Work, Energy and Power
1. Work
The concept of work is introduced as a measure of the action of a force
~F on a body in a given displacement.
Consider a particle moving on a curve (C) under the action of a force
field ~F. Suppose that the position vector ~r of the particle has undergone a
very small change d~r (Figure 1).
Figure 1. As the displacement on the curve tends to zero, it approaches d~r
We define the elementary work dw done by the force ~F on the particle
moving on the curve (C) through a displacement d~r by:
dw = ~F · d~r
(1)
The total work done by the force ~F in any finite displacement from a
point A to a point B along a trajectory (C) is the summation of the elementary works dw, hence:
ˆ B
~F · d~r
WA → B =
(2)
A
2. Kinetic energy and work - energy theorem
The basic dynamic characteristics of the particle motion are: linear momentum and kinetic energy. The linear momentum being the vector quantity:
~p = m · ~v
(3)
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We define the kinetic energy of the particle by the scalar quantity:
T = 12 m · v2
(4)
2.1. Work-Energy theorem
For a particle of mass m and velocity ~v, its kinetic energy is:
T = 12 m · v2 = 12 m · ~v · ~v
(5)
Then the work done on the particle from point A to point B is given by
equation 2:
ˆ B
~F · d~r
WA → B =
A
Where ~F = ∑ F~ext , that is ~F is the sum of all external force excerted on
the particle.
Developing the expression of the work (equation 2) and using Newton’s second law (~F = m ddt~v ):
ˆ B
ˆ B
ˆ B
ˆ B
~
~
d
v
d~r
d
v
~F · d~r =
· d~r =
WA → B =
m · d~r =
m d~v
m
dt
dt
dt
A
A
A
A
But
d~r
dt
= ~v, then:
ˆ
WA → B =
v~2
v~1
m~v · d~v =
h
iv~
2 2
1
~
m
v
2
v~1
= 21 mv22 − 12 mv21
(6)
Therefore
WA→ B = T2 − T1
(7)
Keeping in mind that WA→ B is the algebraic sum of all the external
works, i.e. ∑ Wext , done by the external forces, then we can write:
∑ Wext = T2 − T1
(8)
Therefore, the change of kinetic energy of a particle during a certain interval
of time is equal to the algebraic sum of all external works done by all the external
forces on the particle during that interval of time.
3. Power
The power is the rate at which work is done. That is, the work done by
a force per unit time. Its unit in SI system is the Watt "W" (Watt = J/s =
kg · m2 /s3 ).
3.1. Average Power
If a work ∆W is done during a time interval ∆t, then the a verage power
is given by:
∆W
(9)
Pav =
∆t
2
3.2. Instantaneous Power
If an elementary work dw is done during a very small time interval dt,
then the instantaneous power is given by:
dW
∆W
=
dt
∆t→0 ∆t
Pinst = lim
(10)
Which can be considered as the limit of the average power as ∆t tends
to zero.
From now onwards, for simplicity, we will write P instead of Pinst .
The power can be expressed in terms of the force ~F acting of the particle
and the velocity of the particle ~v:
~
d F · d~r
dW
=
(11)
P=
dt
dt
If the force ~F is constant, then:
~F · d~r
d
dW
d~r ~
P=
=
= ~F ·
= F · ~v
dt
dt
dt
(12)
4. Conservative force fields and conservation of energy:
A force field ~F is said to be conservative if one of the following conditions is satisfied:
1.
2.
3.
4.
~ × ~F = ~0
∇
´B
The integral A ~F · d~r is independent of the path followed from A to B.
¸
The integral ~F · d~r along any simple and closed curve is zero.
~ U ( U is called potential
There exists a function U such that ~F = −∇
function – in this case it represents the potential energy). Then
ˆ
B
ˆ
B
~ U · d~r
~F · d~r =
−∇
A
A
ˆ B ∂U~ ∂U~ ∂U~
~
~
~
i+
j+
k · dxi + dy j + dzk
=
−
∂x
∂y
∂z
A
ˆ B
ˆ B ∂U
∂U
∂U
=
−
dx +
dy +
dz =
−dU
∂x
∂y
∂z
A
A
= U A − UB
(13)
WA → B =
⇒ WA → B
Which is independent of the path from A to B.
So, in the case of conservative force:
W = U1 − U2
(14)
W = T2 − T1
(15)
and:
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Combining equation 14 and equation 15, we obtain in the case of conservative forces:
T2 − T1 = U1 − U2
⇒ T1 + U1 = T2 + U2
(16)
(17)
Therefore for conservative forces the quantity T + U remains constant
throughout the motion. So we can write:
T+U = E
(18)
E is called the mechanical energy and is only defined for conservative
forces.
Physics Zone by Farid Minawi
www.physics-zone.com
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