As a result of friction, the angular speed of a wheel changes with time according to

Question

d(theta)/dt = w0 e^-(aplpha*t)
where w0 and alpha are constants. The angular speed changes from an initial angular speed of 4.87 rad/s to 4.27 rad/s in 33.93s.
A) Determine the magnitude of the angular acceleration after 2.46s. answer in rad/s^2
B) How many revolutions does the wheel make after 2.59 s? answer in rev
C) find the number of revolutions it makes before coming to rest, answer in rev.

any help would be appreciated, thank you in advance

Answer 1

angular speed=dTheta/dt

angular acceleration=d"theta/dt"
= -alpha*wo e^{-alpha*t)

now given: speed(t=0)=wo=4.87
speed(t=33.93)=4.87e^(-alpha*33.93) solve for alpha
take the ln of each side,
ln (4.27)=ln(4.87)-alpha*33.93
solve for alpha

for b, integrate dTheta/dt from t=0 to t=2.57

for c. rest means dTheta/dt is zero, so solve for t when dTheta/dt=0, notice that it NEVER gets to rest, so tf=infinity.