Question

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

Answer 1

Answer:

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

Answer 2

The angular acceleration is  -2.1 rad/s². The average angular velocity is  1.8 rad/s. The time taken is 3.1 s.

Circular motion

If we assign the  counterclockwise to be positive and vice versa.

Then we find the angular acceleration:

ω₀ = 5.0 rad/s

ω = -1.5 rad/s

Δθ = 5.5 rad

From;

ω² = ω₀² + 2αΔθ

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

α = -2.1 rad/s²

Next we obtain the average angular velocity from;

ωavg = ½ (ω + ω₀)

ω = final angular velocity

ω₀ = initial angular velocity

ωavg = ½ (-1.5 rad/s + 5.0 rad/s)

ωavg = 1.8 rad/s

To obtain the time taken;

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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