First
First Year
Year University
University Physics
Physics 1A
1A
Topic
Topic 1:
1: Mechanics
Mechanics
Particle
Particle Kinematics
Kinematics In
In One
One Dimension
Dimension ((
§2.1§2.1‐‐
2.6)2.6) DisplacementDisplacement
Displacement
Displacement isis thethe distancedistance anan objectobject isis fromfrom thethe origin,origin, irrespectiveirrespective of of thethe routeroute taken
taken toto thatthat position.position. BeingBeing aa vector,vector, itit willwill alsoalso stipulatestipulate directiondirection fromfrom thethe origin.origin. The
The displacementdisplacement of of anan objectobject underunder constantconstant velocityvelocity cancan bebe determineddetermined asas follows:follows:
(Where(Wherev v x x isis constant)constant)Velocity
Velocity andand AccelerationAcceleration
Velocity
Velocity isis thethe raterate of of changechange of of displacementdisplacement withwith respectrespect toto time.time. AverageAverage velocityvelocity is
is thethe changechange inin displacementdisplacement divideddivided byby thethe changechange inin time.time. InstantaneousInstantaneous velocityvelocity can
can bebe determineddetermined accordingaccording toto thethe followingfollowing equation:equation:
lim
lim
∆
∆
∆∆
∆∆
Acceleration
Acceleration isis thethe raterate of of changechange of of velocityvelocity withwith respectrespect toto time.time. AverageAverage acceleration
acceleration isis thethe changechange inin velocityvelocity divideddivided byby thethe changechange inin time.time. InstantaneousInstantaneous acceleration
acceleration cancan bebe determineddetermined byby thethe followingfollowing equation:equation:
lim
lim
∆
∆
∆∆
∆∆
Total
Total accelerationacceleration isis givengiven by:by:
Motion
Motion withwith ConstantConstant AccelerationAcceleration
In
In casescases wherewhere aa particleparticle movesmoves underunder constantconstant acceleration,acceleration, itsits averageaverage accelerationacceleration will
will bebe numericallynumerically equalequal toto itsits instantaneousinstantaneous accelerationacceleration atat anyany givengiven point/spoint/s inin time.
time. AsAs such,such, aa graphgraph of of velocityvelocity vsvs timetime willwill produceproduce aa straightstraight line.line.
1122
22
Non
Non
‐‐
UniformUniform AccelerationAccelerationThis
This occursoccurs whenwhen anan objectobject changeschanges velocityvelocity atat differentdifferent ratesrates atat anyany givengiven pointpoint inin time.
time.
Motion
Motion in
in Two
Two and
and Three
Three Dimensions
Dimensions ((
§3.1§3.1‐‐
3.4,3.4, 4.14.1‐‐
4.6)4.6) VectorsVectors
~Vector
~Vector arrowsarrows indicateindicate thethe magnitudemagnitude andand directiondirection of of thethe vectorvector ~The
Resolution
Resolution andand UnitUnit VectorsVectors
Vectors
Vectors areare resolvedresolved onon thethe basisbasis of of theirtheir directiondirection fromfrom thethe origin.origin. AA guideguide forfor thethe notation
notation isis providedprovided below.below.
Unit
Unit vectorsvectors allall havehave aa magnitudemagnitude of of 1.1. TheyThey areare alwaysalways underlinedunderlined andand indicatedindicated byby a
a ^^ signsign aboveabove thethe wordedworded vectorvector (eg.(eg.
))Polar
Polar CoordinateCoordinate SystemsSystems
This
This isis basebase onddon thethe principleprinciple thatthat anyany pointpoint rr of of coordinatescoordinates (x,(x, y)y) willwill consistconsist of of 22 component
component indicatingssindicating itsits location.location. TheseThese componentscomponents are:are:
VectorResolution
Resolution andand UnitUnit VectorsVectors
Vectors
Vectors areare resolvedresolved onon thethe basisbasis of of theirtheir directiondirection fromfrom thethe origin.origin. AA guideguide forfor thethe notation
notation isis providedprovided below.below.
Unit
Unit vectorsvectors allall havehave aa magnitudemagnitude of of 1.1. TheyThey areare alwaysalways underlinedunderlined andand indicatedindicated byby a
a ^^ signsign aboveabove thethe wordedworded vectorvector (eg.(eg.
))Polar
Polar CoordinateCoordinate SystemsSystems
This
This isis basebase onddon thethe principleprinciple thatthat anyany pointpoint rr of of coordinatescoordinates (x,(x, y)y) willwill consistconsist of of 22 component
component indicatingssindicating itsits location.location. TheseThese componentscomponents are:are:
VectorVector
Vector Subtraction:Subtraction:
Δ
Δvv == vv
‐
‐
uu ToTo workwork outout vectorvector “v“v
‐‐
u”u” wewe mustmust reversereverse “u”“u” andand addadd toto “v”.“v”. LetLet usus imagineimagine thatthat aa carcar isis movingmoving atat 20m/s20m/s NN andand itit turnsturns aa cornercorner soso itit isis nownow movingmoving 20m/s20m/s W.W. ItIt isis possible
possible toto useuse aa vectorvector diagramdiagram toto determinedetermine thethe changechange inin velocity.velocity.
Step
Step 1:1: DrawDraw originaloriginal vectorvector diagramdiagram StepStep 2:2: ReverseReverse “u”“u” StepStep 3:3: SolveSolve forfor x.x.
√ √ 800
800
Therefore Therefore 20
20√ √ 2 2
Or Or28.28ms28.28ms‐‐11SWSW (2(2 DecimalDecimal Places)Places)
Equating
Equating VectorsVectors
The
The processprocess of of equatingequating vectorsvectors isis achievedachieved byby ensuringensuring allall componentscomponents areare equalequal and
and thenthen ensuringensuring thatthat allall forcesforces areare equalequal (as(as forcesforces areare vectorsvectors inin themselves).themselves). AsAs such
such thethe magnitudemagnitude andand directiondirection of of thethe vectorsvectors shouldshould bebe equal.equal.
Equations
Equations of of MotionMotion inin VectorVector FormForm
∆∆
∆∆∆∆
∆
∆
∆∆
∆∆
∆∆∆∆
∆
∆
∆∆
∆∆
Projectile
Projectile MotionMotion
Projectile
Projectile motionmotion cancan bebe thoughtthought of of asas consistingconsisting of of horizontalhorizontal andand verticalvertical components
components whenwhen withinwithin thethe Earth’sEarth’s gravitationalgravitational field.field. TheThe horizontalhorizontal componentcomponent involved
involved thethe projectileprojectile travellingtravelling atat constantconstant velocity.velocity. TheThe verticalvertical componentcomponent involved
involved thethe projectileprojectile experiencingexperiencing aa forceforce of of gravity,gravity, pullingpulling itit downdown atat aa constantconstant acceleration.
acceleration. TheseThese componentscomponents areare resolvedresolved asas vectorsvectors whichwhich areare independentindependent of of one
one another,another, involvinginvolving vectorvector additionaddition toto determinedetermine motion.motion. At
At anyany givengiven time,time, thethe diagramdiagram belowbelow isis indicativeindicative of of thethe componentscomponents of of projectile
projectile motion.motion. TheThe horizontalhorizontal componentcomponent isis equalequal toto VcosVcosθθ,, whilewhile thethe verticalvertical component
component isis equalequal toto VsinVsinθθ.. TheThe equationsequations forfor calculatingcalculating thethe displacement,displacement, accelerations
accelerations andand velocitiesvelocities of of thethe individualindividual componentscomponents areare providedprovided below.below.
θ θ
V
V
V
V
yy= V sin
= V sin θ
θ
OtherOther equationsequations usedused inin projectileprojectile motionmotion include:include:
V
V
xx= V cos
= V cos θ
θ
22∆
∆
∆
∆
∆∆
WhenWhen dealingdealing withwith thethe projectileprojectile motionmotion of of 11 particleparticle inin 3D,3D, itit isis possiblepossible toto alteralter thethe axes
axes toto createcreate aa 2D2D problem.problem. ThisThis ensuresensures thatthat thethe projectileprojectile fliesflies inin thethe planeplane of of thethe page,
page, butbut willwill notnot workwork inin situationssituations of of 22 oror moremore projectiles.projectiles.
Uniform
Uniform CircularCircular MotionMotion
Uniform
Uniform circularcircular motionmotion isis achievedachieved byby aa constantconstant angularangular velocity.velocity. For
For thisthis toto exist,exist, thethe accelerationacceleration vectorvector mustmust havehave aa componentcomponent whichwhich isis perpendicular
perpendicular toto thethe path,path, oror inin otherother words,words, itit pointspoints toto thethe centrecentre of of thethe circle.circle. The
The followingfollowing equationsequations allowallow thethe determinationdetermination of of velocityvelocity andand anglesangles beingbeing sweptswept out: out:
2
2
2
2
NoteNote thatthat whenwhen questionsquestions requestrequest displacementdisplacement of of particlesparticles inin uniformuniform circularcircular motion
motion inin polarpolar form,form, itit isis inin thethe followingfollowing form:form:
Centripetal Acceleration and Period of Rotation
This refers to an acceleration which is centre seeking, and is calculated according to the equation:
Note that centripetal acceleration is never constant owing to the persistent changing of velocity.
The period of rotation can be calculated according to the equation:
2
Example
A plane travels in a horizontal circle, speed v and radius r. For a given v, what is the r for which the normal force exerted by the plane on the pilot is twice her weight. What is the direction of this force?
Centripetal Force:
Vertical Forces:
By eliminatingθ ,
30° , Tangential and Radial Acceleration
Tangential acceleration is the component which will cause a change in the velocity of the particle. It has parallels with instantaneous velocity and is given by:
Radial acceleration is derived from the changes in direction of the velocity, and is given by:
Because radial and tangential acceleration are perpendicular component vectors of acceleration, the magnitude of total acceleration can be given by:
Relative Motion
All measurements are made relative to a frame of reference. When describing the positing or motion of a moving object, we need to state clearly the frame of
reference we are using for our observations.
The velocity of an object, as measured by a moving observer, is referred to as relative velocity. Relative Velocity is the difference between the velocity of the object, relative to the ground, and the velocity of the observer relative to the ground.
When the objects are travelling in the same direction, relative velocity can be calculated by the formula:
V1 –V2
When the objects are travelling in opposite directions, relative velocity can be calculated by:
V1+ V2
Particle Dynamics (
§5.1‐
5.8, 6.1) Newton’s Laws of MotionNewton’s First Law of Motion (Law of Inertia)
An object will remain at rest or travel with a constant velocity unless acted upon by a net force.
Newton’s Second Law of Motion
The force required to move an object is proportional to its mass.
Σ
Note that this law only applies in inertial frames and is a sum of all forces acting on an object.
Newton’s Third Law of Motion
For every action, there is an equal and opposite reaction.
Mass
This is the amount of matter within a given object. A key property is that the mass is proportional to be resistance the object offers when attempting to alter its velocity (for instance, causing a tennis ball to accelerate is easier than causing a train to accelerate). As a consequence, the acceleration of the object is inversely
proportional to the mass of an object when a fixed force is applied.
Applications of Newton’s Laws
Newton’s laws can be applied in terms of the tension in cables. The particles which are applying a tension to the cable can be in equilibrium or under a net force. If the particle is in equilibrium,
∑
0
as there is no force in the x‐
direction. Additionally,∑
0
0
, such that Fgis the gravitational force and T is theupward force provided by tension.
If the particle is under a net force, then
∑
, wherethe force is being exerted in the horizontal plane. This principle can be applied to the y
‐
plane if the force is applied vertically. However, if there is no force appliedvertically, then:
∑
. In other words, the normal force has the same magnitude but opposite direction to gravitational force.
0
Contact Forces
When an object moves while it is in contact with another medium, there is a force exerted which resists the motion. This is because of the interactions between the object and its surroundings; resultant of the jagged nature of objects. These forces are known as contact forces.
The normal component of a contact force is the ‘normal force’ (N). The component which occurs in the plane of contact is the friction force (Ff ).
The normal force is at right angles to the surface and results from deformation. If there exists relative motion, there is kinetic friction (which opposes motion). If there is no relative motion, then there is static friction, which opposes any applied force.
Below are equations which display the ratio of friction forces to normal forces.
(As such, the normal force is proportional to the kinetic friction force)
(As such, the normal force is greater than the static frictional force, where friction can be 0)A generalised law involving friction states that kinetic and static friction are roughly independent of the normal force and of contact area.
A comparative large scale example would be plate tectonics. The jagged nature of plates causes them to lock periodically, and this locking continues until such time as enough force is exerted to break free of this locking. This locking causes static friction due to the lack of relative motion and opposes any applied force to unlock the plates. The breaking free causes a deformation in the plates and then interacts with its surrounds resulting in kinetic friction.
What becomes clear is the force required to keep an object moving is less than that to initially move an object.
Examples
Dynamics of Circular Motion
In circular motion, there exists a radial component of acceleration, nut also a tangential component with a magnitude of
|
|
, therefore the force on the particlehas a radial and tangential component. As such,
and hence∑
∑
∑
Example
A civil engineer wishes to redesign a curved roadway in such a way that a car will not have to rely on friction to round a curve without skidding. In other words, a car
moving at the designated speed can negotiate the curve even when it is covered in ice. Such a road is banked. Suppose the designated speed is 13.4m/s and the curve radius is 35m. Find the angle the road should be banked by.
On a banked road, the normal force has a horizontal component which points to the centre of the curve. However, since the force of static friction is 0, the only
component which can cause centripetal acceleration is
.
. Therefore:∑
.
(1)∑
. 0
.
(2) (1)/(2)=
.
.
27.6°
Work and Energy (
§7.1‐
7.4, 15.1, 7.5‐
7.8, 8.1‐
8.2, 8.5) Mechanical WorkThe work W done on a system by an agent exerting a constant force on the system is equal to:
∆.
where θ is the angle between the force and the displacement vectors.Deforming Springs
Based on the diagram and Hooke’s Law, the work done by the spring on the block is equal to
.
Whereas the work done applied onto the spring is equal to:
Vector Dot Product
This is also known as the scalar product (being the product of 2 scalar quantities). This is due to the notation for the multiplication of scalar properties being (a.b), while (axb) is used for vectors.
Because of this, we can derive scalar products by components. Hence:
.
.
.
.
.
.
.
… . …
.
Since ., . . all equal 0,
.
This can be applied to the problem below:
ariable Forces V
Hooke’s Law
Hooke’s Law deals with the behaviour of products which display linear elasticity. It is based on the principle that the force applied to an object will be proportional to any deformation.
However, he law has a major flaw with it; it only applies to a small portion of the of tive
h below: t
graph of force vs deformation (or intermolecular separation, as this is an indicator deformation). In deformation by bending, some separations are stretched, while others are compressed. There also exists a neutral position where there is no rela stretching or compression. When this stretching or compression exceeds the limitations of the object, the law fails. As such, it can only be used as an approximation.
Clearly the repulsive forces must be substantially stronger overall as it is difficult to compress an object beyond a few percent.
Also quite clear is that at rest, force is 0.
Another clear observation is that there are several ranges over which force is proportional to deformation are very small (where Hooke’s law will apply). Also, each region has its own approximation and hence constant.
Kinetic Energy
Using Newton’s second law, we can derive the following equation for the net work on an object:
.
·
.
However,
Therefore,
∆
Work Energy Theorems
“When work is done on a system and the only change in the system is in its speed, the net work done on the system equals the change in the kinetic energy of the system.”
As such, this theorem indicates that the speed of a system will increase if the net work done on it is positive as the final kinetic energy will be greater. Conversely, the speed of a system will decrease if the work done on it is negative.
Potential Energy
Potential energy is based on the concept of getting energy bac . Not all forces cank store energy however; friction energy cannot be recovered, whereas that in a
compressed spring can be recovered (likewise for work done in a gravitational field). There is a minimum of potential energy at the equilibrium point.
For any conservative force (where work done against is W=W(r)), it is possible to define a potential energy U as
∆
. That is,∆
.
For instance, using gravity, where an object of mass m is being slowly lifted (with no acceleration) from a height yito a final height of yf , it is found that the work done on
the object as the displacement increases is a product of the applied force and the displacement, such that
∆ ∆̂
:
.∆ ̂.
̂
Asa result of this, it is possibleto determine that gravitational potential energy is equal to:
and enceh
∆
ison, a spring, which possesses elastic potential energy:
By compar
∆
arbitrary and can be anything. For instance, with the The choice of zero for U is
example of the spring, U=0 at x=0, and so
Conservative Forces
These forces are where the work done around a closed loop is equal to zero. Such forces have 2 key properties:
• The work done by a conservative force on a particle moving between any 2 points is independent of the path taken by a particle
• The work done by a conservative force on a particle moving through any closed path is zero (where the beginning point and endpoint are identical) An example of such a force is g n
. ravity. Based o
, it becomes clear that only the initial and final co‐
ordinates matter and hence over any closed path the work done will be zero. This is similar for elastic systems. The work of a conservative force is generally represented by
and generally
∆
Non
‐
Conservative ForcesThese are forces where the work done in a closed look cannotequal zero. An example of this would be friction. As such, the mechanical energy is defined as:
(wh ere K is kinetic energy and U is potential energy). Such forces will cause a change in the mechanical energy of the system. For instancbook along a non
‐
idealised table, the kinetic energy is converted toe when sliding a internal energy as heat. Furthermore, the path taken in a closed loop will determine how much
tion.
onservation of Mechanical Energy
kinetic energy is converted to internal energy; the longer the path, the more fric
C
Mechanical energy is generally not conserved. However, if non
‐
conservative forces do no work, therefore mechanical energy is conserved.pplications A
Power
This is the rate of doing work. Generally speaking, W is used for work, while
| |
is used for weight. It is worth noting that when displacing in the horizontal direction, there is no work being done against gravity. Additionally, velocity is normallyf constant. The SI unit is joules per second, or watts. Positive work is that done in the same direction as the force. Negative force is work done opposite to the direction o the force. Average Power:
∆
Instantaneous Power:
Example:Gravitation (
§13.1, 13.4‐
13.6) Newton’s Law of GravitationNewton’s Law of Universal Gravitation states that every mass in the universe is ttracted to every other mass in the universe by a force of gravitation.
a 2 2 1 d G F =
Motion of Planets and Satellites Slingshot Effect
Satellites must have a certain velocity in order to stay in orbit. If not they will either crash into Earth or go out into space and never return. This velocity depends on the force of gra
m
vity acting on the satellite. Satellites paths of motion are affected by seen by orbits and the slingshot affect.
a Motion
law states that all planets orbit the sun in an elliptical orbit where the ted at one of the foci.
Kepler’s second law is a consequence of conservation of angular momentum and
m
gravity, this can be
Kepler’s Laws of Planet ry
Kepler’s first sun is loca
states that a planet will sweep equal areas in equal amounts of time. As such:
where L and M are constants.Kepler’s third law shows the relationship between the period and radius of orbits.
G and Variation
The value of gravity varies as the distance away from the centre of gravity increases. As such, the value of gravity can be seen to be:
where r is the radius of the the surface. As such, a gravitational field exists, and lthough at large distances, the force is negligible. planet and h is the height aboveextends infinitely into space, a Based on
,
̂
wherê
is a unit vector pointing radially outwardfrom the planet, the negative sign indicates the field points to the centre of the planet.
ential Energy Gravitational Pot
While
is a reasonable estimate of the potential energy of an object, it can only apply close to the surface of a planet. Where a more general equation isrequired,
will apply.Escape Velocity
This is when the kinetic energy of an object allows it to escape from the gravitationa field of any object (usually a plane
l t). Hence:
Orbits and Energy
Since non
‐
conservative forces do no work, mechanical energy is conserved. Hence
, where E is the mechanical energy of the satellite. As such,
. By removing velo city andconsidering the circularorbit:
.
Since
Therefore:
As such, a smallradius will result in a very negative potential energy and a very large kinetic energy. Hence the inner planets are faster than the outer planets. Based on this, largeorbits require alarge amountof work to reach the required altitude, where its velocity will decrease at these higher altitudes.
mes evident with spacecraft in orbit. In order to speed up and catch
it will travel faster until it catches up with the second spacecraft. It will then fire its engines forward to slow down and
ence climb up to its original, slower orbit. This beco
up to another spacecraft, a spacecraft will fire its engines backwards, losing energy and doing negative work on itself. This has the effect of more negative mechanical energy, causing it to fall to a lower orbit where
h
Momentum and Collisions (
§9.1, 9.3‐
9.6) Conservation of Linear MomentumTotal Momentum Prior = Total Momentum Following M1U1+ M2U2+… = M1V1+ M2V2+ …
Momentum is conserved in collisions because of Newtons Third Law. This is because there is an equal and opposite reaction to every action done. Because of this, the reaction forces must equal the action forces.
The change in momentum of an object is referred to as Impulse. It is defined by the following equation:
∆
Collisions in One Dimension
Elastic collisions in 1 dimension are those where the total kinetic energy, and hence momentum, of the system is the same before and after the collision. In reality, no collisions are perfectly elastic because objects will deform slightly during the collision as well as a small amount of energy being converted to other forms, such as heat and sound. Based on this, the following apply:
and
Based on this, inelastic collisions are those where the total kinetic energy in a system is different before and after a collision, even if momentum is conserved. This can happen when objects stick together (known as a perfectly inelastic collision) or when an object is substantially deformed (as with a ball) when being bounced. In elastic collisions are generally hard to analyse without additional information.
In a perfectly inelastic collision, both objects will travel with a common
velocity after impact. Being an isolated system, momentum is conserved and as such, the final velocity can be determined by the following:
.Collisions in Two Dimensions
In collisions in two dimensions, momentum is conserved in both the x and y axes independently as
and
Given the initial y component of momentum in a 2 particle system is zero (given the direction can be taken as the x
‐
axis):
cos
cos
Where θ is the angle object 1 flies off at, and φ is the angle at which object 2 flies off at (where object 1 collides with object 2), and:0
sin
sin
If the collision is elastic, then:
. Kineticenergy will not be conserved if the collision is inelastic.
Centre of Mass
The x
‐
coordinates of the centre of mass of a series of particles can be found by the following equation:
∑
∑
∑
∑
Where xiis the x
‐
coordinate of the i particle and the total mass isth
∑
, w the sum runs over n particles. The y and z coordinates of the centre of mass can also be derived by similar equations:here
∑
∑
∑
∑
∑
∑
∑
∑
The vector position of the centre f mass of an extended object can be expressed in the form:
The centre of mass of any symmetrical object li es on the axis of symmetry . For instance, the centre of mass for a sphere is at its and on any plane of symmetry
geometric centre.
The work done by the centre of mass can be given by the change in kinetic energy of the centre of mass.
Example:
any
‐
re of mass of such a system is given by:
M Particle Systems
The velocity of the cent
∑
∑
The total momentum of a system of particles is given by:
∑
∑
he acc
T eleration of the centre of mass can be given by:
∑
∑
As such, the sum of all external forces can be given by:
∑
Rotation (
§10.1, 10.3‐
10.6)Angular Velocity and Acceleration
Angular displacement:
Angular velocity:
Angular acceleration:
Δ
2.Δ
Δ
Note that the above equations are comparable to the projectile equations, however, the respective components of displacement, velocity and acceleration have been divided by r to obtain their angular equivalents.
Angular Quantities Linear Angular
.sin
Example: A bicycle wheel has a radius of 40cm. What is its angular velocity when the wheel travels at 40kmh.
.
.
/
.
28 .
Rotational Kinetic Energy
energy is derived om the sum of the individual kinetic energies of id object.
As seen above, rotational inetic energy is equal to
Rotational kinetic fr
. However, since the particles within a rigk
∑
,
∑
.The total kinetic energy of rolling object is:a
Moment of Inertia
This is a measure of the ability of an object to resist changes in its rate of rotation. It is also referred to as the rotational analogue of mass.
For a system of masses, this is equal to:
∑
For a continuous body:
ecomes that I will depend on the total mass, distribution of mass and shape, as we as the axis of rotation.
What b evident is ll
The momen of inertia can so be defined by the equation:t al
where n is a number.Itcan also be efined asd
where √
. For a hoop,
. Other objects (for which k does not need to be remembered):• Disc:
√
• Solid sphere:
It is worth noting that whenan object has a lower moment of inertia, there is less rotational kinetic energy, and hence more translational kinetic energy.
Torque
Torque is the turning oment of force. Torque that is about an axis of rotation is distance between the end of the beam and the point
m a
equal to the product of the
through which the force is applied and the component of force perpendicular to the beam.
System of Particles
Example: Person on rotating seat holds two 2.2kg masses at arms length and draws them towards their chest. What is the increase in agular momentum? Is K
First Year University Physics 1A
Topic 2: Thermal Physics
Temperature
Temperature and Thermal Equilibrium
Temperature is the concept by which an object is perceived to be hot or cold. The definition of temperature is dependent upon the concepts of thermal contact and thermal equilibrium. At higher temperatures, particles will possess more energy and hence have more kinetic energy and collisions.
Thermal contact is where two objects are able to exchange energy between themselves, either via heat or EMR. This is a consequence of a temperature
difference between the two objects. When particles collide with a wall, it can
transfer energy to this wall, which in turn can transfer the energy to the particles on the other side of the wall.
Thermal Equilibrium is where two objects exchange no net energy when placed in thermal contact. As stated above, this contact does not need to be physical as energy can be transferred via EMR. A key law related to thermal equilibrium is the Zero
‐
th Law of Thermodynamics. This law states that if any two objects A and B are separately in thermal equilibrium with a third object C, then A and B are in thermal equilibrium with each other. Consequently, if A and B are brought together, no net energy will be exchanged.Temperature can then be defined using thermal equilibrium. It can be
thought of to be a property which determines whether or not an object is in thermal equilibrium with another object/s. If two objects are in thermal equilibrium, then they are the same temperature. Quite evidently, the temperature is now defined by energy.
Measuring Temperature
The temperature of an object is measured with a thermometer. They are based on the principle that some physical property of a system changed as the temperature of the system changes. A temperature scale can be based on any of the following
properties:
• Volume of a liquid (Mercury or alcohol)
• Dimensions of a solid (such as in the expansion/contraction of train tracks) • Pressure of a gas at a constant volume (Ideal gases)
• Volume of a gas at a constant pressure (Ideal gases) • Electrical resistance of a conductor
(As heat rises, conductivity decreases proportionally to the temperature) • Colour of an object (as with Black Bodies)
In calibrating a thermometer, it must be placed in thermal contact with some natural system that remains at a constant temperature (such as the triple point of water). The Celsius scale is based upon the ice point of water being at 0 and the boiling point at 100 with 100 increments in between these points.
A key problem with thermometers, especially liquid in glass variants (such as those involving alcohol or mercury), is that the thermometer may only agree with the calibration point/s. As a consequence, there can be large discrepancies when the temperature is far from these points (the scale between calibration points may not be linear). Thermometers also have a limited range; for instance, mercury cannot operate below
‐
30°C and alcohol cannot work above 85°C.The constant volume gas thermometer is based upon the effects on the pressure of a fixed volume of gas as the temperature changes. An increase in the temperature causes greater pressure, which pushes the mercury out. The height difference is proportional to the temperature.
When comparing temperature, it is important to use the Kelvin scale because this is a true representation of the kinetic energy of an object. For instance, water at 100°C is not twice as hot as water at 50°C. However, a flame at 233°C is twice as hot as an ice cube at
‐
20°C.Absolute Zero
This concept is based upon the classical physics principle that at 0K, particles will have zero kinetic energy. It would follow logically that the molecules would then settle out on the bottom of the container. However, quantum physics shows that some residual energy will remain, and this is called zero
‐
point energy. Furthermore, the concept of absolute zero is a theoretical concept. This is because for a substance to be at absolute zero, the container must be at absolute zero because the two objects would be in thermal contact. This problem continues to compound, and hence for an object to be absolute zero, everything must be at absolute zero.Thermal Properties of Matter
When an object is heated, it will expand. Consequently, the joints in many objects allow room for this in the form of expansion joints. This expansion is a consequence of the change in average separation between atoms in an object. If this expansion is small relative to the original dimensions, then a good approximation of this change in dimensions is this is proportional to the change in temperature.
Linear Expansion
The coefficient of linear expansion is as follows:
It should be noted that some, but not all, materials can expand in one direction while contract in another as temperature increases. Also as the linear dimensions change, the surface area and volume will also change.
Volume Expansion
This is based on the principle that the change in volume is proportional to the original volume and the change in temperature.
∆
∆
Where β is the coefficient of volume expansion and Viis the initial volume.
In solids,
3
. However, the formula assumes the material is isotropic, or the same in all directions.The principle of volume expansion can be exploited in thermometers. The greater volume a liquid will expand, the greater the accuracy of the thermometer.
Area Expansion
This is based on the principle that the change in area is proportional to the original area and change in temperature.
∆ 2
∆
Where Aiis the initial area.
Bimetallic Strips
These are strips of metal containing two different metals physically bonded to one another. At a certain temperature, the strip will be perfectly straight. However, as one strip expands more than the other, it will bend as the temperature changes. An application for such an object is in thermostats.
Water
Water, unlike other substances, will increase in density as its temperature rises from 0°C to 4°C. Above 4°C, water will behave like any other substance. This is a
consequence of the hydrogen bonds between molecules.
Changes in Volume
While the volume expansion equation requires an initial volume for temperature change, there is no equilibrium separation for the atoms in a gas. In other words, there is no standard volume for any fixed temperature, and hence the volume depends sole on the container. Consequently, the volume for gases is variable. And the change in volume is considered.
Kinetic Theory and the Ideal Gas
Macroscopic Properties of a Gas Equation of State for a Gas
This describes how the volume, pressure and temperature of a gas of mass m are related.
The Mole
The amount of gas within a given volume can be expressed in the number of moles. One mole of any substance contains Avogadro’s number of constituent particles. The number of moles is calculated by
Boyle’s Law
When a gas is kept at a constant temperature, its pressure is inversely proportional to its volume.
Charles and Gay
‐
Lussacs’s LawWhen a gas is kept at a constant pressure, its volume is directly proportional to its temperature.
Ideal Gas Law
The equation of state for an ideal gas is:
Where n is the number of moles, R is a constant called the Universal Gas Constant (8.314J/mol . K), T is temperature, P is pressure, V is volume, and the units for PV is Joules.
This law is often stated in terms of the total number of molecules present, hence:
Where kBis Boltzmann’s constant of
and
1.38 10
/
.It is common to call P, V and T the thermodynamic variables of an ideal gas. This law helps explain that the pressure a gas exerts on the walls of a container are a consequence of the collisions of gas molecules with the wall.
Ideal Gas Law Assumptions
1. The number of molecules in the gas is large and the average separation between the molecules is large compared with their dimensions. Such molecules occupy a negligible volume in the container.
2. The molecules obey Newton’s laws of motion, but as a whole they move randomly. Consequently, any molecule can move in any direction with any speed. At any given moment, a certain percentage of molecules move at high speeds and a certain percentage will move at slow speeds.
3. The molecules interact only by shot range forces during collisions. Hence there are no attractive or repulsive forces between them.
4. Molecules make elastic collisions with the walls
5. The gas under consideration is a pure substance. In other words, all the molecules are identical (not entirely true in reality due to isotopes) The first 3 assumptions are the most important however.
Molecular Model of the Ideal Gas
The molecular model for a gas was developed by Brown in 1801 after observing that pollen suspended in water moved in an irregular pattern. He thought that the pollen contained some life
‐
force, however, it is now known that this was from watermolecules bumping into the pollen randomly. This was the first evidence of atomisation which was an observation rather than a deduction.
Kinetic Interpretation of Temperature Pressure and Kinetic Energy
Note the use of
, which is the mean value of the speed squared. This is aconsequence of the large number of particles in a gas, and hence it is impossible to refer to a specific particle. Furthermore, total velocity is zero as there as many vector components in one direction as the other.
By comparing the above equation with that for the Ideal Gas Law we find:
Hence, the temperature is a direct measure of the average molecular kinetic energy. Simplifying the equation gives:
And given this can be applied in any direction:
The
is a consequence of each component (x, y and z) are
of the overall equation.Using this, it becomes apparent that the translational degree of freedom contributes an equal amount of energy to the gas (given each direction is independent of the others). This concept of each component contributing equally to the energy of the system can be referred to as the Theorem of the Equipartition of Energy.
Total Kinetic Energy of a Gas
The total kinetic energy is just N times the kinetic energy of each molecule:
Noting that
If the gas has only translational energy, the this is the internal energy of the gas. Based on this, the internal energy of an ideal has depends solely on temperature.
Root Mean Square (RMS) Speed
This is the square root of the average of the squares of the speeds. Hence:
Where M is the molar mass.
Heat and the First Law of Thermodynamics
Heat and Internal Energy of Ideal Gases
In 1850, Joule discovered a link between the transfer of energy by heat in thermal processes and the transfer of energy by work in mechanical processes. This led the concept of energy to be generalised to include internal energy.
Internal energy is the sum of all energies possessed by a particle. For
instance, in a gas, this includes gravitational potential, vibration, rotational, random translational, chemical potential and rest mass energies. The kinetic energy due to motion through space is NOT included. Internal energy can be changed by both the application of heat (or flow of energy) or work (applying a force).
Thermal energy refers to the sum of gravitational potential, rotational,
vibrational and random motion kinetic energies. This is represented by the symbol Q. Heat can be interpreted to be a flow of energy between two or more
Heat Capacity
This is defined as the amount of energy required to raise the temperature of a sample by 1°C. It can be depicted by the following equation.
∆
.Specific Heat for Solids and Ideal Gas
Specific heat is the heat capacity per unit area, or
. The equation for specificheat is given by
∆
. The specific heat is a measure of how insensitive an object is to temperature change as a greater specific heat will require more energy to change that substance’s temperature.Sign conventions for heat capacity and specific heat are:
• If temperature increase, Q and ΔT are positive as energy transfers into the system
• If temperature decreases, Q and ΔT are negative as energy transfers out of the system.
The specific heat of water is rather large compared to many other substances. The consequences of this are various weather phenomenon, such as moderated
temperatures along the coast and sea breezes.
Calorimetry
This is a technique for measuring the specific heat of a substance. It involves heating a material, adding it to a sample of water, and then recording the final temperature. Assuming the system of the sample and the water is isolated, conservation of energy requires that the amount of energy which leaves the sample to be the same as the energy which enters the water.
This minus sign is important, and is indicative of the sample losing energy, which the water gains energy.
Phase Changes
This refers to the change of physical state of a substance, such as solid to liquid. During a phase change, there is no change in temperature of the substance. The energy required to effect this change is called Latent Heat.
Latent Heat
This is the amount of energy required to cause a substance to change state. It is equal to
where m is the mass of the sample. The latent heat of fusion is the energy required to change from solid to liquid, while the latent heat ofvaporisation is the energy required to change from liquid to gas. A positive sign will be used to indicate energy being transferred into the system. Conversely, a negative sign will indicate that energy is lost by the system. An important concept in
calculating latent heat is that the temperature will not change until the sample has completely changed state.
Work Done on an Ideal Gas
The state variables will describe the macroscopic state of a system. In an ideal gas, these are pressure, temperature, volume and internal energy. However, this
macroscopic state can only be specified if the system is in thermal equilibrium. Transfer variables describe the changes in state. They are zero unless a process occurs to cause the transfer of energy across a system boundary. For example, heat and work are transfer variables. For instance, heat can only be assigned a value if it crosses a boundary.
The work done on an ideal gas can be given by:
. . .
Since P=F/AThe change in volume is given by
.
and hence the work done is .
.The total work done is given by:
PV Diagrams
These are diagrams showing the correlation between pressure and volume to allow a determination of the work done on an ideal gas. The work done on such a diagram is very dependent on the path taken.
The above diagram indicates that the volume has been reduced before the pressure is increased by heating.
The above diagram shows that the pressure was firstly increased before the volume was decreased.
The above diagram shows that the pressure and volume continually changed. While
is an applicable equation for the first 2 cases, in diagram 3, the evaluation of work requires the P(V) function to be known.Conversion of Work to Thermal Energy
If a piston compresses a gas, the kinetic energy of the particles will be increased. This is through conservation of momentum, where the moving piston supplies kinetic energy to the particles, thereby increase thermal energy.
First Law or Thermodynamics
This is a special case of conservation of energy taking into account the change in internal energy through energy transfers in work and heat. The law states that:
A key consequence of this law is that there must exist an internal energy which is determined by the state of the system. For infinitesimal changes:
Applications of the First Law of Thermodynamics
The Adiabatic process is where no energy enters of leaves the system by heat. This is achieved by insulating the system, or having the system proceed quickly enough that no heat can be exchanged. Since
0,∆
. If the gas is compressed in this manner, then W is positive, so internal energy is also positive and hencetemperature increases.
The isobaric process is one which occurs at constant pressure. The work done is
, where P is constant.The isovolumetric process takes place in constant volume. Since there is no change in volume, W=0. Hence in internal energy equals Q. Additionally, if any heat is added, since the volume is constant, all of the energy transferred results in an increase in internal energy.
The isothermal process occurs at constant temperature. Since temperature doesn’t change, internal energy equals zero. Any energy that enters must leave the system. Its PV graph is as follows:
Since
, the equation forms a parabola. Also, since the gas is an ideal gas and the process is quasi‐
static, .log
If the gas expands, then Vf >Vi, and hence the work done is negative.
The Transfer of Heat
Cyclic processes are where the system starts and ends at the same state. Such a process is not isolated. On a PV diagram, this would be indicated by a closed curve. Because the internal energy is a state variable, there is no change in internal energy. Hence, if
∆
0,
. In such processes, the net work done per cycle is equal to the area enclosed by the curve on a PV diagram.Heat is typically transferred by conduction, convection or radiation. In
conduction, particles become energised and collide with other particles. An increase in kinetic energy therefore will transfer through to the other particles, thus
conducting heat. However, the conduction process can only take place if there is a difference between two parts of the conducting medium. The rate of transfer can be given by
where κ is a constant of thermal conductivity, P ispower, dx is the thickness and Ta and Tbare the temperatures of 2 different
materials, where substance A is hotter than B.
Convection refers to energy transfer be moving a substance. It is often associated with changes in density, such as in air. This is referred to as natural convection. Forced convection is achieved with fans or pumps. Convection results from the heating of air, such that it expands and rises, while cooler air cycles in. Thus a continuous current is established.
Radiation does not require physical contact. It is a result of the IR emissions of a warm body. The rate of radiation can be given by Stefan’s law stating:
, where P is the rate of transfer (in Watts), σ is a constant(
5.669610
), A is the surface area, e is the constant of emission or emissivity, and T is the temperature in K.The rate at which an object radiates heat is determined by its surrounds, hence
