Modern Engineering Thermodynamics

20 CHAPTER 1: The Beginning
EXAMPLE 1.6 (Continued )
17. A cubical box is measured to three significant figures with a ruler and found to be 1.21 ft on one side, 1.22 ft on
another side, and 1.20 ft on the third side. Determine the volume of the box to the proper number of significant figures.
Answer: 1.77 ft 3 .
18. A shaft is measured with a micrometer and found to have a diameter of 1.735 inches (to four significant figures).
Determine the circumference of the shaft to the proper number of significant figures. Answer: 5.451 inches.
1.11 POTENTIAL AND KINETIC ENERGIES
In classical physics, the term potential energy usually refers to gravitational potential energy and represents the
work done against the local gravitational force in changing the position of an object. It depends on the mass m
of the object and its height Z above a reference level, written as
where k 1 is defined by Eqs. (1.6) and (1.7) as 1/g c (see Table 1.2).
Potential energy ¼ PE ¼ k 1 mgZ = mgZ
g c
(1.10)
Kinetic energy represents the work associated with changing the motion of an object and can occur in two forms:
translational and rotational. The total kinetic energy of an object is the sum of both forms of its kinetic energy. The
translational kinetic energy of an object is the kinetic energy resulting from a translation velocity V, written as
mV
Translational kinetic energy = ðKEÞ trans
= k 2
1
2
= mV 2
2g c
(1.11)
The rotational kinetic energy of an object is the kinetic energy resulting from a rotation about some axis with an
angular velocity ω, written as
Rotational kinetic energy = ðKEÞ rot
= k 1
Iω 2
2 = Iω2
2g c
(1.12)
where I is the mass moment of inertia of the object about the axis of rotation.
The mass moment of inertia of an object is the integral of a mass element dm located at a radial distance r from
the axis of rotation:
Z
I = r 2 dm (1.13)
Table 1.7 provides equations for the mass moment of inertia of various common geometrical shapes with a total
mass m.
Table 1.7 Mass Moments of Inertia of Various Common Shapes
x
x
Slender circular rod
I x = mL 2 /2
L
Thin rectangular plate
I x = m(a 2 + b 2 )/12
I y = ma 2 /12
I z = mb 2 /12
y
z
x
x
Solid rectangular prism
I x = m(a 2 + b 2 )/12
I y = m(a 2 + L 2 )/12
I z = m(b 2 + L 2 )/12
y
z
Thin disk
I x = mR 2 /2
I y = I z = mR 2 /4
y
R
z