Step 2 of 9 (a) For first cylinder the total initial energy is purely potential due to its height and zero speed. Here, g is the accelera tion due to gravity, m is the mass of the cylinder and h i is the initial height of the cylinder. The final energy of this cylinder is again purely potential. Step 3 of 9 By conservation of energy, Hence, initial height of cylinder is equal to the final height of the cylinder. Same result will hold for second cylinder, because initially both cylinders were at same height. Hence proved, both cylinders will reach their initial height. Step 4 of 9 The energy conservation as, Here, is the initial kinetic energy of the cylinder, is the initial gravitation potential energy of the cylinder, is the final kinetic energy of the cylinder and is the final gravitational potential energy of the cylinder. Step 5 of 9 (b) As, kinetic energy of the cylinder on the top of the incline is zero and potential energy is maximum, when it comes down , its potential energy converts into kinetic energy and at the bottom its potential energy becomes zero and the kinetic energy is maximum. Suppose the incline has an angle from the horizontal, and the distance covered by the cylinders on the incline is , so the vertical component i.e. height of the incline is, By conservation of energy for the cylinder which rolls down the incline gives, Here, is the mass of the cylinder, is the gravitational acceleration of the cylinder, is the final linear velocity of the cylinder at the bottom, is the moment of inertia of the cylinder and is the angular speed of cylinder. Substitute for I and v1 for and solve for v1. So, the final velocity of the first cylinder is .