Question

A rotating wheel requires 5 s to rotate 38 revolutions. Its angular velocity at end of time interval 5-s is 79 rad/s. What is the constant angular acceleration of the wheel?
Show your work please

Answer 1

"Rotating wheel" is meant to indcate that the wheel is already rotating at the start. Denote the initial angular velocity by ω₀. Then the angular displacement θ at time t is

θ = ω₀t + 1/2 αt ²

while the angular velocity ω is

ω = ω₀ + αt

It takes 5 s for the wheel to rotate 38 times, or turn a total of (2π rad/rev) (38 rev) = 76π rad, as well as to reach an angular velocity of 79 rad/s, so that

76π rad = ω₀ (5 s) + 1/2 α (5 s)²

79 rad/s = ω₀ + α (5 s)

Solve the second equation for ω₀ and substitute into the first equation, then solve for α :

ω₀ = 79 rad/s - α (5 s)

==>   76π rad = (79 rad/s - α (5 s)) (5 s) + 1/2 α (5 s)²

==>   76π rad = (79 rad/s) (5 s) - 1/2 α (5 s)²

==>   1/2 α (5 s)² = (79 rad/s) (5 s) - 76π rad

==>   α = ((79 rad/s) (5 s) - 76π rad) / (1/2 (5 s)²)

==>   α ≈ 12.5 rad/s²