Question

A string is wrapped several times around the rim of a small hoop with radius 8.00 cm

and mass 0.180 kg
. The free end of the string is held in place and the hoop is released from rest . The hoop has descended 85.0 cm

A string is wrapped several times around the rim of a small hoop with a radius of 0.0800 meters. A hand holds the free end of the string going upward. When the string unwinds, the hoop rotates clockwise.

Calculate the angular speed of the rotating hoop.
Express your answer in radians per second.
Calculate the speed of its center.

Answer 1

The angular speed (ω) of the rotating hoop is approximately 14.42 radians per second, and the speed of its center (v) is approximately 1.153 meters per second.

Certainly! Let's calculate the angular speed (ω) and the speed of the center (v) step by step.

Given data:

Radius of the hoop (r): 0.0800 meters

Mass of the hoop (m): 0.180 kg

Height descended (h): 0.85 meters

Acceleration due to gravity (g): 9.8 m/s²

Step 1: Angular Speed (ω) Calculation:

The formula for angular speed is given by:

ω = √(2gh/r)

Substitute the given values:

ω = √(2 * 9.8 * 0.85 / 0.0800)

ω = √(16.646 / 0.0800)

ω ≈ √208.075

ω ≈ 14.42 rad/s

So, the angular speed (ω) is approximately 14.42 radians per second.

Step 2: Speed of the Center (v) Calculation:

The relationship between linear and angular speed is given by:

v = rω

Substitute the values:

v = 0.0800 meters * 14.42 rad/s

v ≈ 1.153 meters per second

So, the speed of the center (v) is approximately 1.153 meters per second.