Question

A cylinder with moment of inertia I about its center of mass, mass m, and radius r has a string wrapped around it which is tied to the ceiling (Figure 1) . The cylinder's vertical position as a function of time is y(t). At time t=0 the cylinder is released from rest at a height h above the ground.

A) The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate ?and v, the velocity of the center of mass of the cylinder?
B) Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds to positive angular velocity.
C) Suppose that at a certain instant the velocity of the cylinder is v. What is its total kinetic energy, Ktotal, at that instant?
D) Suppose that at a certain instant the velocity of the cylinder is v. What is its total kinetic energy, Ktotal, at that instant?

Answer 1

The relationship between the angular rotation rate and the velocity of the center of mass of a cylinder wrapped with a string is omega = v / r. The total kinetic energy of the cylinder at a certain instant can be calculated by summing the rotational and translational kinetic energies.

Step-by-step explanation:

For a cylinder with moment of inertia I rotating about its center of mass, and a mass m, the string wrapped around it constrains both its rotational and translational motion. The relationship between the angular rotation rate (omega) and the velocity of the center of mass (v) is given by the equation:

omega = v / r

Where r is the radius of the cylinder.

At any instant with a velocity v, the total kinetic energy (Ktotal) of the cylinder is given by the sum of its rotational kinetic energy (Krot) and translational kinetic energy (Ktrans):

Ktotal = Krot + Ktrans

Rotational kinetic energy is given by the equation:

Krot = (1/2) I omega^2

Where I is the moment of inertia of the cylinder. Translational kinetic energy is given by:

Ktrans = (1/2) m v^2

Thus, the total kinetic energy can be calculated by substituting the values of I, v, and r into the respective equations.