Modern Engineering Thermodynamics

48 CHAPTER 2: Thermodynamic Concepts
EXAMPLE 2.3 (Continued )
5. Develop a balance equation for the conservation of electric charge in a system. Answer: The conservation of charge
balance equation is
Net electric charge
Net change in electric
=
transported into the system charge in ðor onÞ the system
From the resulting conservation of energy and momentum balance equations developed in Example 2.3 and
its Exercise 1, we can investigate the technology of the ballistic pendulum shown in Figure 2.10. The ballistic
pendulum was developed in 1740 by the English mathematician and engineer Benjamin Robins (1707–1751)
and operates on the principle that the deflection of the pendulum after impact is directly proportional to
the projectile’s impact velocity. Since the projectile is imbedded in the pendulum after impact, we choose to
view this as a closed system consisting of the projectile and the pendulum. Since the system is closed, there
can be no mass transport across the system’s boundaries; and because momentum transport requires the mass
to cross the system boundary, there is also no momentum transport in this system. Then, the conservation
of momentum equation for a closed system reduces to {Net change in momentum of the system} = 0. Therefore,
the initial momentum of the projectile/pendulum system must equal to the final momentum of this
system, or
m projectile V projectile
initial =
m
projectile⁡ + m pendulum⁡ Vpendulum
After impact, the pendulum/projectile system swings through an angle θ, raising the center of gravity by
an amount h = R(1 – cos θ). Since the initial kinetic energy and the final potential energy of the
pendulum/projectile system must be equal, we can write
or
m projectile + m pendulum
!
Vpendulum
2
2g c
final
gh
= m projectile + m pendulum⁡ g c
V pendulum = ½2ghŠ 1/2 = ½2gRð1 − cos⁡yÞŠ 1/2
Then, by combining these equations, we can determine the impact velocity of the projectile as
V projectile =
m
projectile + m pendulum
m projectile
V pendulum =
1 + m
pendulum
½2gRð1 − cos yÞŠ 1/2 (2.19)
m projectile⁡
θ
R
Projectile of mass m projectile
Pendulum of mass m pendulum
FIGURE 2.10
The operation of a ballistic pendulum.