A solid disk with a mass of 100kg and a radius of 0.2m, turns clockwise through an angular displacement 10 rad when starting from rest to attain its maximum angular speed, of 1 revolution every 0.5s. What is the angular acceleration that the wheel experienced in order to attain maximum speed? How much time did it take for the disk to go from rest to maximum angular speed?
To find the angular acceleration of the disk, you can use the equation
angular acceleration (α) = (final angular velocity (ω) - initial angular velocity (ω₀)) / time taken (t)
Given:
- Final angular velocity (ω): 1 revolution every 0.5s. Since there are 2π radians in a revolution, we can convert this to radians per second by multiplying by 2π: (1 rev/0.5s) * (2π rad/1 rev) = 4π rad/s.
- Initially, the disk is at rest, so the initial angular velocity (ω₀) is 0 rad/s.
Substituting the values into the equation, we have:
α = (4π rad/s - 0 rad/s) / t
Now, let's find the time taken (t) for the disk to go from rest to maximum angular speed. The angular displacement (θ) of the disk is given as 10 rad.
We can use the equation for angular displacement to find the time taken:
θ = ω₀t + (1/2)αt²
Simplifying the equation, we get:
10 rad = 0 rad/s * t + (1/2)αt²
10 rad = (1/2)αt²
Now, we can solve for t:
t² = (20 rad) / α
t = sqrt((20 rad) / α)
Now you can substitute this value of t into the equation for angular acceleration to calculate the angular acceleration (α).