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.Kinetic Energy
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 9
Work and Energy Chapter IV
The Kinetic Energy Theorem
Ek B - Ek A = ?WAB F
?ext
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 IACHACHENE FARIDA FHC-UMBB-2024-2025 18
Work and Energy Chapter IV
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 19
Work and Energy Chapter IV
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 20
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 x 10 x 0.5 = 1.54 J
Hence, the total mechanical energy at equilibrium: ET = Ep = 1.Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 5
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
?No Work Done
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 4
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.Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 15
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.Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 12
Work and Energy Chapter IV
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 13
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.Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 1
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
?0 5 10 15 20 25 30
0,0
0,5
1,0
x(m)
F(N)
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 14
Work and Energy Chapter IV
4.Potential energy
The kinetic energy Ek of a particle is associated with its motion.dr
->
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 6
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.Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 8
Work and Energy Chapter IV
Kinetic energy Ek
, is the energy possessed by an object due to its motion.= m
dV
dt
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 10
Work and Energy Chapter IV
We substitute the expression for F
->
into equation (*):
(*) => dW = m
dV
dt ?< 0 negative work
Dr IACHACHENE FARIDA FHC-UMBB-2024-2025 3
Work and Energy Chapter IV
?acting through an
infinitesimal displacement dr
?.Introduction
2.1.?0  0
??2
< ?