ME 232 Homework Problems

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 / = -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