Work and Potential Energy
Work is a scalar quantity. It is the product of a force and the corresponding
displacement. Potential energy is the capacity of a system to do work on
another system. These concepts are advantageous in the analysis of equilibrium
of complex systems, in dynamics, and in mechanics of materials.
Work of a Force
The work U of a constant force F is
U = Fs
where s = displacement of a body in the direction of the vector F.
For a displacement along an arbitrary path from point 1 to 2, with dr tangent
to the path,
In theory, there is no work when:
1. A force is acting on a ?xed, rigid body (dr = 0, dU = 0).
2. A force acts perpendicular to the displacement (F ? dr = 0).
Work of a Couple
A couple of magnitude M does work
U = M?
where ? = angular displacement (radians) in the same plane in which the
couple is acting. In a rotation from angular position ? to ?,
Virtual Work
The concept of virtual work (through imaginary, in?nitesimal displacements
within the constraints of a system) is useful to analyze the equilibrium of
complex systems. The virtual work of a force F or moment M is
expressed as
?U = F. ?r
?U = M. ??
There is equilibrium if
where the subscripts refer to individual forces or couples and the corresponding
displacements, ignoring frictional effects.
Mechanical Ef?ciency of Real Systems
Real mechanical systems operate with frictional losses, so
input work = useful work + work of friction
(output work)
The mechanical ef?ciency ? of a machine is
Gravitational Work and Potential Energy
The potential of a body of weight W to do work because of its relative
height h with respect to an arbitrary level is de?ned as its potential energy.
If h is the vertical (y) distance between level 1 and a lower level 2,
the work of weight W in descending is
= potential energy of the body at level 1 with respect to level 2
The work of weight W in rising from level 2 to level 1 is
= potential energy of the body at level 2 with respect to level 1
Elastic Potential Energy
The potential energy of elastic members is another common form of
potential energy in engineering mechanics. For a linearly deforming helical
spring, the axial force F and displacement x are related by the spring
constant k,
F = kx (similarly, M = k? for a torsion spring)
The work U of a force F on an initially undeformed spring is
In the general case, deforming the spring from position x1 to x2,
Notation for Potential Energy
The change in the potential energy V of a system is
U = ??V
Note that negative work is done by a system while its own potential energy
is increased by the action of an external force or moment. The external
agent does positive work at the same time since it acts in the same direction
as the resulting displacement.
Potential Energy at Equilibrium
For equilibrium of a system,
where q = an independent coordinate along which there is possibility of
displacement. For a system with n degrees of freedom,
Equilibrium is stable if (d2V/dq2) > 0.
Equilibrium is unstable if (d2V/dq2) < 0.
Equilibrium is neutral only if all derivatives of V are zero. In cases of complex
con?gurations, evaluate derivatives of higher order as well.
Moments of Inertia
The topics of inertia are related to the methods of ?rst moments.
They are traditionally presented in statics in preparation for application
in dynamics or mechanics of materials.
Moments of Inertia of a Mass
The moment of inertia dIx of an elemental mass dM about the x axis
(Figure 1.2.28) is de?ned as
dlx = r2 dM = (y2 + z2) dM
where r is the nearest distance from dM to the x axis. The moments of inertia
of a body about the three coordinate axes are
FIGURE 1.2.28 Mass element dM in xyz coordinates.
Radius of Gyration. The radius of gyration rg is de?ned by rg = ?lx M,
and similarly for any other axis. It is based on the concept of the body of mass M
being replaced by a point mass M (same mass) at a distance rg from a given axis.
A thin strip or shell with all mass essentially at a constant distance rg from the
axis of reference is equivalent to a point mass for some analyses.
Moment of Inertia of an Area
The moment of inertia of an elemental area dA about the x axis
(Figure 1.2.29) is de?ned as
dIx = y2 dA
where y is the nearest distance from dA to the x axis. The moments of inertia
(second moments) of the area A about the x and y axes
(because A is in the xy plane) are
FIGURE 1.2.29 Area A in the xy plane.
The radius of gyration of an area is de?ned the same way as
it is for a mass: 
Polar Moment of Inertia of an Area
The polar moment of inertia is de?ned with respect to an axis
perpendicular to the area considered. In Figure 1.2.29 this may be
the z axis. The polar moment of inertia in this case is
Parallel-Axis Transformations of Moments of Inertia
It is often convenient to ?rst calculate the moment of inertia about a
centroidal axis and then transform this with respect to a parallel axis.
The formulas for the transformations are
where I or JO = moment of inertia of M or A about any line 
IC or JC = moment of inertia of M or A about a line through the mass
center or centroid and parallel to
d = nearest distance between the parallel lines
Note that one of the two axes in each equation must be a centroidal axis.
Products of Inertia
The products of inertia for areas and masses and the corresponding
parallel-axis formulas are de?ned in similar patterns. Using notations
in accordance with the preceding formulas, products of inertia are
Parallel-axis formulas are
Notes: The moment of inertia is always positive. The product of inertia
may be positive, negative, or zero; it is zero if x or y (or both) is an axis of
symmetry of the area. Transformations of known moments and
product of inertia to axes that are inclined to the original set of axes are
possible but not covered here. These transformations are useful for
determining the principal (maximum and minimum) moments of inertia
and the principal axes when the area or body has no symmetry.
The principal moments of inertia for objects of simple shape are
available in many texts.
By : E-book Mechanical_Engineering_Handbook
