Angular/Rotational Kinematics. complicated problem! As a result of friction, the angular speed of a wheel c...

As a result of friction, the angular speed of a
wheel c hanges with time according to
dθ / dt = ω(initial)* e^(−a t)
,
where ω(initial) and a are constants. The angular
speed changes from an initial angular speed
of 4.89 rad/s to 3.59 rad/s in 3.92 s .

1. Determine the magnitude of the angular
acceleration after 2.22 s.
Answer in units of rad/s^2

2. How many revolutions does the wheel make
after 2.47 s ?
Answer in units of rev

3. Find the number of revolutions it makes before coming to rest.
Answer in units of rev

ωi = 4.89 rad/s
ωe = 3.59 rad/s
te = 3.92 s

dθ/dt = ωi * e^(−a*t)
(dθ/dt) / ωi = e^(−a*t)
ln( (dθ/dt) / ωi ) = -a*t
a = ln( (dθ/dt) / ωi ) / -t = ln( ωe / ωi ) / -te = 0.079

1)
d^2θ/dt^2 = -a*ωi * e^(−a*t)
d^2θ/dt^2 (t=2.22) = -0.32 rad/s^2

2)
θ = integral(dθ/dt)dt = -ωi/a * e^(−a*t) + C
θ (t=2.47) = -ωi/a * (e^(−a*2.47)-e^(0)) = 11.0 rad = 1.75 rev

3)
dθ/dt = 0 as t -> infinity
Therefore, find lim(t->inf) for θ(t)
lim(t->inf) θ(t) = -ωi/a * (e^(−a*inf)-e^(0)) = -ωi/a * (0-1) = ωi/a = 62.0 rad = 9.87 rev