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

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

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

Mechanics- To study dynamic systemsStudy of Objects at Rest (Valence of applied forces and Moments)

Description of the Object’s Motion(Position, Velocity, & Acceleration)

Origin of the Object’s Motion (Force, Momentum, & Energy)

Kinematics Kinetics

Mechanics: Physics of Behaviors of the Objects

Statics

• Kinematics: How to describe the object’s motion Where is the object? PositionHow 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

Types of motions

• Translational motionmotion by which a body shifts from one point in space to another– e.g., the motion of a bullet red from a gun

• Rotational motionmotion 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 motionmotion which continually repeats in time with a fixed period– e.g., the motion of a pendulum in a grandfather clock

• Circular motionmotion by which a body executes a circular orbit about another fixed body – e.g., the (approximate) motion of the Earth about the Sun

• 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.

Angular displacement

Planar, rigid object rotating about origin O.

For full circle: 22

r

r

r

s

Full circle has an angle of 2 radians,

Thus, one radian is 360°/2

s

radianmeasurer

rs

Angular displacement:

Average angular speed:

Instantaneous angular speed:

Average angular acceleration:

Instantaneous angular acceleration:

o

oavg

ot t t

dt

d

tt

0

lim

oavg

ot t t

dt

d

tt

0

lim

Angular velocity and acceleration

• rotates through the same angle,

• has the same angular velocity,

• has the same angular acceleration.

Every particle (of a rigid object):

characterize rotational motion of entire object

Linear motion

(linear acceleration, a)

ov v at

21

2o ox x v t at

1

2( )o ox x v v t

o t

21

2o ot t

1

2( )o o t

2 2 2 ( )o ov v a x x 2 2 2 ( )o o

Rotational motion ( rotational acceleration, )

Linear and angular quantities

Tangential speed of a point P:

Tangential acceleration of a point P:

Arc length s: rs

rv

.rat rr

vat

22

A grindstone rotates at constant angular acceleration = 0.35 rad/s2. At time t = 0, it has an angular velocity of o= - 4.6 rad/s2 and a reference line on it is horizontal, at the angular position o=0.

(a) At what time after t=0 is the reference line at the angular position = 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?

Example 1

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

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 must be in radian unit. The SI unit for I is

kg.m2 and it is a scalar.

Rotational inertia

2i

ii rmI

Note that the moments of inertia are different for different axes of rotation (even for the same object)

dVrdmrmrIi

iimi

222

0lim

2

3

1MLI

2

12

1MLI

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 = m k2 or m

Ik

• Note that the moments of inertia are different for different axes of rotation (even for the same object)

2comI I M h parellel axis theorem

• 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

( )a ( )b

• For example, we can apply parallel axis theorem in the case of (a) and (b) above.

Parallel Axis Theorem

232

MrI212

MrI

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

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

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.

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).

Mathematically, Newton’s second law of motion can be writtenF = 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 for rotation

• Note that must be in radian.

t tF ma

t tF r ma r

,ta r 2( ) ( )m r r mr Since

The quantity in parentheses is the moment of inertia of the particle about the rotation axis, therefore I

Proof :

Newton’s 2nd Law for Rotation

Inet

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

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.

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?

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

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

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

Banking Angle Your car has m and is

traveling with a speed V around a curve with Radius R

What angle, , should the road be banked so that no friction is required?

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?

Conical PendulumA 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 sin.

1. What is the velocity of the ball?

2. Calculate the period of the ball

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 = mgwhere 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

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

Example 2A 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?

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? 

Example 4How 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?

Example 5How 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

Example 6A 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?

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?

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.

1. Newton’s second law can be written in mathematical form as F = ma. Within the summation of forces F, ________ 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 Fi = 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.

3. The block (mass = m) is moving upward with a speed v. Draw the FBD if the kinetic friction coefficient is k.

A) B) C) D) None of the above.kmg

mg

N

mg

N

kN

N

mg

kN

v

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.

A) B)

C) D)

= •

may

max

W

Ry

Rx

= • max

W

Ry

Rx

= •

may

max

W

Ry

= •

may

Ry

y

x

5. Internal forces are not included in an equation of motion analysis because the internal forces are_____.

A) equal to zeroB) equal and opposite and do not affect the calculations C) negligibly smallD) not important

6. A 10 N block is initially moving down a ramp with a velocity of v. The force F is applied to bring the block to rest. Select the correct FBD.

A) B) C)k10

10

N

F

k10

10

N

F

kN

10

N

F

F

v

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.

7. When a pilot flies an airplane in a vertical loop of constant radius r at constant speed v, his apparent weight is maximum at

A) Point A B) Point B (top of the loop)

C) Point C D) Point D (bottom of the loop)

r A

B

C

D

9. For the path defined by r = , the angle at rad is

A) 10 º B) 14 º

C) 26 º D) 75 º

10. If r = 2 and = 2t, find the magnitude of r and 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

···