Question

A simple yo-yo consists of a hoop of radius R with a string wound around the rim several times. The free end of the string is held fixed, and the hoop is released from rest. The string unwinds without slipping. Calculate the hoop’s linear speed and the hoop’s angular speed when the hoop has descended a distance D.

Answer 1

To calculate the linear speed and angular speed of the yo-yo, we use the relationship between linear speed and angular speed. The linear speed is given by the product of the angular speed and the distance from the axis of rotation.

Step-by-step explanation:

To calculate the linear speed and angular speed of the yo-yo, we need to use the relationship between linear speed and angular speed. The linear speed of a point on a rotating object is given by the product of the angular speed and the distance from the axis of rotation.

First, we need to determine the distance D traveled by the yo-yo. Since the hoop descends, the string unwinds, and the circumference of the hoop is equal to 2πR, we can use the formula D = 2πR.

Next, we can calculate the linear speed by multiplying the angular speed by the radius. The angular speed (ω) is given by the equation ω = v / R, where v is the linear speed. Rearranging the equation gives us v = ωR.

So the linear speed of the hoop is v = ωR, and the angular speed is ω.