Page 352 - Dynamics of Mechanical Systems

0593_C10_fm Page 333 Monday, May 6, 2002 2:57 PM

Introduction to Energy Methods 333

O P(m)

h

FIGURE 10.7.1
A particle released from rest in a S
gravitational field.

Because gravity is the only force applied to P and because the path of movement of P
is parallel to the weight force through a distance h, the work done W is simply:

W = mgh (10.7.1)

Because P is released from rest, its initial kinetic energy is zero. When P reaches S, its
kinetic energy may be expressed as:

K = 1 mv 2 (10.7.2)
2

Then, from the work–energy principle of Eq. (10.6.5), we have:

W = ∆ K or mgh = 1 mv 2 (10.7.3)
2

Solving for v, we obtain the familiar result:

v = 2 gh (10.7.4)

10.8 Elementary Example: The Simple Pendulum

Consider next the simple pendulum depicted in Figure 10.8.1 where the mass m of the
pendulum is concentrated in the bob P which is supported by a pinned, massless rod of
length as shown. Let θ measure the inclination of the pendulum to the vertical. Suppose
the pendulum is held in a horizontal position and released from rest. The objective is to
determine the speed v of P as it passes through the equilibrium position θ = 0. If we
consider a free-body diagram of P as in Figure 10.8.2, we see that of the two forces applied
to P the tension of the connecting rod does no work on P because its direction is perpen-
dicular to the movement of P. From Eq. (10.2.18), we see that the work W done by the
weight force is:

W = mgl (10.8.1)