Problem 2/132 Page 72, Engineering Mechanics - Dynamics, Meriam and Kraige, 4th Edition
Given: A rod of varying length and making a varying angle with the x axis. The length and its first two time derivatives at the instant of interest are given below. The angle and its first two time derivatives are given below.
Find: The velocity and acceleration of the end of the rod at the instant described.
Parameter values:
r = 31 ft
dr/dt = 1.5 ft/s
d2r/dt2 = -4 ft/s2
q = 20 degrees = p/9
radians
dq/dt = 5 degrees/s = p/36
radians/s
d2q/dt2 = 2 degrees/s2
= p/90 radians/s2
Recalling the definitions of velocity and acceleration in polar coordinates:
v = vr er + vq eq
vr = dr/dt = 1.5 ft/s
vq = r dq/dt
= 2.71 ft/s
v = 1.5 ft/s er + 2.71 ft/s eq
a = ar er + aq eq
ar = d2r/dt2 - r (dq/dt)2
= -4.24 ft/s2
aq = r d2q/dt2
+ 2 dr/dt dq/dt = 1.344 ft/s2
a = -4.24 ft/s2 er + 1.344 ft/s2 eq