15. A wheel rotates with a constant angular acceleration of 3.5 rad/s². (a) If the angular speed
of the wheel is 2 rad's at t=0, through what angular displacement does the wheel rotate in 2
s?
b) Through how many revolutions has the wheel turned during this time interval?
c) What is the angular speed of the wheel at = 2 s?
(a) To find the angular displacement, we can use the equation:
θ = ω₀t + (1/2)αt²
where θ is the angular displacement, ω₀ is the initial angular speed, α is the angular acceleration, and t is the time.
Substituting the given values:
θ = (2 rad/s)(2 s) + (1/2)(3.5 rad/s²)(2 s)²
θ = 4 rad + 7 rad
θ = 11 rad
Therefore, the wheel rotates through an angular displacement of 11 radians in 2 seconds.
(b) To find the number of revolutions the wheel has turned, we need to convert the angular displacement from radians to revolutions. One revolution is equal to 2π radians.
Number of revolutions = (11 rad) / (2π rad/rev)
Number of revolutions ≈ 1.75 revolutions
Therefore, the wheel has turned approximately 1.75 revolutions during this time interval.
(c) To find the angular speed at t = 2 s, we can use the equation:
ω = ω₀ + αt
Substituting the given values:
ω = (2 rad/s) + (3.5 rad/s²)(2 s)
ω ≈ 9 rad/s
Therefore, the angular speed of the wheel at t = 2 s is approximately 9 rad/s.