Question

A yo-yo is constructed of three disks: two outer disks of mass m, radius r, and thickness d, and an inner disk (around which the string is wrapped) of mass m, radius r, and thickness d, the yo-yo is suspended from the ceiling and then released with the string vertical. Calculate the tension in the string as the yo-yo falls. Note that when the center of the yo-yo moves down a distance y, the yo-yo turns through an angle y/r, which in turn means that the angular speed w is equal to v_cm/r. The moment of inertia of a uniform disk is 1/2mr��. (Use the following as necessary: m, r, d, m, r, and g for the acceleration due to gravity.)

Answer 1

To calculate the tension in the string as the yo-yo falls, we can use the equations Tsinθ = mrω² and Tcosθ = mg. By solving these equations simultaneously, we can find the tension in the string. The moment of inertia of a uniform disk can be calculated using the equation I = 1/2mr², and the angular speed can be calculated using the equation ω = v_cm/r.

Step-by-step explanation:

To calculate the tension in the string as the yo-yo falls, we need to consider the forces acting on the yo-yo. The tension in the string provides the centripetal force, which can be calculated using the equation Tsinθ = mrω², where T is the tension, m is the mass, r is the radius, ω is the angular speed, and θ is the angle of rotation. The component of the tension that is vertical opposes the gravitational force, and can be calculated using the equation Tcosθ = mg, where g is the acceleration due to gravity.

By solving these equations simultaneously, we can find the value of T, which represents the tension in the string. The moment of inertia of a uniform disk can be calculated using the equation I = 1/2mr². The velocity of the center of mass can be calculated using the equation v_cm = ωr. By substituting the value of v_cm in the equation for angular speed, we can find the value of ω.