Engineering

1. Kinetics of particles
Energy and momentum method

2. • Work done by Force
• Potential and kinetic energy of particles
• Principles of work and energy and its application
• Power and Efficiency
• Conservation of energy
• Principle of impulse and momentum
• Impulsive motion and impact
• Direct central and Oblique impact

3. Work done by a force
• Consider a particle which moves from point A to A’ .
• r is the position vector of A and dr is change in position.
• The magnitude of vector dr which joins A
and A’ be ds.
• When force F is applied on the particle, the
work done by force during the displacement ds be
du = component of force along the
Direction of motion* distance travelled
or, du = Fcosαds

4. 4
Work of a force during a finite displacement:
Work is represented by the area under
the curve of Ft plotted versus s.
  







2
1
2
1
2
1
cos
2
1
s
s
t
s
s
A
A
ds
F
ds
F
r
d
F
U




5. Work of a weight
• The work du of the weight is equal to the product of W and
the vertical displacement of cg of the body
5
  y
W
y
y
W
dy
W
U
dy
W
dU
y
y












1
2
2
1
2
1
• Work of the weight is positive when y < 0,
i.e., when the weight moves down.
• -ve sign define that work is done on the
body

6. Work of the force exerted by spring
• Consider a spring is attached to the fixed point B
at one end and to the body to another end.
• At initial position the spring is undeformed
• When the spring is deformed, the magnitude of
the force exerted by a spring is proportional to the
elongation of the spring (deflection)
• Work of the force exerted by spring,
2
2
2
1
2
1
2
1
2
1
2
1
kx
kx
dx
kx
U
dx
kx
dx
F
dU
x
x










F= Kx
Where, k = spring constant
Finite work done,

7. Work of a gravitational force
Work of a gravitational force (assume particle M
occupies fixed position O while particle m follows path
shown),
1
2
2
2
1
2
2
1
r
Mm
G
r
Mm
G
dr
r
Mm
G
U
dr
r
Mm
G
Fdr
dU
r
r











8. Kinetic energy of particle(Work and energy
principle)

10. Application of work and energy principle

12. Power and Efficiency
• rate at which work is done.
v
F
dt
r
d
F
dt
dU
Power










• Dimensions of power are work/time or force*velocity.
Units for power is
s
m
N
1
s
J
1
(watt)
W
1 


•
input
power
output
power
input work
k
output wor
efficiency





13. • A car weighing 1814.4 kg is driven down a 5o incline at a
speed of 96.54 km/h. when the brakes are applied causing
a constant total braking force of 6672N. Determine the
distance traveled by the car as it comes to stop. Use the
principle of work and energy.

15. • Two blocks are joined by an inextensible cable as
shown. If the system is released from rest, determine
the velocity of block A after it has moved 2 m.
Assume that the coefficient of friction between block
A and the plane is mk = 0.25 and that the pulley is
weightless and frictionless.

16. Potential energy
• Let us consider a body of weight W that moves along a curve
path as shown in figure.
Therefore, the work done is equal to minus
the change in potential energy
• Work is independent of path followed; depends
only on the initial and final position of W and is
called potential energy
Wy
Vg 
   2
1
2
1 g
g V
V
U 


• Units of work and potential energy are the same:
J
m
N 


Wy
Vg
• Work of the weight during displacement from y1 to y2
2
1
2
1 y
W
y
W
U 



17. Potential energy of gravitational force
• Work of the gravitational force when body displaced from A1
to A2.
• Work at any position
• We also know that
U = Vg So,
Vg =
Again we have
F =W =
GMm =WR2
Vg= -WR2/r
1
2
2
1
r
Mm
G
r
Mm
G
U 


r
Mm
G
U 

r
Mm
G

2
R
Mm
G

18. Potential energy due to spring
• Work of the force exerted by a spring depends
only on the initial and final deflections of the
spring,
2
2
2
1
2
1
2
1
2
1 kx
kx
U 


• Note that the preceding expression for V is
valid only if the deflection of the spring is
measured from its undeformed position.
•This expression states that work of a force is equal
to minus the change in P.E
• The potential energy of the body with respect
to the elastic force,
   2
1
2
1
2
2
1
e
e
e
V
V
U
kx
V





19. Conservation of energy
• Work of a force is equal to minus of potential energy
)
(
).........
( 1
2
2
1 i
V
V
U 



• Work of a force is equal to change in kinetic energy,
)
(
..........
1
2
2
1 ii
T
T
U 


• From (i) and (ii)
2
1
2
2
1
1
1
2
2
1
E
E
T
V
T
V
T
T
V
V







• When a particle moves under the action of
conservative forces, the total mechanical energy is
constant or conserved.

20. Example of conservation of energy

21. A 20 N collar slides without friction
along a vertical rod as shown. The
spring attached to the collar has an
undeflected length of 4 cm and a
constant of 3 N/cm.
If the collar is released from rest at
position 1, determine its velocity after
it has moved 6 cm. to position 2.

22. Principle of impulse and momentum

23. An automobile weighing 4000 N is
driven down a 5o incline at a speed of
88 m/s when the brakes are applied,
causing a constant total braking force of
1500 N.
Determine the time required for the
automobile to come to a stop.

24. A 0.5 kg baseball is pitched with a
velocity of 80 m/s. After the ball is hit
by the bat, it has a velocity of 120 m/s
in the direction shown. If the bat and
ball are in contact for 0.015 s, deter-
mine the average impulsive force
exerted on the ball during the impact.

25. Impulsive motion and impact
• When a very large force is acted upon a particle for very short
interval of time and produce a definite change in momentum
then the force is known as impulsive force and the motion is
called impulsive motion.
2
1 v
m
t
F
v
m





 
When impulsive forces act on a tennis ball:
Non-impulsive forces: forces for which is small, and
therefore can be neglected (such as weight)
F t


26. • Impact: Collision between two bodies which occurs during a
small time interval and during which the bodies exert large
forces on each other.
• Line of Impact: Common normal to the surfaces in contact
during impact.
Impact
•Line of contact: The line tangential to the contact surface

27. • Central Impact: Impact for which the mass centers of the two
bodies lie on the line of impact, it is called central impact.
otherwise, it is an eccentric impact.
• Direct Impact: Impact for which the velocities of the two
bodies are directed along the line of impact.
• Oblique Impact: Impact for which one or both of the bodies
move along a line other than the line of impact.
Types of impact
1. Direct central impact
2. Oblique central impact
3. Direct eccentric impact
4. Oblique eccentric impact

28. Direct Central Impact
• Bodies moving in the same straight line:
• Upon impact the bodies undergo a
period of deformation, at the end of
which, they are in contact and moving at
a common velocity.
• A period of restitution follows during
which the bodies either regain their
original shape or remain permanently
deformed.
• Wish to determine the final velocities of
the two bodies. The total momentum of
the two body system is preserved,
A A B B A A B B
m v m v m v m v
 
  
A B
v v


29. Direct Central Impact
• Period of deformation: u
m
Pdt
v
m A
A
A 
 
• Period of restitution:
A
A
A v
m
Rdt
u
m 

 
1
0 









e
u
v
v
u
Pdt
Rdt
n
restitutio
of
t
coefficien
e
A
A
Considering impact on A and its motion

30. • A similar analysis of particle B yields
B
B
v
u
u
v
e




• Combining the relations leads to the desired second
relation between the final velocities.
 
B
A
A
B v
v
e
v
v 





31. Coefficient of restitution
• It is the ratio of the final and initial relative speed
between two objects after they collide.
• It is denoted by e
• Elastic impact (e = 1): In a perfectly elastic collision, no
energy is lost and the relative separation velocity equals the
relative approach velocity of the particles. In practical
situations, this condition cannot be achieved.
In general, e has a value between 0 and 1.
Total energy and total momentum conserved.
Initial K.E = Final K.E
B
A
A
B v
v
v
v 





32. v
v
v A
B 




e depends on materials, impact velocity, shape & size of colliding surfaces.
 v
m
m
v
m
v
m B
A
B
B
A
A 



• Plastic impact (e = 0): In a plastic impact, the relative
separation velocity is zero. The particles stick together
and move with a common velocity after the impact.

33. Oblique central impact
• When the velocities of two colliding bodies are not directed
along the line of impact, it is called oblique impact.
• Along line of contact(tangential direction):
Momentum of each particle is conserved
       t
B
t
B
t
A
t
A v
v
v
v 




34. • Along line of impact(normal direction):
Total momentum of the particles is conserved
       n
B
B
n
A
A
n
B
B
n
A
A v
m
v
m
v
m
v
m 




• Normal components of relative velocities
before and after impact are related by the
coefficient of restitution.
       
 
n
B
n
A
n
A
n
B v
v
e
v
v 





35. A ball is thrown against a frictionless, vertical wall. Immediately
before the ball strikes the wall, its velocity has a magnitude v and
forms angle of 30o with the horizontal. Knowing that e = 0.90,
determine the magnitude and direction of the velocity of the ball as it
rebounds from the wall.

36. The magnitude and direction of the velocities of two identical
frictionless balls before they strike each other are as shown.
Assuming e = 0.9, determine the magnitude and direction of the
velocity of each ball after the impact.

37. Two blocks of mass 25Kg and 15 Kg moving with velocities
of 8m/s and 10m/s respectively. For oblique impact of these
two masses, find the final velocities after the impact. Given,
e= 0.5 and impact occurs along y-axis as in figure.