3.41 Figure P3.41 illustrates a pendulum with a base that moves. The base
acceleration is $a(t)$. Derive the equation of motion in terms of $\theta$ with $a(t)$ as the
input. Neglect the mass of the rod.
3.42 The overhead trolley shown in Figure P3.42 is used to transport beams in a
factory. The beam is rectangular, with a length of $L$. It is desired to limit the
trolley horizontal acceleration $a$ so that the beam does not swing too much.
The beam starts from rest with $\theta(0) = 0$. (a) Use a small-angle approximation
and determine the maximum value of $\theta$ if $L = 3$ m and $a = 2$ m/s$^2$. Once you
obtain an answer, use it to check the validity of the small-angle approximation.
(b) Solve the problem given in part (a), using symbolic values for $L$, $a$, and $g$.
Does the answer depend on all three variables? If not, explain why.
Figure P3.40
Figure P3.41
Figure P3.42
