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Two heavy disks are connected by a short rod of much smaller radius. The system is placed on a ramp so that the disks hang over the sides as in Fig The system rolls down the ramp without slipping. (a) Near the bottom of the ramp the disks touch the horizontal table and the system takes off with greatly increased translational speed. Explain why. (b) If this system raced a hoop (of any radius) down the ramp, which would reach the bottom first?

Two heavy disks are connected by a short rod of much smaller radius. The system is placed on a ramp so that the disks hang over the sides as in Fig The system rolls down the ramp without slipping. (a) Near the bottom of the ramp the disks touch the horizontal table and the system takes off with greatly increased translational speed. Explain why. (b) If this system raced a hoop (of any radius) down the ramp, which would reach the bottom first?

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
8 years ago
The figure below shows the heavy disk connected by a short rod and their axis of rotation as they roll down the ramp.
232-1408_1.PNG
The dash line in the figure above is the axis of rotation of the disks-rod system.
(a) It is important to note that when the disks-rod system is up the ramp, they have gravitational potential energy, but as they roll down the inclined plane, this energy is converted into the rotational and translational kinetic energy.
It can be seen from the figure above that up the ramp the frictional force exist between the surface of rod and the inclined plane surface. The larger friction results during the motion of disks-rod system on the ramp because the frictional force acts along the entire length of the rod.
However, when the disks-rod system reaches the bottom, the frictional forces exist between the disk and the horizontal surface and not between the rod and horizontal surface. This lower frictional loss on the horizontal surface and the energy possessed by the disks-rod system together becomes responsible for increasing the speed relatively.
(b) The hoop will reach the bottom first because the frictional force between the hoop and the ramp would be very less relative to that between the rod and the ramp. Therefore the hoop is likely to win.
But, the rotational inertia of the hoop will play a major role in deciding the time it takes to reach at bottom. A hoop with large radii has large rotational inertia, and is expected to move slowly.

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