How long will it take for a basketball spinning on someone’s finger to stop if it undergoes and angular acceleration of -0.15 rad/s and is traveling at 27 rad/s initially? How many revolutions will it make?

The basketball's angular velocity ω at time t is

ω = 27 rad/s + (-0.15 rad/s²) t

It comes to a stop when ω = 0, which happens for

0 = 27 rad/s - (0.15 rad/s²) t

t = (27 rad/s) / (0.15 rad/s²)

t = 180 s

In this time, the ball would undergoes an angular displacement θ of

θ = (27 rad/s) t + 1/2 (-0.15 rad/s²) t ²

Plug in t = 180 s and solve for θ :

θ = (27 rad/s) (180 s) - 1/2 (0.15 rad/s²) (180 s)² = 2430 rad

One complete revolution corresponds to a turn of 2π rad, so the ball makes

(2430 rad) / (2π rad/rev) ≈ 386.747 rev

or about 387 revolutions as it slows down.