Problem 5/6 Page 334, Engineering Mechanics - Dynamics, Meriam and Kraige, 4th Edition
Given: Angular deceleration proportional to the square of the angular speed.
Find: Determine the angular displacement and time elapsed when the angular velocity has been reduced to one third of its original value.
a = - k w2
k = 0.1
w0 = 12 rad/s = Original speed.
The angular acceleration is the time derivative of the angular speed.
a = dw
/ dt = - k w2
Separating variables and integrating:
d w / w2 = - k
dt
-1 / w = -k t + C
At t=0, the angular speed is known to be w0.
Thus:
-1 / w0 = C
-1 / w = -k t - 1 / w0
Setting the final speed to 1/3 of w0:
-3 / w0 = - k tf - 1 / w0
tf = 1/k ( 2 / w0) = 1.667 s
To determine the angular displacement, recall:
a = dw
/ dt = dw / dq
dq / dt = dw
/ d q w = -
k w2
Separating variables and integrating:
dw / w
= -k dq
ln (w / D) = - k q
When the angular displacement is zero, the angular speed is know to
be w0. Thus:
ln ( w0 / D ) = 0
D = w0
ln (w /w0 ) =
- k q
Evaluating when the angular speed has been reduced to 1/3 of its original
value:
ln ( 1/3 ) = - k qf
qf = 1/k ln(3) = 10.99 rad