This chapter discusses dynamic engineering systems including uniform acceleration, energy transfer through various forms like potential and kinetic energy, and oscillating mechanical systems. It covers concepts like Newton's laws of motion, conservation of energy, and how energy is transferred and stored in linear and rotating systems, as well as damped oscillatory motion. Simple harmonic motion of linear and transverse systems is also qualitatively examined.

1. Chapter 2- Dynamic Engineering Systems
2.1 Uniform acceleration
• linear and angular acceleration
• Newton’s laws of motion
• mass, moment of inertia and radius of gyration of rotating components
• combined linear and angular motion
• effects of friction
2.2 Energy transfer
• gravitational potential energy
• linear and angular kinetic energy
• strain energy
• principle of conservation of energy
• work-energy transfer in systems with combine linear and angular motion
• effects of impact loading
2.3 Oscillating mechanical systems
• simple harmonic motion
• linear and transverse systems;
• qualitative description of the effects of forcing and damping

2. Potential energy
• Stored energy that an object has due to its
position or condition
• When work gets done on an object, its potential
and/or kinetic energy increases
• There are different types of potential energy:
1. Gravitational energy
2. Elastic potential energy (energy in an stretched
spring)
3. Others (magnetic, electric, chemical, …)

3. Conservative and non-conservative forces
• A force for which W12= - W21 is called a conservative forces.
• The net work done by a conservative force around any
closed path (return back to the initial configuration) is zero.
• A force that is not conservative is called a nonconservative
force.
• We cannot define potential energy associated with a
nonconservative forces.
– Examples of conservative forces
• gravitational force
• spring force
– Examples of non-conservative forces
• fluid drag force

4. Gravitational potential energy
In general :
x y z
∆U = − W = − ∫ Fx dx − ∫ Fy dy − ∫ Fz dz
f f f
xi y i z
i
Gravitational potential energy, ∆Ug
y y r
∆U g = − Wg = − ∫ Fg dy = − ∫ (−mg ) dy Q Fg has only y-component: (-mg ) ˆ
f f
j
y i y
i
= mg  y 
 
yf
yi
⇒ ∆U g = mg ( y − y ) = mg ∆yf i
Rewrite Uf as U and yf as y and take U i to be the
reference level and assign the values U i =0 and yi =0.
⇒ U ( y ) = mg y
4/8

5. Elastic potential energy
xi xf
1
∆U s = −W s = k (x f − x i )
2 2
2
Elastic potential energy stored in a spring:
1 2 The spring is stretched or
U s = kx compresses from its
equilibrium position by x
2

6. Exercise
• What P.E. is gained when a
100. kg object is raised 4.00 m
straight up?
• What PE would be gained if an
object were moved 4 m to the
right?

7. Kinetic energy
• Energy due to motion of the object
• The unit of kinetic energy is Joules (J).
• Kinetic energy is a scalar (magnitude only)
• Kinetic energy is non-negative (zero or positive)
• Basically divided into two
– Linear Kinetic energy
linear kinetic energy is given by:
m = mass
K = 1 mv 2
2
v = speed (magnitude of velocity)
v2 = vx2+ vy2
– Angular Kinetic energy
angular kinetic energy is given by: K=1/2 Iω2
ω = angular speed
I = inertia

8. Remember work-kinetic energy theorem for linear motion:
1 1
Wnet = mv f − mvi 2
2
2 2
Net work done on an object changes its kinetic energy
There is an equivalent work-rotational kinetic energy theorem:
1 1
Wnet = I ω f − I ωi 2
2
2 2
Net rotational work done on an object changes its rotational kinetic energy

9. Conservation of energy
• Energy cannot be created or destroyed (Energy
conservation law)
• Energy can be transformed from one form to
another form
• The total energy of an isolated system cannot
change.
• When we stated the conservation of energy for a
system two conditions are applied:
• Isolated system (no external forces)
• Only conservative forces in the system

10. Work
• How do we give an object kinetic energy? By doing work!
How do we do work? By means of a force acting on the
object!!
• Work W is energy transferred to or from an object by means
of a force acting on the object.
– Energy transferred to an object is defined as positive work
– Energy extracted from an object is defined as negative work
– ‘Work ’ ‘transferred energy’
– ‘Doing work’ ‘the act of transferring energy’
• Work = (force along displacement)X(displacement)
or mathematically
W = F . d = Fd cos(φ )
• The work done by the force F is zero if:
– d = 0: displacement equal to zero
φ = 90deg. : force perpendicular to displacement

11. Illustration
Definition:
The work done on an object by an external CONSTANT
force is- the product of the component of the force in the
direction of the displacement and the magnitude of the
displacement.
r r
W = F ×d ×cos θ = F .d = F|| ×d

12. • Work is a scalar
– Work has only magnitude, no direction.
– The value of W is independent of how we draw the
coordinate system (unlike the components of r or F)
• The SI unit of work is the Joule (J)
1 Joule ≡ 1 Newton·meter
1 J = 1 Nm = 1 kg m2/s2
• Work can be positive, negative or zero depending on
the angle between the force and the displacement

13. Exercise
F = 10 N
What is the work done by
- the gravitational force θ = 60°
d = 10m
- the normal force
- the force F
when the block is displaced
along the horizontal.

14. Exercises
• What is the potential energy of a 10kg mass:
1. 100m above the surface of the earth
2. at the bottom of a vertical mine shaft 1000m deep
• Find the work done in raising 100 kg of water
through a vertical distance of 3m.
• A car of mass 1000 kg travelling at 30m/s has its
speed reduced to 10m/s by a constant breaking
force over a distance of 75m. Find:
1. The cars initial kinetic energy
2. The final kinetic energy
3. The breaking force

15. Combined linear and angular motion
• Two bicycles roll down a hill that is 20m high. Both
bicycles have a total mass of 12kg and 700 mm diameter
wheels (r=.350 m). The first bicycle has wheels that weigh
0.6kg each, and the second bicycle has wheels that weigh
0.3kg each. Neglecting air resistance, which bicycle has
the faster speed at the bottom of the hill? (Consider the
wheels to be thin hoops).

17. Effects of impact loading
Impulse
• Accumulated effect of force exerted on an object for a
period of time
• Impulse = (force)(time)
• Increase in F or t ⇒ increase in I
• Vector
• Equal to the change in momentum of the system
Impact
• Collision characterized by the exchange of a large force
during a short time period
• Impact Types
– Perfectly elastic
– Perfectly plastic
– Elastic
• Characterized by Coefficient of Restitution [COR]
(e=1, 0<e<1, e=0)

18. Strategies in Sports
Giving impulse:
– Maximize (batting, throwing,
etc.)
Receiving impulse:
– Minimize (landing, receiving,
etc.)
– Reduce impulse force by
increasing time

19. Applications of impact
• Ball drop and COR
• COR vs surfaces

20. Exercises
• An object weighing 15. N is lifted from the ground
to a height of 0.22 m. Find object's change in PE
• In the figure, a 2.0 kg package of tamale slides along a
floor with speed v1 = 4.0 m/s. It then runs into and
compresses a spring, until the package momentarily
stops. Its path to the initially relaxed spring is
frictionless, but as it compresses the spring, a kinetic
frictional force from the floor, of magnitude 15 N, acts on
it. The spring constant is 10,000 N/m. By what distance
d is the spring compressed when the package stops?

21. Exercises
• A roller coaster cart of mass m = 300 kg starts at rest at
point A, whose height off the ground is h1 = 25 m, and a
little while later reaches point B, whose height off the
ground is h2 = 7 m. What is the potential energy of the
cart relative to the ground at point A? What is the speed
of the cart at point B, neglecting the effect of friction?

22. Exercises
• Consider the diagram. The mass of block A is 75 kg and
the mass of block B is 15 kg. The coefficient of static
friction between the two blocks is µ= 0.45. The
horizontal surface is frictionless. What minimum force F
must be exerted on block A in order to prevent block B
from falling?

23. Exercises
• A pirate drags a 50 kg treasure chest over the rough
surface of a dock by exerting a constant force of 95N
acting at an angle of 15 above the horizontal. The chest
moves 6m in a straight line, and the coefficient of kinetic
friction between the chest and the dock is 0.15. How
much work does the pirate perform? How much energy is
dissipated as heat via friction? What is the final velocity of
the chest?