Initially, a ball has an angular velocity of5 rad/s counterclockwise. some time later, after rotating through a total angle of 5.5 radians, the ball has an angular velocity of 1.5 rad/s clockwise. (a) what is the angular acceleration? (b) what is the average angular velocity? (c) how much time did it take to change the speed

a) 2.1 rad/s² clockwise

b) 1.8 rad/s counterclockwise

c) 3.1 s

Explanation:

Take counterclockwise to be positive.

Given:

ω₀ = 5.0 rad/s

ω = -1.5 rad/s

Δθ = 5.5 rad

a) Find α

ω² = ω₀² + 2αΔθ

(-1.5 rad/s)² = (5.0 rad/s)² + 2α (5.5 rad)

α = -2.1 rad/s²

b) Find ω_avg

ω_avg = ½ (ω + ω₀)

ω_avg = ½ (-1.5 rad/s + 5.0 rad/s)

ω_avg = 1.8 rad/s

c) Find t

Δθ = ½ (ω + ω₀) t

(5.5 rad) = ½ (-1.5 rad/s + 5.0 rad/s) t

t = 3.1 s

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