What would happen if we put a round plate in the bicycle instead of a ring?
Answers
Answer:
If your question is stated correctly, then because of the word “slide”, we should expect that they reach the bottom at the same time. There is no rotation, and so no need to consider rotational kinetic energy. Both objects have the same mass, therefore the same gravitational acceleration.
However, I guess that your question should read “… roll down an inclined plane…”
In that case, the disk will reach the bottom first.
There are many ways to justify this. One way is to resort to moment of inertia calculations. You’ll find that the moment of inertia for the ring is larger than that of the disk, and consequently the ring will accelerate more slowly down the slope.
Here’s an easier way to understand this.
Initially, both the ring and the disk have zero kinetic energy, and non-zero gravitational potential energy relative to the bottom of the slope. When the ring and the disk are released, the gravitational energy is converted to kinetic energy of translation and kinetic energy of rotation. At the bottom of the plane, all of the energy is kinetic energy.
Now since the ring has its mass concentrated far away from the axis of rotation, whereas the disk has its mass uniformly spread out, the amount of rotational kinetic energy stored in the ring will be greater than that stored in the disk for the same rate of rotation. This means that the ring must be moving at a slower speed than the disk when they reach the bottom.
This principle holds at each point on the plane. Hence the disk reaches the bottom first.