Question

A 60-cm string is wrapped around the 2-cm radius of a yo-yo. Starting from rest, the yo-yo descends with a constant angular acceleration of 4rad/s^2. How long will it take the string to completely unwind?

Answer 1

  Let’s determine the circumference of the yo-yo.
C = 2 * π * 0.02 = 0.04 * π
As the yo-yo rotates one time, this much string unwraps from the yo-yo. Let’s determine the number of times the yo-yo rotates as 60 cm of string unwraps.

Number of times = 0.6 ÷ (0.04 * π) = 15/π
This means the yo-yo rotates approximately 4.77 times as it moves down 60 cm.

1 time = 2π radian
Number of radians = 2π * 15/π = 30 rad
This is the total angle that the yo-yo rotated. Use the following equation to determine the time.

θ = ωi * t + ½ * α * t^2, ωi = 0
30 = ½ * 4 * t^2
t = √15
The time is approximately 3.87 seconds

Let’s convert the angular 0.08 m/s^2
Let’s use the following equation to determine the time for the yo-yo to fall 60 cm

d = vi * t + ½ * a * t^2, vi = 0
0.6 = ½ * 0.08 * t^2
t = √(0.6 ÷ 0.04) = √15
The time is approximately 3.87 seconds