A wheel starts from rest and in 10.65 s is rotating with an angular speed of 5.810 π rad/s.
(a) Find the magnitude of the constant angular acceleration of the wheel. rad/s^2
(b) Through what angle does the wheel move in 6.085 s? rad
(a) clearly the acceleration is
(5.810π rad/s)/(10.65s) = 1.714 rad/s^2
(b) starting from when? t=0, or after reaching the final speed?
just as with linear motion, the "distance" (total radians) is 1/2 at^2
To find the magnitude of the constant angular acceleration of the wheel (a), we can use the formula:
ω = ω₀ + αt
where:
ω is the final angular speed (5.810π rad/s)
ω₀ is the initial angular speed (0 rad/s since the wheel starts from rest)
α is the angular acceleration (unknown)
t is the time (10.65 s)
Rearranging the formula to solve for α, we have:
α = (ω - ω₀) / t
Substituting the given values, we get:
α = (5.810π rad/s - 0 rad/s) / 10.65 s
Now, let's calculate α:
α = 5.810π rad/s / 10.65 s
Using a calculator, we can find that α ≈ 1.724 rad/s².
Therefore, the magnitude of the constant angular acceleration of the wheel is approximately 1.724 rad/s².
Now, let's move on to part (b) of the question.
To find the angle through which the wheel moves in 6.085 s (θ), we can use the formula:
θ = ω₀t + (1/2)αt²
where:
θ is the angle (unknown)
ω₀ is the initial angular speed (0 rad/s since the wheel starts from rest)
α is the angular acceleration (1.724 rad/s², as calculated previously)
t is the time (6.085 s)
Substituting the given values, we have:
θ = 0 rad/s * 6.085 s + (1/2) * 1.724 rad/s² * (6.085 s)²
Now, let's calculate θ:
θ = 0 rad + 0.5 * 1.724 rad/s² * (6.085 s)²
Using a calculator, we find that θ ≈ 118.98 rad.
Therefore, the wheel moves through an angle of approximately 118.98 rad in 6.085 s.