A ring takes time t? to slide down and time t? to roll down an inclined plane of length L. What is the ratio of t? : t??

To understand this question, we compare two motions of a ring on an inclined plane: sliding and rolling.

When the ring slides down, it moves only because of gravity. The acceleration is given by g sinθ, where g is gravity and θ is the angle of the incline. In this case, there is no rotation, so all energy is used for forward motion. Because of higher acceleration, the ring reaches the bottom faster.

When the ring rolls down, it both moves forward and rotates. Some energy is used in rotation, so the acceleration becomes smaller. For a ring, the acceleration is reduced to (g sinθ) / 2. Because of this lower acceleration, the ring takes more time to reach the bottom.

Using motion formulas, we find that time depends on acceleration. When we calculate the ratio of time taken in both cases, we get:

t? : t? = 1 : √2

Final Answer: t? : t? = 1 : √2