Work and Energy Chapter IV
Work and Energy
The objective of this chapter is to introduce the energy tools used in mechanics to solve
problems. Indeed, sometimes the fundamental principle of dynamics is not sufficient or
not appropriate to reach a solution.
- Introduction
- Work Done by a Force
Work done by a force measures the energy transfer when the force causes an object to
move.
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Work and Energy Chapter IV
A- Work done by a constant force
work done by a constant force 𝐹
→
during a rectilinear displacement 𝐴𝐵, is defined as
the scalar product of the force 𝐹
→
and the displacement 𝐴𝐵.
𝑊𝐴𝐵(𝐹
Ԧ
) = 𝐹
Ԧ. 𝐴𝐵
𝑊𝐴𝐵 𝐹
Ԧ
= 𝐹 𝐴𝐵 𝑐𝑜𝑠 𝛼
Where :
𝑊𝐴𝐵 𝐹
Ԧ
: is the work done in joules (J).
𝐴𝐵 : is the displacement of the object in meters (m).
: Angle between 𝐹
→
and 𝐴𝐵.
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Work and Energy Chapter IV
b. Nature of work
= 0
90o
✓ Positive work
𝐹
→
motive force 0 ≤ 𝛼 <
𝜋
2
𝑐𝑜𝑠 𝛼 > 0
𝑊 𝐹
Ԧ
= 𝐹 𝐴𝐵 𝑐𝑜𝑠 𝛼 > 0 positive work
✓ Negative work
𝐹
→
resistive force
𝜋
2
< 𝛼 ≤ 𝜋 𝑐𝑜𝑠 𝛼 < 0
𝑊 𝐹
Ԧ
= 𝐹 𝐴𝐵 𝑐𝑜𝑠 𝛼 < 0 negative work
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Work and Energy Chapter IV
= 90o
𝐹
→
force with zero net work 𝛼 = 90°
𝑊 𝐹
Ԧ
= 𝑜 No work done
✓ No Work Done
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Work and Energy Chapter IV
Exercise 1
A body weighing 4 kg ascends a ramp inclined at 30o over a distance of 15 m.
The driving force is F = 30 N.
Calculate the work done by each force acting on this body. Given static and
dynamic friction coefficients, respectively μs = 0.4 and μd = 0.2.
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Work and Energy Chapter IV
B- Work of a variable force
When force is variable, the calculation of work for this variable force involves first
defining the increment of work, denoted as dW, done by a force 𝐹
Ԧ
acting through an
infinitesimal displacement 𝑑𝑟
Ԧ.
Vectors used to define
work.
B
A
Path
𝑑𝑊 = 𝐹
Ԧ ⋅ 𝑑𝑟
Ԧ
= 𝐹 𝑑𝑟 cos 𝛼
The total work done by the force during the displacement
from point A to point B (refer to Figure opposite) is
expressed as:
𝑊𝐴𝐵 𝐹
→
= න
𝐴
𝐵
𝐹
→
⋅ 𝑑𝑟
→
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Work and Energy Chapter IV
Exercise 2:
A body m is subjected to a force 𝐹
→
,
moving along the trajectory OABCO,
as shown in figure opposite.
Given that: 𝑭
→
= −𝒚𝒊
Ԧ + 𝒙𝒋
Ԧ
,
Calculate the work done by this force for the body to move
from O → A→B→ C→O.
0,5 1,0 1,5 2,0
0,0
0,5
1,0
1,5
2,0
A
B
O
y=x
y=x
2
/2
x(m)
y(m)
C
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Work and Energy Chapter IV
Concept of Power: The power p of a force 𝐹
→
represents the work done by this
force per unit of time. In simple terms, power measures how quickly work is
done.
Average power
𝑝 =
𝑊(𝐹
Ԧ
)
𝛥𝑡
Instantaneous power :
𝑝 =
𝑑𝑊(𝐹
Ԧ)
𝑑𝑡
=
𝐹
Ԧ
𝑑𝑟
Ԧ
𝑑𝑡
= 𝐹
Ԧ
ณ
𝑑𝑟
Ԧ
𝑑𝑡
𝑉
⇒ 𝑃 = 𝐹
Ԧ. 𝑉
The SI unit for power is the watt (W), equivalent to one joule per second.
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Work and Energy Chapter IV
Kinetic energy Ek
, is the energy possessed by an object due to its motion.
Mathematically, it is expressed as:
𝑬𝑲 =
𝟏
𝟐
𝒎𝑽
𝟐
Where:
Ek
is the kinetic energy,
m is the mass of the object,
and V is its velocity.
- Kinetic Energy
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Work and Energy Chapter IV
The Kinetic Energy Theorem
𝐸𝑘 𝐵 − 𝐸𝑘 𝐴 = 𝑊𝐴𝐵 𝐹
Ԧ𝑒𝑥𝑡
Demonstration
Consider a particle moving under the action of a resultant force 𝐹
→
between A and B. The work done by 𝐹
→
for an elemental displacement 𝑑𝑟
→
is :
𝑑𝑊 = 𝐹
→
⋅ 𝑑𝑟
→
(∗)
We apply the second law of Newton : 𝐹
Ԧ
= 𝑚𝑎
Ԧ 𝑎𝑛𝑑 𝑎
Ԧ
=
𝑑𝑉
𝑑𝑡
⇒ 𝐹
Ԧ
= 𝑚
𝑑𝑉
𝑑𝑡
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Work and Energy Chapter IV
We substitute the expression for 𝐹
→
into equation (*):
(∗) ⇒ 𝑑𝑊 = 𝑚
𝑑𝑉
𝑑𝑡 ⋅ 𝑑𝑟
→
𝑎𝑛𝑑 𝑉 =
𝑑𝑟
→
𝑑𝑡 ⇒ 𝑑𝑟
→
= 𝑉𝑑𝑡
𝑑𝑊 = 𝑚
𝑑𝑉
𝑑𝑡 𝑉𝑑𝑡 ⇒ 𝑑𝑊 = 𝑚𝑉𝑑𝑉
The work done by 𝐹
→
from A to B :
��
𝐵
𝑑𝑊(𝐹
Ԧ
�𝑉� = (
𝑉𝐵 𝑚 𝑉
ถ
𝑑𝑉
𝑉𝑑𝑉 𝑐𝑜𝑠 0
�𝑉� =
𝑉𝐵 𝑚𝑉𝑑𝑉 = ⇒
𝑊(𝐹
Ԧ
) =
1
2
𝑚𝑉
2
𝑉𝐴
𝑉𝐵
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Work and Energy Chapter IV
so : 𝑊𝐴𝐵(𝐹
Ԧ
) =
1
2
𝑚𝑉𝐵
2
𝐸𝑘(𝐵)
−
1
2
𝑚𝑉𝐴
2
𝐸𝑘(𝐴)
Hence:
𝑊𝐴𝐵(𝐹
Ԧ
) = 𝐸𝑘(𝐵) − 𝐸𝑘(𝐴)
it is he Kinetic Energy Theorem.
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Work and Energy Chapter IV
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Work and Energy Chapter IV
Exercise 4 :
A particle of mass m moving along a straight trajectory is subjected to a
force F(x), the variations of which
with respect to x are depicted
in figure below.
Calculate the change in kinetic energy
of this particle between
the positions x=0 and x=30 (m).
0 5 10 15 20 25 30
0,0
0,5
1,0
x(m)
F(N)
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Work and Energy Chapter IV
4.Potential energy
The kinetic energy Ek of a particle is associated with its motion. There is another
form of energy that is associated with its position; this energy is called potential
energy. It is energy stored by a body and can later be transformed, for example, into
kinetic energy when the body is set in motion.
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Work and Energy Chapter IV
Examples of potential energy
a) Gravitational potential energy
𝐸𝑝 = 𝑚𝑔ℎ + 𝐶𝑠𝑡
If we set Ep
=0 for h=0,
then the constant (𝐶𝑠𝑡) is equal to 0
Gravitational potential energy
m
h
It is the energy that a body possesses due to its
position in a gravitational field
Work and Energy Chapter IV
b- Elastic Potential Energy
It is potential energy stored as a result of deformation of an elastic object such as
spring.
When a spring is compressed or stretched,
it stores elastic potential energy.
This energy is released when
the spring returns to its original shape.
𝐸𝑝 =
1
2
𝑘𝑥
2 𝑥 = 𝐿 − 𝐿0
x: The elongation of the spring.
k: is the stiffness constant.
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Work and Energy Chapter IV
- Mechanical Energy of a System
The mechanical energy of a system, denoted as ET
, is the sum of its kinetic
energy and potential energy.
𝐸𝑇 = 𝐸𝑘 + 𝐸𝑝
If the mechanical energy of a system increases or decreases, it means that it has
received or released energy from the external environment. If there is no exchange of
energy between the system and the external environment, the system is said to be
isolated. In this case, its energy 𝐸𝑇 remains constant.
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
Solution
Calculation of the total mechanical energy at equilibrium:
We know that :
𝐸𝑇 = 𝐸𝑘 + 𝐸𝑝
At equilibrium:
- the velocity of the system is zero 𝑉 = 0 ⇒ 𝐸𝑘 = 0
- and the potential energy of the system 𝐸𝑝 = 𝐸𝑝(𝑒𝑙𝑎𝑠𝑡𝑖𝑐) + 𝐸𝑝(gravitational)
So: 𝐸𝑝 =
1
2
𝑘𝑥
2 + 𝑚𝑔ℎ
𝑁𝐴: 𝐸𝑝 =
1
2
200(0.02)
2 + 0.1 × 10 × 0.5 = 1.54 𝐽
Hence, the total mechanical energy at equilibrium: 𝑬𝑻 = 𝑬𝒑 = 𝟏. 𝟓𝟒 𝑱.
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025
Work and Energy Chapter IV
- Conservative force
Potential energy exists exclusively for a specific category of forces
known as conservative forces, or forces that can be derived from a
potential function.
Properties of a Conservative Force
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
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Work and Energy Chapter IV
Work and Energy Chapter IV