WORK : Figure shows a force acting on a particle at A which

moves along the path. The position vector measured from

some convenient origin O locates the particle as it passes

point A, and is the differential displacement from A to A.

The work done by the force during the displacement is

defined as

rdFdU

F

r

rd

F

The magnitude of this dot product is dU=Fdscosa, where ds

is the magnitude of . This expression may be interpreted

as the displacement multiplied by the force component

Ft=Fcos a in the direction of the displacement.

rd

With this definition of work, it should be noted that the

component Fn=Fsina normal to the displacement does no

work.

The SI units of work are those of

force (N) times displacement (m)

of N.m. This unit is given Joule (J).

Calculation of Work

During a finite movement of the point of application of a force,

the force does an amount of work equal to

In order to carry out this integration, it is necessary to know the

relations between the force components and their respective

coordinates or the relation between Ft and s.

dzFdyFdxFrdFU zyx

or

dsFU t

We consider the common linear spring of stiffness k where the force required to stretch or

compress the spring is proportional to the deformation x. We wish to determine the work

done on the body by the spring force as the body undergoes an arbitrary displacement

from an initial position x1 to a final position x2.

x1 x x2

F=kx P

F=kx

F

Undeformedposition

Stretched

x1 x x2

F=kx

F

Undeformedposition

Compressed

F=kxPx x

(a)(b)

21

2221

21

2

12

1

xxkdx.kxU

FdxU

x

x

Work done on the block by the spring is negative since spring force is in the

opposite direction of the displacement.

Work done on the block by the spring

We assume that g is constant, that is the altitude variation is sufficiently small so that the

acceleration of gravity may be considered constant.

1221

2

1

yymgmgdyU

y

y

If the body rises (perhaps due to other forces not shown), then ( y2 y1) > 0

and this work is negative. If the body falls, ( y2 y1) < 0 and the work is positive.

Work done on the body by the weight mg as the

body is displaced from an altitude y1 to a final

altitude y2 is

We now consider the work done on a particle of mass m moving along acurved path under the action of the force F, which is the resultant of allforces acting on the particle.

The work done by F during a finite movement of the particle from point 1to point 2 is

2

1

2

1

21 dsFrdFU t

When we substitute Newton’s second law , the expression becomes

2

1

2

1

21 rdamrdFU

where at is the tangential component of the acceleration. In terms of the

velocity atds=vdv. Thus the expression for work becomes

dsarda t

2

1

21

22

2

1

212

1v

v

vvmmvdvrdFU

The kinetic energy T of the particle is defined as2

2

1mvT

and is the total work which must be done on the particle to bring it from a

state of rest to a velocity v. Kinetic energy T is a scalar quantity with the units

of “N.m” or “Joule (J)” in SI units. Kinetic energy is always positive, regardless

of the direction of the velocity.

Expression for the work may be written as

TTTU 1221

which is the work-energy equation of particle. The equation states that the

total work done by all forces acting on a particle as it moves from point 1 to

point 2 equals the corresponding change in kinetic energy of the particle.

Although T is always positive, the change T may be negative, positive or zero.

We consider first the motion of a particle of mass m in close proximity to the

surface of the earth, where the gravitational attraction (weight) mg is essentially

constant. The gravitational potential energy Vg of the particle is defined as

mghVg

hmghhmgVg 12

The change in potential energy is

The second example of potential energy occurs in the deformation of an

elastic body, such as a spring. Elastic potential energy is

2

02

1kxkxdxV

x

e

21

22

2

1xxkVe

The change in elastic potential energy is

With the elastic member included in the system, we may write the work-energyequation as

where U1-2 term is for the work of all forces other than gravitational forces andspring forces.

22221111 egeg VVTUVVT or

eg VVTU 21

We may rewrite the alternative work-energy equation as

eg VVTU 21

Where E=T+Vg+Ve is the total mechanical energy of the particle. For problemswhere U1-2 term is zero, and the energy equation becomes

0E or E=constant

This equation expresses the law of conservation of dynamical energy.

The capacity of a machine is measured by the time rate at

which it can do work or deliver energy. The total work or

energy output is not a measure of this capacity since a motor,

no matter how small, can deliver a large amount of energy if

given sufficient time. Thus, the capacity of a machine is rated

by its power, which is defined as the time rate of doing

work.

Power (GÜÇ)

When a force does work an amount U, the power P will be

Power is a scalar quantity.

In SI system it has the units of N∙m/s=Joule/s=Watt (W).

MP

vFP

vdt

rd

dt

rdF

dt

dUP

F

or