To determine the number of revolutions the wheel makes in the given scenario, we need to calculate the angular displacement of the wheel during the acceleration period and then convert it to revolutions.
The angular velocity, final angular velocity (ωf), is given as 30 rad/s, and the initial angular velocity (ωi) is 0 rad/s because the wheel starts from rest. The time taken (t) is given as 6 seconds.
Using the equation for angular acceleration:
ωf = ωi + α * t
Since ωi = 0 rad/s, we can simplify the equation to:
ωf = α * t
Solving for α (angular acceleration):
α = ωf / t = 30 rad/s / 6 s = 5 rad/s²
Now, we can calculate the angular displacement (θ) during the acceleration period using the equation:
θ = ωi * t + (1/2) * α * t²
Since ωi = 0 rad/s:
θ = (1/2) * α * t² = (1/2) * 5 rad/s² * (6 s)² = 90 rad
To convert the angular displacement to revolutions, we need to know the relationship between radians and revolutions. There are 2π radians in one revolution.
1 revolution = 2π radians
Therefore:
90 rad * (1 revolution / 2π rad) ≈ 14.32 revolutions
So, the wheel makes approximately 14.32 revolutions during the acceleration period.