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Engineering Physics Partial Lecture Notes

Mechanics-1

Syllabus 2001 Wk 1 Intro

Chpt 1 Measurement Wk 2 Error Propagation Wk 3

Chpt 2 - Motion in 1[D] Wk 4

Chpt 4 - Motion in 2[D] Wk 5 QUIZ 1

Chpt 5 Laws of Motion Wk 6

Chpt 5, 6 - Circ. Motion & apps of Newtons Laws

Mid Semester Break Wk 7 ANZAC DAY Wk 8

Chpt 7,8 - Work & Energy Wk 9 QUIZ 2

39.7 - Relativistic Energy Wk 10

Chpt 9,10 - Momentum, Collisions, Rotation Wk 11

Chpt 40(Brief), 41 - QM Chpt 42, 43(Brief) - Atomic to Solids

Wk 12 Chpt 44 - Nuclear & Radiation

Wk 13 QUIZ 3 + Feedback + Exam Prep


Mechanics-2

Projectile Motion Maximum Height

At the peak position vy = 0 = viy-gt1

Projectile Range Range = distance in x-dir’n during time t2

t2=2t1

Range is a max when sin2 =1 ie. =45o

Angles for the same range 2=90o-1

g

vt i

sin1

2

211

sin

2

1sin.sin

2

1

g

vg

g

vvgttvH ii

iiy

g

vH i

22 sin

2

1

g

vR i

2sin2

g

v

g

vvtvR ii

iix

cossin2.sin2.cos

2

2


Mechanics-3

Kinematics - description of motion. Consider particles initially - point-objects (no size).

Displacement, Velocity and Speed (2.1) Motion involves a change in position with time. Graphically, plot position (x in 1D) versus time (t):

Particle moves from xi at time ti to xf at time tf.

x xf xi is its displacement. SI unit: metre (m).

t tf ti is the time interval. SI unit: second (s).

Average velocity SI unit: metre per

second (m/s)

t

xv

x

t

Motion in One Dimension


Mechanics-4

Graphically [Fig. 2.3a], is slope of chord of x versus t graph.

NOTES:

1. Depends only on initial and final positions, not on actual distance travelled between them.

2. Can be +ve (net movement in +ve x direction), ve (in ve x direction), or zero.

3. Average speed = is always +ve.

Instantaneous Velocity and Speed (2.2) As tf ti (= t), or t 0, instantaneous velocity v at

time t.

i.e.

Graphically [Fig. 2.3b], v is slope of tangent to curve at time t.

NOTES:

1. Can be +ve, ve or zero.

2. ‘Velocity’ alone means instantaneous.

3. Instantaneous speed = magnitude of instantaneous velocity

v

timetotal

distance total

v

dt

dx

t

xt

v

0

lim


Mechanics-5

Can be thought of as infinitesimal displacement dx divided by corresponding infinitesimal time interval dt.

Thus denotes any change in a quantity, and

d denotes infinitesimal change in a quantity. We often separate parts of derivatives and write (e.g.):

dx = v dt

(infinitesimal displacement = velocity infinitesimal time interval).

Acceleration (2.3) If velocity changes from vi at time ti to vf at tf, change in

velocity is v vf vi and

average acceleration SI unit: metre per

second squared (m/s2)

Graphically, slope of chord of v vs t graph. [Fig. 2.5]If t 0, obtain

instantaneous acceleration

NOTES:1. Can be +ve, ve or zero.

2. ‘Acceleration’ alone means instantaneous.

3.

dt

dxv

2

2

dt

xd

dt

dx

dt

d

dt

dva

t

va

dt

dv

t

v

ta

0

lim


Mechanics-6

1[D] Motion With Constant Acceleration (2.5)

and

If then:

(2.8)

Also, may write [Fig. 2.10a]

Since x = xf xi then:

(2.10)

Substituting for vf from (2.8) into (2.10):

i.e. (2.11)

Substituting for t from (2.8) into (2.10):

if

if

tt

vv

t

va

aa

ttt fi 0

t

vva if atvv if

t

xvvv fi

2

tvvx fi 21

tvvxx fiif 21

tatvvxx iiif 21

221 attvxx iif

a

vv

a

vvvvxx

if

iffiif

2

22

21


Mechanics-7

(2.12)

Freely Falling Bodies (2.6) All objects fall with same constant downward acceleration

due to gravity, of magnitude g = 9.8 m/s2 (neglecting air resistance).

Replacing x by y, choosing +ve as up, setting a = g get:

ififiif

fiifif

yygvvgttvyy

tvvyygtvv

222221

21

ifif xxavv 222


Mechanics-8

Forces & Motion 1st Law of Motion

“In the absence of a net force, an object at rest remains at rest and a moving object continues with a constant velocity”

Inertia is the resistance an object offers to a change in

it’s motion Mass - is a measure of the inertia of an object

2nd Law of Motion

acceleration is in same direction as the force

Weight w=mg

3rd Law of Motion

“When an object exerts a force on another object, the second object exerts an equal force on the first but in the opposite direction”

amF forcesF

Force due to gravity

BAAB FF


Mechanics-9

Tension Tension is a force !

Force Units are Newton’s (N) 1N=force which gives a 1kg mass an

acceleration of 1ms-2

Friction Frictional forces impede motion

Coefficient of Friction, Friction depends on what kinds of surfaces

make contact & the perpendicular force with which either surface is pressed against the other, the Normal Force, FN

m

Fg

Tension in string

Nf FF

Static frictionSliding friction

Starting friction

Fapplied

Ffr

icti

on


Mechanics-10

Example Coefficient of Friction

Wood on wood coeff. of static friction S = 0.5 coeff. of sliding friction = 0.3

Q: What is the minimum force required to start a 100kg wardrobe moving on a wooden floor?

nb: box only moves when force is > 49N

Q: What force is required to keep the wardrobe moving at constant speed?

What happens if the force is > 294N?

Q: If we maintain a force of 49N the box will accelerate. What would the acceleration be?

NmgFF SNSf 4908.91005.0

NmgFF Nf 2948.91003.0

NNFFF fapplied 196)294490(

296.1100

196 mskg

NamaF


Mechanics-11

Examples Newton’s 2nd Law

Newton’s 3rd Law

Static situation

Q: What happens if there is no friction with the tabletop?

m

Fg

Tension in string

m

Fg=mga=g

Can you see another pair of equal &

opposite forces?

m1

F1=m1g

F1=m1g

m2

T

T B

A


Mechanics-12

Example Static situation


Mechanics-13

m

Fg

Tension in string

Static frictionSliding friction

Starting friction

Fapplied

Ffr

icti

on

m1

F1=m1g

F1=m1g

m2

T

T B

A


Mechanics-14

Circular Motion Uniform Circular Motion

object moving in a circle at constant speed velocity changes due to direction changes hence acceleration (centripetal acceleration)

r

vac

2

r

mvmaF cc

2

Fc

v

Fc

v

Period of orbit How long does it take for an object to complete

one orbit? Let period = T, the distance travelled is equal

to the circumference of the circle, so the period can be described in terms of v and r.

v

rT

T

r

t

sv

rs

2

2

2


Mechanics-15

Circular Motion (cont.) Non-Uniform Circular Motion

object moving in a circle at varying speed hence has both centripetal acceleration and a

non-zero tangential component of acceleration examples include pendulums and vertical

circular paths (non- constant velocity)

Example A yo-yo of mass m is attached to the end of a

cord of length R and swung in a vertical circle. i.e. velocity is non-uniform because there is a tangential component of acceleration due to the weight of the yo-yo.

Fc

v

Fc

v

Fw = mg

Fw = mg

cos

cos

sin

sin

2

2

gR

vm T

R

mvmg T F

g a

ma mg F

r

t

t t

gR

vm T

gR

vm T

bot

top

2

2

Limiting cases


Mechanics-16

Example

A car rounds a curve of radius 50m at constant velocity. If the friction between the car and the road is 0.5, how fast can the car travel?

The centripetal force that keeps the car on the road and in a circular path is static friction Fs

The maximum speed is when the car is almost skidding.

hr km s m

rg v

r

mvF mg Fc s

/ 56 / 16

8. 9 50 5. 0

2

max

What happens when the car goes faster than this speed?What if the car is travelling slower than this speed?


Mechanics-17

Example

Sum of forces in x direction

Sum of forces in the y direction

solving for v

r

mvn Fx

2

sin

mg n Fy cos

hr km s m

rg v

mgr

mv

n

n

/ 48 / 13

20 tan 8. 9 50 tan

cos

sin

0

2

Effect of Banking What if the curve was banked to 200? How fast

could the car travel now (not including friction)?

n

mg

ncos

nsin

n

mg

What if friction is included? How fast can the car travel now?How do racing cars use centripetal acceleration to their advantage?

What happens when the car goes faster than this speed?What if the car is travelling slower than this speed?


Mechanics-18

Gravitation Newtons law of gravitation

between two masses

G is the universal constant of gravitation 6.67 x 10-11 Nm2kg-2

direction of F is along line between the two centres of mass

Example If a person weighs 800N on Earth, how high do

they have to be to weigh only 200N?

22 1

r

m mG Fg

21

1

E

E

r

m mG F

21

2 R

mmGF E

E

E

EEE

E

rR

r

R

r

R

mm

R

r

mm

F

F

2

200

8002

2

2

2

1

2

21

2

1

What would they weigh if they were at the centre of the Earth?

Frm2

m1


Mechanics-19

Satellites Example

What velocity is required to keep a satellite in orbit?

The weight of the satellite is balanced by the centripetal force.

Note that GME/R2 is equal to g (9.8m/s2).

Hence, GME=gR2.

What distance above the earths surface would a geostationary satellite be?

A geostationary orbit is above a fixed point on the Earths surface (spy, weather and communications satellites). Therefore, the period of rotation is 24hours.

h R

GM

r

GMv

r

M mG

r

mvF

E E

Er

21

2

Alternative derivation

from previous slider

g Rv

E2

3/2

2/3

2

222

E

EE

GMT

r

rGM

rGM

r

v

rT

3/ 22

2

g R T

rE

Using the second formula


Mechanics-20

Terminal Velocity An object moving through a medium will

experience a resistive force R which depends on the speed of the object, the shape of the object and the density of the medium.

Where is the density of air, A is the cross-sectional area of the falling object measured in a plane perpendicular to its motion and D is a dimensionless empirical quantity called the drag coefficient (0.5 for spheres but up to 2 for odd shapes).

2

2

1Av D R


Mechanics-21

Example Terminal Velocity

A skydiver jumps out of a plane. What is the acceleration she reaches? What is the terminal

speed at this point?

Terminal speed is when the net force on the skydiver is zero (when R = -mg).

Fg=mg

R=Fdrag

v A D

mgv

vm

A Dg

a let vm

A Dg a

Av D mg R mg F

t

t

net

2

02

0 _2

2

1

2

2

2


Mechanics-22

Work, in the physical sense, is how effective a force is at moving an object.

Work can be defined as the product of the component of a force in the direction of an objects displacement and the magnitude of the displacement. Units of newton.meter (N.m) or joule (J)

Work is positive when the applied force is in the same direction as the movement.

Work is negative when the applied force is in the opposite direction to the movement. Incorporated into cos

Work can also be considered as an energy transfer. If energy is transferred to the system W is positive; if energy is transferred from the system W is negative.

Chapter 7: Work

d F W

Fd W

cos

FsinF

mg

n

d

Fcos

What is the work done to lift a book?What is the work done on the book by gravity?


Mechanics-23

Work done by a Spring The work done by a varying or non-

constant force is equivalent to the integral of the force over the range of the displacement.

A spring is a common physical system in which the force varies with position. It can be described by Hookes’ Law. The force required to stretch or compress a

spring depends is proportional to the amount of stretch or compression (x) with the proportionality constant k known as the spring constant.

What is the work done on a spring to move it from an initial distance xi to a final distance xf?

f

i

f

i

f

i

f

i

x

xx

x

xx

x

xx

x

x

xx

x

dx F W

dx F x F

x F W

x F W

0lim

kx Fs

2 2

2

1i f

s

x x k W

kxdx dx F W


Mechanics-24

Energy Energy can be defined as the ability to

do work.

There are different forms of energy. Kinetic energy is due to the motion of an

object.

Potential energy is due to the state, shape or position of an object.

Chemical energy results from the positions of electrons about the nucleus. A chemical reaction can convert this potential or stored energy into kinetic energy.

Thermal energy stored in molecules is emitted as thermal radiation. The thermal energy is from the vibrations of the molecule and can be considered as the average kinetic energy of the molecule.

Electrostatic energy potential energy stored in an electric field by virtue

of the position of charged particles. kinetic energy of charged particles moving in space

or through a conductor

Mass energy E = mc2

latent energy stored in masses


Mechanics-25

The energy possessed by a moving object is its Kinetic Energy.

The sum of the work done on an object can be described by the change in kinetic energy of the object. Work-Kinetic Energy theorem.

If friction is included, the theorem is modified.2

2

1. .mv E K

Kinetic Energy

2 2

2

1

2

1i f i fmv mv KE KE KE W

f k other iKE d f W KE

d f KEk friction


Mechanics-26

Power Power is the rate at which work is done.

Measured in watts: 1W = 1J/s = 1kgm2/s3

Examples Two bullets, one twice the mass of the other,

are fired so that they have the same speed. Which bullet has the greater kinetic energy?

An older model car accelerates from 0m/s to vm/s in 10s while a new car accelerates to 2vm/s in the same amount of time. What is the ratio of power of the two vehicles?

vdt

d

dt

dWP F

sF


Mechanics-27

Potential energy is energy an object possesses due to it’s state, shape or position.

Gravitational potential energy energy due to an objects position in a

gravitational field eg. height

Elastic potential energy energy due to a spring’s position

Chapter 8: Potential Energy

mgh PEg

2

2

1kx PEs

In bookh = y and PE = U


Mechanics-28

Forces A Force is called conservative if the

work done to move an object between two points is independent of the path taken. This can also occur if an object moves through a closed path and returns to the same initial point, whereupon the work on the particle is zero.

A Force is called non-conservative if the work done to move an object between two points is dependent on the path taken. For example friction is a non-conservative force.

A

B

A

BFriction on surface


Mechanics-29

Energy is conserved if no external forces do work on the system and if no non-conservative forces act on the objects in the system.

Energy can never be created or destroyed.

Energy may be transformed from one from to another, but the total energy of an isolated system is always constant.

The total energy in the universe is constant

f f i iPE KE PE KE

Conservation of Energy

If friction acts on the objects in the system then energy is not conserved.


Mechanics-30

Examples: work A tugboat exerts a constant force of

5000N on a ship moving through a harbour. How much work does the tugboat do if it moves the ship 3.00km?

A sledge loaded with bricks has a total mass of 18.0kg and is pulled at constant speed by a rope. The rope is inclined 200 to the horizontal and the sledge moves a distance of 20.0m on a horizontal surface. The coefficient of friction between the ice and the sledge is 0.500. What is the tension of the rope? How much work is done on the sledge by the

rope? What is the energy lost due to friction?

When a 4.00kg mass is hung vertically on a spring that obeys Hooke’s law, the spring stretches 0.50cm. If the 4.00kg mass is removed, How far would the string stretch if a 1.50kg

mass is hung on it? How much work must an external agent do to

stretch the same spring 4.00cm from it’s equilibrium position?


Mechanics-31

Examples: work If it takes work W to stretch a Hooke’s

law spring a distance d from its equilibrium position determine the extra work required to stretch it an additional distance d?

A crate of mass 10.0kg is pulled up a rough surface with an initial speed of 1.50m/s. The pulling force is 100N parallel to the incline, which makes an angle of 200 to the horizontal. The coefficient of kinetic friction is 0.400, and the rate is pulled 5.00m. How much work is done by gravity? How much energy is lost due to friction? How much work is done by the 100N force? What is the change in kinetic energy of the

crate? What is the speed of the crate after it has been

pulled 5.00m?


Mechanics-32

Examples: energy A 40.0N child s in a swing that is

attached to two ropes 2.00m long. Find the gravitational potential energy of the child-Earth system relative to the child’s lowest position when the ropes are horizontal the ropes make a 300 angle with the vertical the child is at the bottom of the arc.

A bead slides without friction around a loop-the-loop. If the bead is released from a height 3.50R, what is the speed at point A? How great is the normal force on it if the mass is 5.0g?

R

A

h


Mechanics-33

Examples: energy After its release at the top of the first

rise, a roller coaster car moves freely with negligible friction. The roller coaster has a circular loop of radius 20.0m. The car barely makes it around the loop: at the top of the loop the riders are upside down and feel weightless. Find the speed of the roller coaster at the top

of the loop. Find the speed of the roller coaster at the

bottom of the lop and halfway up the loop Find the difference in height between position

1 and 4 if the speed at position 4 is 10.0m/s.

R

3

1

4

2


Mechanics-34

Chapter 39: Relativity Since it was known that sound waves

propagate through a medium, it was assumed that light waves would have the same limitation. The ether was proposed as the medium of free space.

This idea was tested in the Michelson Morley experiment. No change in the interference patten was ever

observed. Light is now known to be a form of

electromagnetic radiation, which needs no medium for propagation.

Mirror 1

Mirror 2

Velocity of the ether

Observer


Mechanics-35

Einstein’s Principle of Relativity The speed of light is constant:

The speed of light in a vacuum has the same value (3.00x108m/s),regardless of the velocity of the observer or the velocity of the source.

The principle of relativity: The laws of physics must be the same in all

inertial reference frames.

Consequences of special relativity Time dilation

time interval measured by an observer moving with respect to the clock is longer than the time measured by an observer at rest with respect to the clock.

Can use example of laser on a spaceship or in a train

Length contraction Length of an object measured y someone in a

reference frame moving with respect to the object is always less than the proper length of an object (measured at rest relative to an object).

Simultaneity Events that are simultaneous for one observer

may not be simultaneous for another observer who is motion relative to the first


Mechanics-36

Twin paradox Twin Speedo goes on a 20ly journey (at about

0.95c) while twin Goslo stays on Earth. When Speedo returns he is 33 years old while Goslo is 62 years old.

Which twin is the traveller? From Speedo’s frame of reference he stayed still while Goslo moved but from Goslo’s frame of reference Speedo was the traveller.

Resolution: Speedo is in a non-inertial reference frame (ie. speeds up and slows down therefore no paradox.

The total energy of a particle is the sum of the kinetic energy and the rest energy.

Special or General relativity: Special relativity deals with reference frames

moving at constant velocity to each other General relativity deals with accelerating

reference frames laws of nature are the same in any frame of

reference whether accelerated or not Principle of equivalence: inertial mass and

gravitational mass are equivalent. A gravitational field is equivalent to an accelerated frame of reference in the absence of gravitational effects.

Chapter 39.7: Relativistic Energy

2 2 2 2 2

mc c p E


Mechanics-37

The Doppler effect that has been discussed in reference to sound waves can also be considered when viewing light from a moving source.

If a light source is moving with respect to an observer the frequency will shift. If the source (eg, a star) is moving away from

the observer at a constant velocity the frequency of light observed will shift towards the red end of the spectrum (red-shift)

If the source is moving towards the observer then the frequency of light will be blue shifted

Most sources (eg galaxies) are found to be moving away. i.e. the universe is expanding

Doppler Effect

receding advancing


Mechanics-38

Momentum The linear momentum of an object is p = mv.

Where p is momentum, m is mass and v is velocity.

p and v are vector quantities

Momentum in an isolated system is always conserved.

Initial momentum = final momentum

Impulse Impulse is the integral of a force applied over

time eg. kicking a football.

The impulse imparted to an object by a force F is equal to the change in momentum of the object.

f

i

t

tdtp F I

Chapter 9: Momentum

Law of conservation of momentum

F

Imagine trying to catch an egg without breaking it .How would you move your hands so that it didn’t break

when you changed its momentum to zero?

t


Mechanics-39

Collisions An inelastic collision is one in which the total

kinetic energy of the system is not conserved.

A perfectly elastic collision is one in which the colliding bodies stick together after collision and the total kinetic energy is conserved.

An elastic collision is when the kinetic energy of the system is constant (i.e. conserved).

BEFORE AFTER

Note that the momentum of the system is always conserved.


Mechanics-40

ii

ii i

CMm

x mx

Center of Mass Center of Mass

The overall motion of a system of particles can be described in terms of the center of mass of the system.

Eg. football, boomerang. The center of mass of an object is the average

location of all the mass within a body. It is sometimes called the balance point since it can be determined empirically by balancing.

It can be determined by hanging the object from any point on the edge and drawing a vertical line through the object.

rdmM

rCM

1

Is the center of mass of an object at the same

point as the center of gravity?

If a cricket bat is cut in half a the center of mass, do both halves have the same mass?


Mechanics-41

Examples Collisions:

Two cars of equal mass approach an intersection. One vehicle is travelling with a velocity of 13.0m/s towards the east while the other vehicle is travelling north with velocity of v2i. Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of 55.00 north of east. The speed limit for both roads is 60km/hr and the driver of the north-bound car claims he was within speed limits when the collision occurred. Is he telling the truth?


Mechanics-42

Examples Momentum

An 80.0kg astronaut is working on the engines of his ship, which is drifting through space with a constant velocity. The astronaut, wishing to get a better view of the Universe, pushes against the ship and much later finds himself 30.0m behind the ship and at rest with respect to it. Without a thruster, the only way to return to the ship is to throw his 0.500kg wrench directly away from the ship. If he throws the wrench with a speed of 20.0m/s relative to the ship, how long does it take the astronaut to reach the ship?


Mechanics-43

Rotation When dealing with the rotation of a fixed

object about a fixed axis it is called PURE rotational motion.

If a particle moves through a circle of radius r, the distance it moves through is r. (s = r).

The angular displacement of a rotating object is:

The instantaneous angular speed of a rotating object is:

The instantaneous angular acceleration of a rotating object is:

Chapter 10: Rotation

if

dt

d

dt

d

is in radians where one radian is the angle subtended

by an arc length equal to the radius of the arc

R

What is the direction of the angular velocity?

Right Hand Rule


Mechanics-44

Equations of Motion of a Rotating Object Equations of Motion

The kinematic equations of motion are similar for a rotating object but velocity is replaced by angular velocity ‘w’ and acceleration is replaced by angular acceleration ‘’.

The linear position, speed and acceleration can be related to their angular counterparts by:

)(2

2

1

22

2

ifif

iif

if

tt

t

ra

rv

rs

t

f


Mechanics-45

Examples An electric motor rotating a grinding

wheel at 100rev/min is switched off. Assuming a constant negative acceleration (i.e. deceleration)of 2.00rad/s2, How long does it take the wheel to stop? How many radians does it go through before

stopping?

A helicopter has a main blade of diameter 7.60m which rotates at 450rev/min and a tail blade of diameter 1.02m which rotates at 4138 rev/min. What is the speed at the tips of each of the

rotors? How does this compare to the speed of sound?


Mechanics-46

Inertia is the resistance an object offers to a change in

it’s motion Mass - is a measure of the inertia of an object

The moment of inertia of a system of particles is:

The moment of inertia of a rigid object is:

The rotational energy can be described by:

Moment of Inertia

ii ir m I2

dm r I2

2

2

1I KR w= rv hence

this relates to KE = 1/2mv2


Mechanics-47

Examples The center of mass of a tennis ball

(3.80cm in radius) moves at 38.0m/s and spins at 125rad/s. What is the ratio of the rotational kinetic

energy to the translational kinetic energy?

A car tyre has dimensions shown in the picture. The tyre has two sidewalls of uniform thickness 0.635cm and a tread wall of 2.50cm and width 20.0cm. Suppose it’s density is uniform with a value 1.10x103kg.m3. Find the moment of inertia through its center perpendicular to the plane of the sidewalls.sidewall tread

30.5cm

33.0cm16.5cm


Mechanics-48

Torque Torque is the tendency of a force to change

the rotation of an object about some axis. The force F, acts on an object from a distance d, exerts torque.

Torque is defined only when a reference axis is specified.

The net external torque on a rotational system is:

Torque

d F

I

What is the torque on a wrench?

Remember Newtons second law?Apply a force and an object will accelerate.

In this case angular acceleration.


Mechanics-49

A fishing pole (length 2.00m) makes an angle of 200 with the water while the fish applies a force of 100N on the pole at an angle of 370 to the water. What is the torque exerted by the fish about the axis perpendicular to the page and passing through the fisherman's hand?

Examples

200 370

100N2.00m

Nm

r F d F

84

57 sin 00 . 2 100

sin

d

Fr

370200

370

200570

Fr

570


Mechanics-50

Example


Mechanics-51

Power The rate at which work is done by an external

force is rotating an object about a fixed axis is a measure of the power delivered.

Work The net work done by external forces in

rotating a rigid body about a fixed axis equals the rate of change in the rotational kinetic energy of the object.

Power and Work

P

2 2

2

1

2

1i fI I W


Mechanics-52

Examples A mass m1 and m2 are suspended by a pulley

that has radius R and a mass M. The cord has negligible mass and causes the pulley to rotate without slipping. The pulley rotates without friction. The masses start from rest at a distance d apart. Treating the pulley as uniform disk, determine the speeds of the two masses as they pass each other.

M

m1

m2d

R

Two methods of achieving goal.1. Conservation of energy (Example 10.15)

2. Torque and Newton’s second Law

m2

m1

m1g

m2g

+

+

T1

T2


Mechanics-53

Quantum Mechanics

Young’s Double Slit Experiment separates light from a single source into two

beams constructive and destructive interference seen

as light and dark fringes If electrons pass through a double slit then an

interference pattern is also produced. This pattern is the same as that produced by

photons with the same wavelength

Single Slit If the slits are alternately closed so that

electrons only pass through one slit at a time then each slit produces a diffraction pattern

If the single slits are added together they do not give a diffraction pattern such as the double slit.

Therefore in a double slit experiment the two slits cannot be considered independently. It is impossible to determine which slit the electrons went through.

Can electrons really pass through both slits at the same time?

The problem needs to be treated as if the electron simultaneously passes through both slits.


Mechanics-54

Heisenberg Uncertainty Principle It is physically impossible to measure

simultaneously the exact position and exact momentum of a particle.

The error in each of the measured quantities is given by:

This uncertainty is not caused by any experimental inaccuracy (errors) but by the fundamental QUANTUM structure and WAVE nature of matter.

At any given time is is only possible to determine the probability that a particle will be at any given position, not its exact position.

2

p x

This principle can be used to show that the Bohr Model of the atom is incorrect.


Mechanics-55

Wavefunctions Wave function

The Wave Function is a way of describing the dual nature of the electron.

The square of the wave function ||2. is the probability of finding the particle at a given point at some instant in time.

Probability Density If an electron is considered to have a

wavefunction then there is some probability of finding an electron within a volume dx. This is called the probability density and is |x|2.


Mechanics-56

Particle in a box Particle in a box.

Allowed vibrations of a standing wave in a string are a useful model in describing the allowed states for a particle in a box

The energy of the particle is quantized.

L

n=1

n=2

n=3

... 3, 2, 12

2

nn

L

n L

L

x nA x sin ) (

kx A x ysin ) (

22

2

22

2

8

22

2 2

1

nmL

hE

mL

nh

m

pmv E

n

n


Mechanics-57

QM Schrodinger’s equation

Particle in a well particles of sufficient energy can overcome the

well barrier This leads to a description of valence bands

and fermi levels in solids

Tunneling Particles can also tunnel through the wall This leads to applications in physics such as

tunnel diodes Josephson Junction Alpha decay Solar energy quantum traps scanning tunneling microscopy

U Em

dx

d 2 2

22


Mechanics-58

Application Scanning Tunneling Microscopy

based on the tunneling current between a metallic tip, which is sharpened to a single atom point and a conducting sample.

A small bias voltage (mV to 3 V) is applied between an atomically sharp tip and the sample. If the distance between the tip and the sample is large no current flows. However, when the tip is brought very close ( 10 Å) without physical contact, a current (pA to nA) flows across the gap between the tip and the sample.

Electrons can tunnel across the vacuum barrier separating the tip and sample in the presence of small bias voltage. This tunneling current is the result of the overlapping wave-functions between the tip atom and surface atom.

sensitive to the gap distance between the tip and sample, the local density of electronic states of the sample and the local barrier height. As we measure the current with the tip moving across the surface, atomic information of the surface can be mapped out.

The STM measures the surface density of states.


Mechanics-59

STM Quantum corral

iron atoms on a copper surface at 4K are manipulated into a corral 14.3nm across.


Mechanics-60

Thompson model volume of positive charge with electrons

embedded throughout.

Rutherford model scattering experiment showed most of the

atom was empty space planetary model where electrons orbit the

positive charged nucleus at set distances nucleus made of protons and neutrons

difficulties are emission and centripetal acceleration

Bohr model quantized energy levels emission (E = hf) arises from electrons

jumping between levels and emitting photons of particular frequencies

Modifications: electron spin relativity

Chapter 42: Atomic Models


Mechanics-61

Atomic Emission Spectra An atom in an excited state makes a transition

to a lower energy state and emits electromagnetic radiation of a particular frequency.

Atomic Absorption Spectra An atom in a ground state can absorb

electromagnetic radiation at specific wavelengths

Energy Level Diagram transitions occur between energy levels

Chapter 42: Atomic Spectra


Mechanics-62

Quantum numbers allowed wavefunctions depend on four

quantum numbers n is the principle quantum number

n = 1,2,3…. l is the orbital quantum number

l = 0, 1, 2…. n -1

ml is the orbital magnetic quantum number ml = -l, -l+1, ….. l+1, l

ms is the spin magnetic quantum number ms = +1/2, -1/2

Selection rules for allowed transitions between energy levels in an atom

Pauli exclusion principle no two electrons in an atom can be in the

same quantum state ie. no two electrons can have the same set of

quantum numbers

, 0

1

m


Mechanics-63

Atomic Spectra Atomic transitions can occur through

stimulated absorption photon stimulates atom to an excited state

spontaneous emission atom in an excited state emits a photon

stimulated emission atom in excited state is stimulated by a photon

to emit another photon (i.e. two photons emitted).

The above examples are from the hydrogen atom.

Another example is Fraunhofer lines which appear when viewing the sun as the source. Emission occurs in the sun. Absorption occurs both in the outer layers of the sun and also in the Earths atmosphere.

Can other suns also have these lines? What does that tell us about other solar

systems?


Mechanics-64

Atomic Spectra X-ray Spectra

x-rays are emitted when a metal target is bombarded with high energy electrons or other charged particles

most of the bombarding particles are scattered this radiation is called Brehmstahlung radiation scattered elastically and in-elastically

if the energy is sufficient to knock out an electron from the K shell an electron of particular energy will be emitted.

appears as sharp peaks which are characteristic of the material on the continuous background

The atomic transitions (adsorption and emission)

can be used to identify materials

using the techniques of spectroscopy.


Mechanics-65

Spectroscopy Spectroscopy is the measurement of a

spectrum from a source. solar spectrum gas spectrum solid spectra

The source (electrons / x-rays / protons) is directed at a sample which adsorbs some of the energy, emits some energy and reflects or scatters the rest.

Examples are: XPS: X-ray photoelectron spectroscopy PIXE: Proton induced x-ray emission AES: Auger electron spectroscopy Mass Spectrometry does not use energy

spectrum but rather charge to mass ratio to identify elements.


Mechanics-66

Light Amplification by Stimulated Emission of Radiation

Spontaneous emission of light. Enough energy must be supplied to an atom to get it

into an excited state. It will then decay to a ground state and emit light in the process. The emission is called spontaneous because it requires no external stimulus and, for a particular atom, the time interval between absorption and emission is unpredictable. Allowed transitions take about 10-8s while forbidden transitions take up to 105 times longer (10-3)s.

Stimulated Emission If the atom is already in the excited state when a

photon interacts with it, the photon will stimulate the atom to undergo a transition to a lower energy level. A photon of exactly the same energy will be emitted. Hence, photon amplification.

Population Inversion Normally, nearly all the atoms are in their ground

state. Stimulated emission is in direct competition with

absorption. a population inversion (more atoms in an excited

state than in the ground state) must occur for continued emission.

Excited state should also be metastable (stimulated emission occurs before spontaneous emission)


Mechanics-67

Laser Highly monochromatic: narrow bandwidth (range of frequencies),

about 100 times smaller than gas discharge Very coherent: investigated by separating beam into two sub-beams

and measuring interference over a distance Well collimated: angle of divergence is small (1mrad)

used to determine the Earth-moon distance (+/- 10cm) by bouncing laser off reflectors on the moon

Helium-Neon Laser Mixture of helium and neon gas at low pressure with mirrors at each

end. An Electric field is generated and electrons excite He atoms into higher

energy level E3.

Collision transfer with Neon excites it into E2.

E2 –E1 in Neon is forbidden but E1 –E2 is not.

Hence population in E2 builds up and E1 population rapidly depleted. atoms that emit light of the same frequency are stimulated

(resonance) leaky mirror allows about 1% of laser light out

E1

E3 LASERlight

Eg

E2

Eg

Forbidden


Mechanics-68

Chapter 43: Molecules and solids Molecular bonds

can be ionic bonding, covalent bonding, hydrogen bonding or Van de Waal’s

Energy and Spectra of Molecules energy of a molecule is made up of its

electronic energy, vibrational energy, translational energy, and its rotational energy.

When two identical atoms are brought close together the energy levels of the atoms overlap. When large numbers of atoms are considered, such as in a solid, band structures may appear

e.g. sodium (Na)

vib rot trans elE E E E E

3s

2s

2p

1s

3p

Figure 43.19


Mechanics-69

Chapter 43 Fermi energy

The probability that a particular state having energy E is occupied by one of the electrons in a solid is given by:

the fermi energy is the level at which the probability of finding an electron drops to zero

Conduction

Insulation

Density of states

1

1) () (

T kE E

B

f

e

E f

Ef

E0

Conduction Band

Valence Band

Ef

E0

Energy Gap


Mechanics-70

Chapter 44: Nuclear and Radiation Nuclei

atomic number Z is the number of protons in the nucleus

neutron number N is the number of neutrons in the nucleus

mass number A is the number of nucleons (protons and neutrons) in a nucleus (A = N + Z)

Isotopes of an element contain different numbers of neutrons have the same Z value but different N and A

values

The mass of an isotope of 12C is defined as 12u where the u is the atomic mass unit (amu)

u = 1.660540 x 10 -27kg

Fe

XAZ

5626


Mechanics-71

The Nucleus The Nuclear force

is a short range (2fm) [10-15m] and falls to zero when distance between nucleons is > several fm.

Very strong attractive force that acts between all nuclear particles.

Magnitude depends on the relative spin orientations of the nucleons

It keeps the nucleus together despite the repulsive Coulomb forces between like charges.

Independent of the charge of interacting nucleons (there can use high energy e’s to probe nuclei).

Energy The total energy of the nucleus is less than the

combined energy of the separated nucleons. The difference is called the binding energy.

Mass of selected particles table 44.1 use figure 4.9

u MeV M Nm Zm MeV EA n p b/ 494 . 931 ) (


Mechanics-72

Nuclear Models Liquid drop model

nucleons interact strongly with one another and frequently collide analogous to thermal agitation of a liquid drop.

Three major effects influence the binding energy the volume effect

for A>50 BE is proportional to A and therefore to the nuclear volume.

the surface effect surface nucleons have fewer neighbours and

hence reduce the binding energy the coulomb repulsion effect

proton repels proton and therefore reduces the binding energy

atoms with large numbers of neutrons also decrease the binding energy

Independent-particle model (shell model) each nucleon is assumed to exist in a shell,

similar to an atomic shell model. Nucleons exist in quantized energy states and there are few collisions

A

Z NC

A

Z ZC A C A C Eb

2

43

1 33

2

2 1

) ( )1 (


Mechanics-73

Decay processes Types of Decay

alpha decay: 4He X is parent nucleus, Y is daughter nucleus spontaneous decay, therefore relativistic energy

and momentum is conserved

beta decay: electron (e-) or positron (e+). daughter contains same number of nucleons but

the atomic number is changed by 1 another particle had to be included to conserve

momentum

named neutrino and antineutrino; massless (or very small), zero electric charge, spin 1/2, interacts very weakly with matter, therefore difficult to detect.

gamma decay: high energy photon nucleus in an excited state decays to a lower

state and emits a photon. Can occur in conjunction with other processes.

He Y XAZ

AZ

42

42

e Y X

e Y XA

ZAZ

AZ

AZ

1

1

e C N

e N C126

127

147

146

X XAZ

AZ

C C

e X B126

126

126

125

Schrodinger’s Cat


Mechanics-74

Decay Decay process

If a radioactive material contains N0 radioactive nuclei a t=0, then the number of nuclei after time t follows:

The decay process is exponential with a decay constant l.

Half-life The half-life of a substance is how long it takes

for half the initial nuclei to decay.

te N N

0

) 2 ln(2

1

2/ 1

0 02

1

t

e N Nt

Decay of Radioactive Substance

y = 4019.9e-0.2497x

0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12 14

Time( hrs)

Co

un

ts


Mechanics-75

Activity Units of radioactivity

The activity of a sample is the number of nuclei that decay in a certain time period

becquerel (Bq) A becquerel is the SI unit of activity and us defined

a 1 decay per second.

curie The curie (Ci) is defined as the activity of 1g of

radium. Equivalent to 3.7 *1010 decays/second

Example: Smoke Detector A common device that uses radioactive

material is a smoke detector. Generally it utilises alpha-decay of a source

such as 24195Am.

t te R e N N

dt

dNR

0 0


Mechanics-76

Example: Carbon Dating There is a small amount of carbon-14 found in

the atmosphere. Plants contain this same amount of carbon during their lifetime but when they die they no longer obtain carbon -14. I.e. the amount if carbon in the sample is fixed at their death. This carbon -14 decays (to Nitrogen-14). By measuring the proportion of carbon-14 in a sample the age of the sample can be determined.

Example: The activity of atmospheric carbon due to the

presence of carbon-14 is 0.255Bq per gram of carbon. A 400mg piece of charcoal was found to have an 14C activity of 139 decays/hour. The half-life of 14C is 5730yrs. How old is the object?

Solution

y yN N

T

s gs m h

RR

NN

atoms

atomsratio

atoms

atoms mol gg total N

atomss

s RC N

y ss T

s

s m h d y y T

8030 10 21 . 1/ 379 . 0/ ln

379 . 0255 . 0 400 . 0

60 60 139

10 32 . 110 02 . 5

10 65 . 6

10 02 . 5

10 02 . 6 / 0. 120. 1 ) (

10 65 . 610 84 . 3

255 . 0) (

10 21 . 1 10 84 . 310 81 . 1

693 . 0 2 ln

10 81 . 1

60 60 24 25 . 365 ) 5730 ( 5730

4 0

1

1

0 0

1222

10

22

23

101 12

114

1 4 1 1211

21

11

21

What are some of

the problems that can occur

in this technique?


Mechanics-77

Nuclear Reactions Fission

nuclear fission occurs when a heavy nucleus such as 235U splits into two smaller nuclei.

The combined mass of the daughter nuclei is less than the mass of the parent and the mass defect (*c2) is the energy released.

Fusion when two nuclei combine to form a heavier

nucleus the process is called fusion the mass of the final nucleus is less than the

combined mass of the original nuclei and energy is released.

Fusion of hydrogen is the source of the Sun’s energy.

+ HEAT

+ HEAT


Mechanics-78

Radiation Safety Radiation Damage

Radiation can cause damage to matter, the degree and type of damage depends on the type and energy of the radiation, and the properties of the matter.

For example, electromagnetic radiations such as gamma or x-rays strips matter of electrons and causes ionisation.

However, alpha rays cause ten times more biological damage than x-rays

Units of radiation 1rad = the amount of radiation that increases the

energy of 1kg of absorbing material by 1*10-2J. Dosage

Since the dose depends not only on the amount of radiation but also on the type, the RBE, relative biological effectiveness of the different sources must be included

Radiation RBE factor

X-rays and Gamma rays 1

Beta particles 1.0-1.7

Alpha particles 10-20

Thermal neutrons 4-5

Fast neutrons and protons 10

Heavy ions 20

REM, radiation equivalent in man REM = RAD x RBE


Mechanics-79

Applications Uses of Radiation

carbon dating tracing materials analysis therapy food preservation

Example Problem 45.23: Assume that an x-ray

technician takes an average of 8 x-rays a day and receives a dose of 5rem/yr as a result.

Estimate the dose per photograph How does the technicians exposure compare

with low-level background radiation. Solution:

Assuming they work about 50 weeks a year the dose per photograph would be: 5rem/(50*5*8) = 1/400 rem per photo (2.5millirem/photo)

The background radiation is about 0.13rem/yr so the technician receives about 38 times more radiation than background levels

Since the recommended rem limit is about 0.5rem/yr this is still quite high however for whole body exposure the limit is 5rem/yr which is what the technician receives.

Dosages greater than 400rem result in 50% mortality


Mechanics-80

Nuclear Magnetic Resonance Imaging Nuclear Spin

nuclei have spin since their components (neutrons and protons) also have spin and orbital angular momentum.

Nuclear spin has an associated nuclear magnetic moment n.

Precession When in an external magnetic field a nuclear

magnetic moment (as well as electronic magnetic moment) will precess at a frequency proportional to the magnetic field.

The magnetic moment can be parallel or antiparallel to the magnetic field resulting in two states.

It is possible to observe the difference between these states using Nuclear Magnetic Resonance

NMR the application of an external oscillating

magnetic field which is adjusted to match the precessional frequency causes the states to “flip” resulting in an absorption of energy (~10-7eV).

The absorption of energy of a particular frequency can be determined electronically and imaging can occur.

Figure 44.5


Mechanics-81

Example: MRI of Sex Magnetic resonance imaging of male

and female genitals during coitus and female sexual arousal by Willibrord Weijmar Schultz, Pek van Andel,

Ida Sabelis, Eduard Mooyaart

http://www.studentbmj.com/back_issues/0100/papers/1596.html

Fig 3 Midsagittal image of the anatomy of sexual intercourse (experiment 12). P=penis, Ur=urethra, Pe=perineum, U=uterus, S=symphysis, B=bladder,

I=intestine, L5=lumbar 5, Sc=scrotum


Mechanics-82

Exam Preparation


Mechanics-83

figure

d

L

m1g m2g

d1d2

z z y y x x

z y x

z y x

B A B A B A

B B B

A A A

B A

k j i B

k j i A

C A B A C) B( A

A B B A

wavelength

Inte

nsi

ty

penny ep stuff > eng phys serway 2

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