A solid cylinder and a cylindrical shell have the same mass, same radius, and turn on frictionless, horizontal axles. (The cylindrical shell has light-weight spokes connecting the shell to the axle. A rope is wrapped around each cylinder and tied to a block. The blocks have the same mass and are held the same height above the ground, as shown in the figure (Figure 1) Both blocks are released simultaneously. The ropes do not slip. A) Which block hits the ground first? Or is it a tie? Explain. B) Complete the sentences with following terms.(solid) (hollow) , (translational kinetic) (gravitational potential) (rotational potential) (rotational kinetic) By the time the blocks reach the ground, they have transformed identical amounts of energy into energy of the cylinders. energy of the blocks and But the moment of inertia of a cylinder is higher than that of a cylinder of the same mass, so more of the energy of the system is in the form of rotational kinetic energy for the cylinder than for the one. This leaves less energy in the form of translational kinetic energy for the cylinder. But it is the energy that determines the speed of the block. So the block moves more slowly for the system with the cylinder, and so its block reaches the ground last.

The answer to the queations are;

A) The block attached to the solid cylinder would hit the ground first.

B) By the time the blocks reach the ground, they have transformed identical amounts of _gravitational potential_______energy into_____rotational kinetic________ energy of the cylinders. energy of the blocks and _______translational kinetic_____But the moment of inertia of a ____hollow______cylinder is higher than that of a ____solid_______ cylinder of the same mass, so more of the energy of the system is in the form of rotational kinetic energy for the ______hollow_____________cylinder than for the ___solid_______ one. This leaves less energy in the form of translational kinetic energy for the ____hollow________cylinder. But it is the ____translational kinetic________ energy that determines the speed of the block. So the block moves more slowly for the system with the ______hollow______cylinder, and so its block reaches the ground last.

Explanation:

To solve the question, we note that

The total energy of motion of the moving cylinders is equal to

K = 1/2·m·v² + 1/2·I·ω²

Where

m = Mass

v = Velocity

ω = Angular velocity

I = moment of inertia where I for hollow cylinder = MR² and

I for solid cylinder = 1/2·MR².

Therefore we have

K for solid cylinder = 1/2·m·v² + 1/2·I·ω² = 1/2·m·v² + 1/2·1/2·MR²·ω²

= 1/2·m·v² + 1/4·MR²·v²/r² = 1/2·m·v² + 1/4·M·v² = 3/4·m·v²

For the hollow cylinder, we have

K = 1/2·m·v² + 1/2·MR²·ω² = 1/2·m·v² + 1/2·MR²·v²/r² = 1/2·m·v² + 1/2·m·v²

= m·v²

From conservation of energy the initial potential energy is transformed into potential energy as follows

PE = m·g·h

Where:

m = Mas

g = Gravitational acceleration

h = height

Therefore

For the solid cylinder 3/4·m·v² = m·g·h and  v² = and v  =

For the hollow cylinder m·v² = m·g·h and  v² = and v  =

This shows that the solid cylinder has a higher downward velocity and the block attached to the solid cylinder would hit the ground first

B) By the time the blocks reach the ground, they have transformed identical amounts of _gravitational potential_______energy into_____rotational kinetic________ energy of the cylinders. energy of the blocks and _______translational kinetic_____But the moment of inertia of a ____hollow______cylinder is higher than that of a ____solid_______ cylinder of the same mass, so more of the energy of the system is in the form of rotational kinetic energy for the ______hollow_____________cylinder than for the ___solid_______ one. This leaves less energy in the form of translational kinetic energy for the ____hollow________cylinder. But it is the ____translational kinetic________ energy that determines the speed of the block. So the block moves more slowly for the system with the ______hollow______cylinder, and so its block reaches the ground last.