1) Uniform acceleration, energy transfer, and oscillating mechanical systems are examined in Chapter 2 on dynamic engineering systems. 2) Outcomes for Chapter 2 include analyzing dynamic systems involving uniform acceleration and determining the behavior of oscillating mechanical systems. 3) Mechanics involves the study of kinematics (motion), kinetics (forces), and statics (equilibrium) to describe the behavior of objects.

1. Module content
• Chapter 1: Static engineering systems
– Simply supported beams
– Beams and columns
– Torsion in circular shafts
• Chapter 2: Dynamic engineering systems
– Uniform acceleration
– Energy transfer
– Oscillating mechanical systems
• Chapter 3: DC and AC theory
– DC electrical principles
– AC circuits
– Transformers
• Chapter 4: Information and energy control systems
– Information systems
– Energy flow control systems
– Interface system components

2. 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

3. Outcomes and Assessment criteria
To achieve each outcome in chapter 2 a
learner must demonstrate the ability to:
Analyse dynamic engineering systems
– determine the behaviour of dynamic mechanical
systems in which uniform acceleration is present
– determine the effects of energy transfer in
mechanical systems
– determine the behaviour of oscillating mechanical
systems

4. Mechanics- To study dynamic systems
Study of Objects at Description of the Origin of the
Rest (Valence of Object’s Motion Object’s Motion
applied forces and (Position, Velocity, (Force, Momentum,
Moments) & Acceleration) & Energy)
Statics Kinematics Kinetics
Mechanics: Physics of Behaviors of the Objects
• Kinematics: How to describe the object’s motion
Where is the object? Position
How fast the position changes with time? Velocity
How fast the velocity changes? Acceleration
• Kinetics: How to explain the object’s motion
 Intrinsic motion of an object :
Changing its position, Constant velocity, No acceleration
 Change in motion due to an external action
Changing its position, Non-constant velocity, Non-zero acceleration

5. Types of motions
• Translational motion
motion by which a body shifts from one point in space to another
– e.g., the motion of a bullet red from a gun
• Rotational motion
motion by which an extended body changes orientation, with respect
to other bodies in space, without changing position
– e.g., the motion of a spinning top
• Oscillatory motion
motion which continually repeats in time with a fixed period
– e.g., the motion of a pendulum in a grandfather clock
• Circular motion
motion by which a body executes a circular orbit about another fixed
body
– e.g., the (approximate) motion of the Earth about the Sun

6. • The different types of motion stated on the last slide can be
combined
– for instance, the motion of a properly bowled bowling ball consists of a
combination of translational and rotational motion, whereas wave
propagation is a combination of translational and oscillatory motion.
• The above mentioned types of motion are not entirely distinct
– e.g., circular motion contains elements of both rotational and oscillatory
motion.
• statics: i.e., the subdivision of mechanics which is concerned with
the forces that act on bodies at rest and in equilibrium.
– Statics is obviously of great importance in civil engineering
– For instance, the principles of statics were used to design the building in
which this lecture is taking place, so as to ensure that it does not
collapse.

7. Angular displacement
s = r ⋅θ
s
θ= (radianmeasure )
r
s 2πr
For full circle: θ = = = 2π
r r
Planar, rigid object rotating
Full circle has an angle of 2π radians, about origin O.
Thus, one radian is 360°/2π = 57.3°

8. Angular velocity and acceleration
Angular displacement: ∆θ = θ − θo
θ − θ o ∆θ
Average angular speed: ωavg = =
t − to ∆t
∆θ dθ
Instantaneous angular speed: ω = lim =
∆t →0 ∆t dt
ω − ωo ∆ω
Average angular acceleration: αavg = =
t − to ∆t
∆ω dω
Instantaneous angular acceleration: α = lim =
∆t →0 ∆t dt

9. Every particle (of a rigid object):
• rotates through the same angle,
• has the same angular velocity,
• has the same angular acceleration.
θ, ω, α characterize rotational motion of entire object

10. Linear motion Rotational motion
(linear acceleration, a) ( rotational acceleration, α)
v = v o + at ω = ωo + αt
x = x o + 1 (v + v o )t
2
θ = θo + 1 (ω + ωo )t
2
1 2 1 2
x = x o + v ot + at θ = θo + ωot + αt
2 2
v 2 = v o 2 + 2a ( x − x o ) ω 2 = ωo 2 + 2α (θ − θo )

11. Linear and angular quantities
Arc length s: s = r ⋅θ
Tangential speed of a point P: v = r ⋅ω
Tangential acceleration of a point P:
2
v
at = r.α at = =ω r
2
r

12. Example 1
A grindstone rotates at constant angular acceleration α =
0.35 rad/s2. At time t = 0, it has an angular velocity of
ωo= - 4.6 rad/s2 and a reference line on it is horizontal,
at the angular position θo=0.
(a) At what time after t=0 is the reference line at the
angular position θ = 5.0 rev?
(b) Describe the grindstone’s rotation between t = 0 and t
= 32 s.
(c) At what time t does the grindstone momentarily stop?

13. Linear inertia and mass
• Inertia
– The tendency of an object to keep the current
state of motion
– Difficulty in changing the state of motion
• Properties of Inertia
– Static inertia vs. dynamic inertia
– Proportional to mass of the object:
• "The more massive an object, the more it tends to
maintain its current state of motion."
• Mass: measure of inertia in linear motion

14. Rotational inertia
Rotational inertia (or Moment of Inertia) I of an
object depends on:
- the axis about which the object is rotated.
- the mass of the object.
- the distance between the mass(es) and the axis of rotation.
- Note that ω must be in radian unit. The SI unit for I is
kg.m2 and it is a scalar.
I ≡ ∑ mi ⋅ ri
2
i

15. lim ∑ ri = ∫ r 2 dm = ∫ r 2 ρdV
2 ⋅∆
I≡ mi
∆ →
mi 0 i
Note that the moments of inertia are different for different
axes of rotation (even for the same object)
1
I = ML2
3
1
I= ML2
12

17. Radius of Gyration
• The mass moment of inertia of a body about
a specific axis can be defined using the
radius of gyration (k). The radius of gyration
has units of length and is a measure of the
distribution of the body’s mass about the
axis at which the moment of inertia is
defined.
I
I = m k or k =
2
m

18. Parallel Axis Theorem
• Note that the moments of inertia are different for different axes of
rotation (even for the same object)
I = 1 Mr 2
2 I = 2 Mr2
3
(a ) (b)
• Let h be the perpendicular distance between the axis that we need
and the axis through the center of mass (remember these two axes
must be parallel). Then the rotational inertia I about the required
axis is
I = I + M h 2 parellel − axis theorem
com
• For example, we can apply parallel axis theorem in the case of (a)
and (b) above.

19. First law
• A particle originally at rest, or moving in a
straight line at constant velocity, will remain
in this state if the resultant force acting on
the particle is zero
– Newton’s First Law looked at objects at rest or
under constant velocity.
– No net force was acting on these objects

20. Second law
• “A force applied to a body causes an acceleration.”
– Acceleration describes how quickly motion changes.
– Or : acceleration = change in velocity
time interval
• Acceleration is proportional to the force, inversely proportional to
mass.
– Usually there is more than one force acting on an object. The resulting
acceleration of an object is due to the total or NET FORCE on the object
– acceleration ∝ net force
– acceleration ∝ 1 / mass (As more mass is added, the acceleration of
the cart is slowed)
– acceleration = net force or a = F
mass m
– Force = mass x acceleration (work out few examples)
• Direction of the acceleration = direction of the force
• First and second law
– If a force is applied to an object, whether it is at rest or moving, the
motion will change. IT ACCELERATES.
– If the force is removed, the object will continue moving at a constant
velocity

21. The first and third laws were used in developing the concepts
of statics. Newton’s second law forms the basis of the study of
dynamics.
Mathematically, Newton’s second law of motion can be written
F = ma
where F is the resultant unbalanced force acting on the particle,
and a is the acceleration of the particle. The positive scalar m
is called the mass of the particle.
Newton’s second law cannot be used when the particle’s speed
approaches the speed of light, or if the size of the particle is
extremely small (~ size of an atom).

22. Newton’s 2nd Law for Rotation
τ net = Iα Newton ' s second law for rotation
• Note that α must be in radian.
Proof :
Ft = mat
τ = Ft r = mat r
Since at = α r, τ = m (α r) r = (m r 2 )α
The quantity in parentheses is the
moment of inertia of the particle about
the rotation axis, therefore τ = I α

23. Third law
• “For every action, there is an equal and opposite
reaction.”
• “When one body exerts a force on a second, the second
body exerts a reaction force that is equal in magnitude
and opposite in direction on the first.”
• Eg: bullet vs. gun, fist fighting, rocket
• For every interaction, the forces always come in pairs
(twos).
– The ACTION FORCE (Object A exerts a force on object B )and
– The REACTION FORCE (Object B exerts a force on object A )
– They are equal in strength and opposite in direction

24. Action and reaction on different masses
• When a cannon is fired, there is an interaction
between the cannon and the cannon ball.
• The forces the cannon ball and cannon exert on each
other are equal and opposite.
• The cannonball moves fast while the cannon only
Kicks a bit because of the difference in their masses.
– FOR THE CANNON : a = F / M
– FOR THE CANNONBALL : f = F / m
• The force exerted on a small mass produces a greater
acceleration than the same force exerted on a large
mass
Question : Does a stick of dynamite contain force?
Answer : No, force is not something an object has, like mass and
volume. An object may posses the capability of exerting force
on another object but it does not possess force.

25. Combined linear and angular motions
• In reality, car tires both rotate and translate
• They are a good example of something which
rolls (translates, moves forward, rotates)
without slipping
• Is there friction? What kind?

26. Derivation
• The trick is to pick your reference
frame correctly!
• Think of the wheel as sitting still
and the ground moving past it with
speed V.
Velocity of ground (in bike frame)
= -wR
=> Velocity of bike (in ground frame)
= wR

27. Friction
• Force acting at the area of contact between two
surfaces
• Magnitude: proportional to the friction coefficient
and the normal reaction force
• Direction: opposite that of motion or motion
tendency
• Types: sliding and rolling
– Sliding: due to relative motion of the surfaces
– Rolling: due to deformation of the surfaces

28. Friction (continued)
• Static vs. Kinetic Friction
– Max. static friction: max. force required to
initiate a motion
– Kinetic (dynamic) friction: force required to
maintain the motion

29. Banking Angle
Your car has m and is
traveling with a speed
V around a curve with
Radius R
What angle, θ , should
the road be banked so
that no friction is
required?

30. Skidding on a Curve
A car of mass m rounds a curve on a flat
road of radius R at a speed V.
What coefficient of friction is required so
there is no skidding?
Kinetic or static friction?

31. Conical Pendulum
A small ball of mass m is
suspended by a cord of
length L and revolves
in a circle with a radius
given by
r = L sinθ .
1. What is the velocity of
the ball?
2. Calculate the period of
the ball

32. Weight
• When near the surface of the earth, the only gravitational force having any
sizable magnitude is that between the earth and the body. This force is called
the weight of the body
– Gravity acting on a body from the Earth
– Direction: downward
• Mass is an absolute property of a body. It is independent of the gravitational
field in which it is measured. The mass provides a measure of the resistance of
a body to a change in velocity, as defined by Newton’s second law of motion (m
= F/a).
• The weight of a body is not absolute, since it depends on the gravitational field in
which it is measured. Weight is defined as
W = mg
where g is the acceleration due to gravity (weight in mass and earth)
SI system: In the SI system of units, mass is a base unit and weight is a
derived unit. Typically, mass is specified in kilograms (kg), and weight is
calculated from W = mg. If the gravitational acceleration (g) is specified in
units of m/s2, then the weight is expressed in newtons (N). On the earth’s
surface, g can be taken as g = 9.81 m/s2.
W (N) = m (kg) g (m/s2) => N = kg·m/s2

33. Momentum and Impulse
• Momentum
– Amount of motion
– Momentum = (mass)(velocity)
– Important in giving or receiving impact, collision, etc.
– Vector
• Impulse
– Collision characterized by the exchange of a large force during a short
time period
– 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

34. Example 2
A compact disc player disc from rest and
accelerates to its final velocity of 3.50 rev/s in
1.50s. What is the disk's average angular
acceleration?

35. Example 3
The blades of a blender rotate at a rate of
7500rpm. When the motor is turned off
during operation, the blades slow to rest
in 3.0 seconds. What is the angular
acceleration?

36. Example 4
How fast is the outer edge of a CD (at 6.0
cm) moving when it is rotating at its top
speed of 22.0 rad/s?

37. Example 5
How many rotations does the CD from the
first problem make while coming up to
speed from rest to wf = 22.0 rad/sec at a=
14.7 rad/s2

38. Example 6
A wheel with radius 0.5m makes 55 revolutions as it
changes speed from 80km/h to 30 km/h. The wheel
has a diameter of 1 meter. (a) What was the angular
acceleration? (b) How long is required for the wheel
to come to a stop if it decelerated at that rate?

39. Bicycle example
A bicycle with initial linear velocity V0 (at t0=0)
decelerates uniformly (without slipping) to rest over a
distance d. For a wheel of radius R:
a) Total revolutions before it stops?
b) Total angular distance traversed by the wheel?
c) The angular acceleration?
d) The total time until it stops?

40. Figure shows a uniform disk, with mass M = 2.5 kg and
radius R = 20 cm, mounted on a fixed horizontal axle. A
block with mass m = 1.2 kg hangs from a massless cord that
is wrapped around the rim of the disk Find the acceleration
of the falling block, the angular acceleration of the disk, and
the tension in the cord. The cord does not slip, and there is
no friction at the axle.

42. 1. Newton’s second law can be written in mathematical form
as ΣF = ma. Within the summation of forces ΣF,
________ are(is) not included.
A) external forces B) weight
C) internal forces D) All of the above.
2. The equation of motion for a system of n-particles can be
written as ΣFi = Σ miai = maG, where aG indicates _______.
A) summation of each particle’s acceleration
B) acceleration of the center of mass of the system
C) acceleration of the largest particle
D) None of the above.

43. 3. The block (mass = m) is moving upward with a speed v.
Draw the FBD if the kinetic friction coefficient is µk.
mg mg
v
A) µ kN B) µ kN
N N
mg
C) µ kmg D) None of the above.
N

44. 4. Packaging for oranges is tested using a machine that exerts
ay = 20 m/s2 and ax = 3 m/s2, simultaneously. Select the
correct FBD and kinetic diagram for this condition. y
W may x
W
A) B)
= • max = • max
Rx Rx
Ry Ry
C) may D) W may
= • = • max
Ry Ry

45. 5. Internal forces are not included in an equation of motion
analysis because the internal forces are_____.
A) equal to zero
B) equal and opposite and do not affect the calculations
C) negligibly small
D) not important
6. A 10 N block is initially moving down a ramp
with a velocity of v. The force F is applied to F
bring the block to rest. Select the correct FBD.
10 F 10 F 10 F v
A) µ k10 B) µ k10 C) µ kN
N N N

46. B
7. When a pilot flies an airplane in a
vertical loop of constant radius r at C r A
constant speed v, his apparent weight
is maximum at
D
A) Point A B) Point B (top of the loop)
C) Point C D) Point D (bottom of the loop)
8. If needing to solve a problem involving the pilot’s weight at
Point C, select the approach that would be best.
A) Equations of Motion: Cylindrical Coordinates
B) Equations of Motion: Normal & Tangential Coordinates
C) Equations of Motion: Polar Coordinates
D) No real difference – all are bad.
E) Toss up between B and C.

47. 9. For the path defined by r = θ2 , the angle ψ at θ = .5 rad is
A) 10 º B) 14 º
C) 26 º D) 75 º
··
10. If r = θ2 and θ = 2t, find the magnitude of r· and θ when
t = 2 seconds.
A) 4 cm/sec, 2 rad/sec2 B) 4 cm/sec, 0 rad/sec2
C) 8 cm/sec, 16 rad/sec2 D) 16 cm/sec, 0 rad/sec2