A 24 inch string is wrapped around the 1/8 inch radius axis of a yo yo. Starting from rest the yo yo descends with a constant angular acceleration of 4 rad/second squared. How long will it take the string to completely unwind?
Basic equation required:
Arc = rθ
S=ut+(1/2)at²
4 rad/s²
≡ 4r inch/s²
=(1/2) inc/s²
initial velocity, u =0
a = 0.5 in/s²
24 inches = 0*t + (1/2)at²
t=sqrt(2*24/0.5)
=sqrt(96)
=9.8 s
To find the time it takes for the string to completely unwind, we need to consider the angular displacement and angular velocity of the yo-yo.
First, let's find the angular displacement. The string is wrapped around the 1/8 inch radius axis, which means the circumference of the axis is 2 * π * (1/8) = π/4 inches. The total length of the string is 24 inches, which means it will make (24 / (π/4)) = 96 rotations.
Next, we need to find the final angular velocity. We can use the equation:
ω^2 = ω0^2 + 2αθ
where:
- ω is the final angular velocity
- ω0 is the initial angular velocity (which is 0 in this case since the yo-yo starts from rest)
- α is the angular acceleration (given as 4 rad/s^2)
- θ is the angular displacement (96 rotations converted to radians)
Converting rotations to radians:
θ = 96 * 2π = 192π radians
Plugging the values into the equation:
ω^2 = 0 + 2 * 4 * 192π
ω^2 = 3072π
ω ≈ 55.16 rad/s
Finally, we can calculate the time it takes for the string to unwind using the equation:
t = ω / α
t = 55.16 / 4
t ≈ 13.79 seconds
So, it will take approximately 13.79 seconds for the string to completely unwind.