Chapter 4 - UCSB HEP

WORK AND ENERGY
to assure that the direction of the velocity is always tangential to
the predetermined path. Hence, constraint forces change only
the direction of v and do no w0rk.l
Example 4.6
The Inverted Pendulum
A pendvlwm consists of a light rigid rod of length 2, pivoted at one end
and with mass m attached at the other end. The pendulum is released
from rest at angle 40, as shown. What is the velocity of m when the
rod is at angle 4
The work-energy theorem gives
Since vo = 0, we have
To evaluate W+,+,, the work done as the bob swings from do to 4, we
examine the force diagram. dr lies along the circle of radius l. The
I
,' forces acting arc gravity, directed down, and the farce of the rod, N.
Since N lies along the radius, N dr = 0, and N does no work. The work
done by gravity is
= mgl sin p dp
where we have used ldrl = 1 dt$.
FV+,+. = 1': mgl sin 4 dd
= '-mgl cos # 1'60
= mgl (cos +o - cos 4).
The speed at 4 is
v(4) = [ZgE (cos &, - cos 4)]*.
The maximum velocity is obtained by letting the pendulum fall from the
top, $o = 0, to the bottom. 6, = T:
v,,,
= z(g1)).
t
1 We can prove that constraint forces do no work as follows. Suppose that the
constraint force Fonahaint changes the velocity by an amount Av, in time At.
Av, is perpendicular to the instantaneous velocity v. The work done by Fm,,t,sint
is Fmnmwsist Ar = m(Av,/Al) * (v At) = mAv, * r = 0.