which will reach the bottom first, a hallow cylider or a solid cylider wth the same mass in an inclined plane?

For each cylinder

mgh=(mv^2)/2+(Iω^2)/2 =(mv^2)/2+(Iv^2)/2R^2.
For hallow cylinder the moment of inertia is I=mR^2, and for solid cylinder I=(mR^2)/2.
Therefore, the velocities at the bottom are
- for hallow cylinder v=sqroot(gh),
- for solid cylinder v=sqroot(1.33gh).
The solid cylinder will be first.

A solid cylinder rolls faster, because a smaller fraction of the available potential energy gets converted to rotational kinetic energy. That leaves more energy available for forward motion.

This is true even if the masses and radii are different.

For a solid cylinder that has rolled doownhill a vertical distance H,
M g H = (M/2)V^2 + (1/2) I w^2
= (M/2)V^2 + (1/2)(M*R^2/2)(V/R)^2
= (M/2)V^2 + (M/4)V^2
V^2 = (4/3) g H

For a hollow thin-walled cylinder,
M g H = (M/2) V^2 + (M/2)V^2 = M V^2
V^2 = g H

To determine which object will reach the bottom first, a hollow cylinder or a solid cylinder with the same mass on an inclined plane, we need to consider their rolling behavior and their moments of inertia.

The moment of inertia is a measure of an object's resistance to rotational motion. For a solid cylinder, the moment of inertia is given by the expression (1/2) * m * r^2, where m is the mass of the cylinder and r is its radius. For a hollow cylinder, the moment of inertia depends on its thickness and can be more complicated to calculate.

When an object rolls down an inclined plane, two forces act on it: gravity and the normal force. The normal force is perpendicular to the surface of the inclined plane and counteracts the downward force of gravity. The frictional force between the object and the inclined plane opposes the object's motion.

In the case of the hollow cylinder, due to its lower moment of inertia, it accelerates faster than the solid cylinder. This is because the mass is concentrated near its outer radius, resulting in a larger rotational acceleration for the given torque produced by the gravitational force.

On the other hand, the solid cylinder has a larger moment of inertia because its mass is distributed throughout its entire volume. This leads to a smaller rotational acceleration and a slower rolling speed compared to the hollow cylinder.

Based on these considerations, the hollow cylinder will reach the bottom of the inclined plane first as it has a higher rotational acceleration and, therefore, a faster rolling speed.