Final answer:
The angular acceleration of the gyroscope is 80 rad/s², and it makes approximately 1.02 revolutions in the process. The correct multiple-choice answer according to the calculation is 80 rad/s² for angular acceleration, with the number of revolutions being slightly over one, which is not reflected in the provided answer options.
Step-by-step explanation:
Angular Acceleration and Revolutions of a Gyroscope
A gyroscope accelerated from rest to 32 rad/s in 0.40 s can have its angular acceleration calculated using the formula:
Angular acceleration (α) = (ω - ω0) / t
where:
ω = final angular velocity (32 rad/s),
ω0 = initial angular velocity (0 rad/s, because it starts from rest), and
t = time taken to reach final angular velocity (0.40 s).
Plugging the values into the formula gives α = 32 rad/s / 0.40 s = 80 rad/s², which is answer choice 'b'.
To determine the number of revolutions, we need to calculate the angular displacement (θ) during the acceleration using the formula:
θ = ω0 * t + (1/2) * α * t²
Since the initial angular velocity (ω0) is zero, the formula simplifies to:
θ = (1/2) * α * t² = (1/2) * 80 rad/s² * (0.40 s)² = 6.4 rad
To convert radians to revolutions, divide by 2π:
Revolutions = θ / (2 * π) = 6.4 rad / (2 * π) ≈ 1.02 revolutions
Thus, the closest answer, without rounding up, would be 'a', which says 1 revolution, but since this option is not available, and keeping in mind the instructions to ignore any typos or irrelevant parts of the question, there's likely an error in the provided answer choices. The actual number of revolutions is slightly more than one, none of the options 'a' 8 revolutions, 'b' 10 revolutions, 'c' 12 revolutions, or 'd' 16 revolutions is correct.