PHY 143
Basic Physics For Engineers I
E-Notes
Prepared by
Mohd Noor Mohd Ali
Physics Lecturer
Applied Science Department
University Teknologi MARA Pulau Pinang
Offered Since July 2007Physics and measurements (2 hrs) ... 4 Units ... 4 Fundamental Quantities... 4 Derived Quantities ... 4 Prefixes... 4 Significant figure ... 5 Conversion of units ... 5
Scalar s and vectors (3 hrs)... 7
Definition... 7
Vector Notation... 7
Addition (& subtraction) of vector components - Geometrical Method... 7
Addition (& subtraction) of vector components - Unit vector method... 8
Resolution of vectors into x and y components... 8
Kinematics (3 hrs)... 10
Position, displacement and velocity... 10
Instantaneous velocity is defined as the velocity at a particular instance in time. ... 10
Acceleration, dv/dt ... 11
Graph of velocity versus time... 12
Constant acceleration in linear motion... 13
Motion in two dimension... 13
Free Fall motion... 14
Newton’s Laws of Motion, Linear momentum and collisions (5hrs)... 15
Definition... 15
The Force Law (Newton’s Second Law) ... 15
Types of forces (Gravitational, Normal & Friction) ... 15
Free body diagram ... 16
Linear momentum and Collisions... 16
Collecting the components ... 17
Collision between two bodies... 17
One moving, one stationary initially... 17
Both moving with different velocity in the same direction initially... 17
Both objects moving towards each other initially. ... 17
Both objects move with different velocities after collision... 17
One object stationary, one moving after collision. ... 17
Both objects moves together with the same velocity after collision... 17
Work, energy and power (3 hrs) ... 17
Work, energy & power... 17
Conservation of mechanical Energy ... 18
Work-Energy theorem... 18
Equilibrium of a particle... 21
Definition of Torque ... 21
Equilibrium of rigid body... 21
Oscillatory motion (3 hrs) ... 22
Periodic motion... 22
Simple harmonic motion (SHM) ... 22
Simple Pendulum ... 23 Conservation of Energy in SHM... 24 Mechanics of solids (2 hrs) ... 25 Elasticity... 25 Young’s Modulus... 25 Shear modulus... 25 Bulk modulus... 26 Mechanics of fluids (2 hrs)... 27
Density and relative density ... 27
Pressure ... 27
Buoyancy. ... 28
Vibrations and waves (2 hrs)... 30
Introduction... 30
Type of waves... 30
Characteristic of wave motion ... 30
The propagation of Waves... 32
Wave equations... 32
Temperature, Thermal expansion and the Ideal Gas Law (4 hrs)... 34
Temperature and thermometers ... 34
Thermal expansion... 34
The gas laws and Absolute temperature... 37
The ideal gas law ... 38
Heat and The First Law of Thermodynamics (4 hrs)... 40
Heat as energy transfer... 40
Specific heat... 40
Latent heat of fusion and evaporation... 40
The first law of thermodynamics ... 40
Physics and measurements (2 hrs)
Physics is an empirical science, which means that measurements are made. To communicate the measurements, system of units is used. Comparison between the measurements can be made if a common unit of measure is used, or at least a known conversion is know, Units
A common system of units used in physics is the SI (Systeme International d’Unites), which is a revised metric system. This system uses meter for length, kilogram for mass and secaon for time.
Fundamental Quantities
Fundamental quantities are the basic quantity of measure. The fundamental quantites of the SI units, its corresponding symbol, the unit of measure and symbol for unit of measure is given in the table below.
Quantity of measure Unit of measure Symbol for Unit
Length, l meter m
Mass, m kilogram kg
Time, t second s
electrical current, I ampere A
Temperature, T kelvin K
amount of substance mole mol
luminous intensity. candela cd
Derived Quantities
Derived quantities are quantities which can be described using the fundamental quantities. Some examples are given below.
Derived quantities Units Symbol of unit
momentum kilogram – meter per second kg m s-1
Force Newton, kilogram – meter
per second per second
N, kg m s-2
Energy Joule, Newton.meter J, kg m2 s-2
Derived quantities are obtained when physical quantities are multiplied or divided with one another. Subtration or addition will not produce derived quantities.
terra (T) 1012 giga (G) 109 mega (M) 106 kilo (k) 103 deci (d) 10-1 centi © 10-2 milli (m) 10-3 micro (µ) 10-6 nano (n) 10-9 pico (p) 10-12 fempto (f) 10-15 Significant figure
When measurements are made, the accuracy of the measuring instruments determine the number of significant figures (digits) that can be taken for the measurement. This is the number of significant figures due to the precision of measuring instrument.
Example
Meter rule – smallest division is 1 mm, Therefore readings can be made to the nearest mm, e.g. 0.563 m – gives 3 significant figures.
Vernier Caliper – readings can be made to the nearest 1/10 of 1 mm, eg 5.42 cm –3 sig. fig Micrometer screw gauge – readings can be made to 1/100 of 1 mm, eg. 14.56 mm – 4 sig fig When quantities undergo mathematical operation, the number of significant figure will usually decreases. The result of a mathematical operation will usually have the smaller number of significant figure from the significant figures of the quantities that undergo the mathematical operation.
Mathematical operation does not increase the precision of the result. Ex.
0.563 m x 14.56 mm = 8.1536 x 10-3 m2 = 8.15 x 10-3 m2 to 3 significant figures.
0.563 m + 14.56 mm = 0.57756 m = 0.576 m to the decimal place of the coarser instrument (meter rule)
Conversion of units
Conversion of units is done by treating the prefix value factor and units as ordinary variables Conversion due to prefix
123.4 m into cm 100 cm = 1 m
Or 1cm = 1 x 10-2 m
123.4 x 1 cm / 1 x 10-2 m = 1234 cm Conversion due to unit’s system 5 feet 5 inch into m
1 feet = 12 inch 1 inch = 2.54 cm
Scalar s and vectors (3 hrs)
Definition
Scalars are physical quantities which have magnitude only. Vectors are physical quantities which have magnitude and direction.
Example of scalars: distance travelled, time, energy
Example of vectors: force, momentum, velocity, acceleration Vector Notation
A vector is represented graphically by an arrow.
The size (length) of the arrow represents the magnitude of the vector, the direction of the arrow is the direction of the vector.
In written form a vector symbol is written in bold typeface, A , or with an arrow A
r
over the symbol. The direction has to be stated explicitely
Addition (& subtraction) of vector components - Geometrical Method When two vectors are added together, the tail of the second vector is connected to the tip/head of the first vector.
The result of the addition is called the resultant vector.
Note : A + B = B + A
Vector subtraction is the addition of a vector with a negative vector.
A negative vector is a vector having the same magnitude as the original vector but opposite in direction.
Example A - B = A + -B
Note : A + - B = - (B + -A)
Addition (& subtraction) of vector components - Unit vector method
A vector A can be written in the form, A = Ax iˆ + Ay ˆ , where j iˆ and jˆ are unit vectors (
magnitude 1, but no associated unit) and point in the +x and +y direction respectively, and Ax
and Ay are the x and y components of the vectors respectively.
A + B = (Ax iˆ + Ayˆ ) +( Bj x iˆ + By ˆ ) j
= (Ax + Bx) iˆ + (Ay + By) jˆ
Resolution of vectors into x and y components.
We use trigonometry to assist us in working with vectors in number.
A vector can be resolve into components along/parallel to a selected set of axis.
Using x-y axes as reference axes. The vector is resolved into components along the x and the y axis.
Ax = A cos θ1i
The magnitude of the resultant vector C = A + B = SQRT[(A cos θ1 + B cos θ2)2 + (A sin θ1
Kinematics (3 hrs)
Position, displacement and velocity
Position – the location of a point / particle / object, i.e. the labels or coordinate with respect to a system of axes (i.e. x-y-z axes)
Position Vector is the vector from the origin to the position / coordinate point. Distance traveled is the length of the path of the object
Displacement - The length vector along a straight line between two positions.
Normally the reference point is the origin (0, 0) in the y plane (2-dim), or x = 0 on the x-axis.
The position vector is represented by r = x iˆ + y jˆ (2-dim) or r = x iˆ ( 1-dim)
Speed - The rate of change of distance traveled. Speed is a scalar quantity. Average Speed = distance traveled divided by time to cover the distance Instantaneous speed is defined as the speed at a particular instance in time.
Example: A student walks a distance of 100 meters in 20 seconds. Find the average speed of the student. Average speed = 100 meters/20 seconds = 5 meter/second = 5 m/s= 5 ms-1 Velocity - The rate of change of displacement. Velocity is a vector.
Average velocity = the displacement divided by time to cover the distance
i t x i t t x x vavg ˆ ˆ 0 1 0 1 ∆ ∆ = − − = r
Instantaneous velocity is defined as the velocity at a particular instance in time.
t x v t ∆ ∆ =lim∆ →0
The position versus time graph is a plot of the particle position with respect to time. It shows the particle position at some instances in time.
From the definition of velocity, i
t x i t t x x vavg ˆ ˆ 0 1 0 1 ∆ ∆ = − − = r
, the average velocity between two position / time interval can be obtained from the graph.
The average velocity between 0 - 5 sec is the slope of the graph between 0 - 5 sec. The average velocity between 5 - 10 sec, 10 -15 sec, 15 – 20 sec is the slope of the graph between 5 -10 sec, 10 -15 sec, 15 – 20 sec respectively.
The average velocity between 0-10 secs is however is not the slope of the graph, and has to be computed.
Acceleration, dv/dt
Acceleration - The rate of change of velocity. The acceleration is a vector.
Average acceleration = the change in velocity divided by the time the change occur.
0 1 0 1 t t v v a − − = r r r
Along the x axis, i t t v v a ˆ 0 1 0 1 − − = r
The equations of motion.
i t t x x vavg ˆ 0 1 0 1 − − = r 0 1 0 1 t t v v a − − = r r r
substitute replace Rewriting v−u=at, and t s u v+ = 2 Then, as u v t s at u v u v 2 ) 2 )( ( ) )( ( 2 2 = + = + − v2 =u2+2as
Graph of velocity versus time
i t x vavg ˆ ∆ ∆ = r i t s vavg= ˆ r t v i s avg r = ˆ i t v i sˆ= avg ˆ t v s= avg t u v a r r r − = i at i u i vˆ= ˆ+ ˆ i t u v i aˆ= − ˆ at u v= + at u v= + 2 u v vavg = + t v s= avg t u v s 2 + = t u at u s 2 + + = 2 2 2 at ut s= + 2 2 1 at ut s= +
The velocity versus time graph gives us the the velocity at some instances in time. In the above example the particle has a constant velocity of 4 m s-1 between 0 – 10 sec. Its velocity decreases to 2 m s-1 in the next 5 sec (5 – 10 sec interval), then increases until it reaches a velocity of 6 m s-1 at t = 20 s.
The particle has zero acceleration in the time interval 0 – 5 sec, undergoes deceleration (acceleration is negative) between 5 – 10 sec, and acceleration during the time interval 10 -20 sec.
Constant acceleration in linear motion
A particle or body moving in a straight line, for example along the x- axis is said to be moving in linear motion. The path of the particle can be represented by a straight line. To simplify analysis, we will only consider the particle to be moving at constant acceleration over a certain time. However, the acceleration of the particle can be of a different value over another time range. A graph of velocity versus time will show straight line segments, similar to the velocity versus time graph above. The kinematics equations can be use over the time when the acceleration is constant.
Example:
A particle moves in a straight line during its motion. It moves from rest (velocity = 0), with an acceleration of 0.5 m s-2 for 10 seconds. Its acceleration then becomes zero for the next five seconds. It then decelerates and stops moving in another 15 seconds. Find
a) The velocity of the particle at 12 seconds. b) The distance it moves during its motion. c) The deceleration of the particle.
Motion in two dimension
Particles or bodies which move in a plane, for example on the x-y plane is said to undergo motion in two- dimension. The velocity of the particles / bodies can be resolved along two perpendicular directions (x-y axes).
An airplane is moving with a velocity of 300 km hr-1 in the direction 030o. It flew for 45 minutes before it changes direction to 110o. It then flew for another 30 minutes with a speed of 350 km hr-1. What is the distance it traveled?
Particles and bodies in projectile motion also move in two dimensions – the horizontal and the vertical direction.
Example:
An arrow is shot at angle of 450 with respect to the ground. The arrow leaves the bow at 50 ms-1. What is the horizontal distance it traveled when it return to its initial height from the ground? (Assume there is no air resistance and the acceleration due to gravity is 9.8 ms-2) Free Fall motion
When an object is release from a height above ground, it will fall to the ground with acceleration due to gravity, g which is 9.8 ms-2. The acceleration due to gravity is always pointing towards the ground. If an object is thrown vertically upwards with a certain speed, it will also fall to the ground with the acceleration due to gravity.
Example:
1. An object is released from rest from a height of 10 m above the ground. Calculate its velocity as it hits the ground.
2. A ball is thrown upwards at a speed of 30 ms-1. How high does the ball travel, before it returns to the ground? Calculate the total time it takes during flight.
Newton’s Laws of Motion, Linear momentum and collisions (5hrs)
Definition
Newton’s First Law states that an object at rest will remain at rest and an object in motion will remain in motion unless act upon by a force.
An object’s velocity does not change if and only if the net force acting on the object is zero. Newton’s First Law is also called the law of inertia. Inertia is the resistance to changes in velocity. Inertia is a property of matter. Inertia is proportional to the mass of the object. An object with a larger mass has a larger inertia than a less massive object.
Newton’s Third Law states that for every action there is an equal and opposite reaction. Forces acts in pairs. Every force is part of the interaction between two objects, and each of the objects exerts a force on the other.
Example: When we sit on a chair, we exert a force on the chair. Conversely, the chair exerts an equal but opposite force on us.
The Force Law (Newton’s Second Law)
Newton’s Second Law states that the acceleration of an object is proportional to the net force on the object and inversely proportional to the mass of the object.
In mathematical representation
Fnet = ΣF = ma, ΣF is the sum all forces acting on the body of mass m, and a the acceleration.
Example:
A constant force of 10 N is applied to a body of mass 2 N which is initially at rest. What is the acceleration on the body? Calculate its velocity after 5 seconds.
Types of forces (Gravitational, Normal & Friction)
A force is a push or pull. Forces can be categorized in several forms, by how it acts on another object and by the source of the force.
Contact forces are forces acting on objects because the objects are in touch with one another. For example the force from a chair holding up a sitting person is a contact force.
Long range forces are forces which do not require the objects to be in touch. For example the force the sun exerts on the earth to keep it rotating around the sun. The force between two magnets is also an example of a long range force.
The Gravitational force is an example of long range force. The gravitational force is given as 2
r GMm
F = , where G is the universal gravitational
constant, G = 6.674 x 10-11 Nm2 kg-2, M and m are the masses of the objects in interaction, and r the distance between the two objects.
Near the surface of the earth the gravitational force is given as F = mg, where g = 9.81 N kg-1
The normal force is the reaction force exerted by a surface in the direction perpendicular (normal) to the surface. The force the seat of a chair exerts on a person sitting on it is a n example of normal force.
Friction or frictional force is a force which acts between two sliding surfaces and opposes the direction of motion of the sliding surfaces. Friction exists because surfaces are not
microscopically smooth. The irregularities between the surfaces produce friction. Friction can be reduced between sliding surfaces by polishing the two surfaces smooth, applying lubricant between the surfaces or inserting balls or rollers (bearing).
The magnitude of the frictional force is propotional to the normal force, f =uN. The
direction is always opposite the direction of motion. µ is the coeficient of friction
(static/dynamic) Free body diagram
A free body diagram is a diagram of an object which shows all forces acting on the object. In a many body problem, free body diagrams are drawn for each body in the problems. All the forces acting on that particular object are drawn on the object. A free body diagram helps in analyzing a kinematics problem.
∑
=∑
i l after l before i p prThe total linear momentum of an isolated system remains constant (is conserved). An isolated system is one for which the vector sum of the external forces acting on the system is zero. A simple system contains two object which can interact with each other. The masses of the objects remain unchanged, but their velocities might change during the interaction.
By The Principle of Conservation of Momentum, the total momentum, (ie the sum of the momentum of the two objects) before the interaction is equal to the total momentum, (ie the sum of the momentum of the two objects) after the interaction.
Sum of momentum before interaction = sum of momentum after interaction.
The interaction between the objects is normally called collision. Solution
From,
Collecting the components
Collision between two bodies
One moving, one stationary initially
Both moving with different velocity in the same direction initially. Both objects moving towards each other initially.
Both objects move with different velocities after collision. One object stationary, one moving after collision.
Both objects moves together with the same velocity after collision
Work, energy and power (3 hrs)
Work, energy & power
Work is the product of the displacement due to the force with the component of the force along the direction of the displacement.
W = (F cos θ) s, where F is the force, s is the displacement and θ is the angle between the
force and displacement.
∑
=∑
i l l l i iu mv mr r 2 2 1 1 2 2 1 1u mu mv m v m r + r = r + r 2 2 1 1 2 2 1 1u m u mv m v m r + r = r + r ) ˆ ˆ ( ) ˆ ˆ ( ) ˆ ˆ ( ) ˆ ˆ ( 1 1 2 2 2 1 1 1 2 2 2 1 u i u j m u i u j m v i v j m v i v j m x + y + x + y = x + y + x + y ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( 1 2 2 1 1 2 2 1 u i m u i m v i m v i m x + x = x + x ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( 1 2 2 1 1 2 2 1 u j m u j m v j m v j m y + y = y + y x x x x m u mv mv u m1 1 + 2 2 = 1 1 + 2 2 y y y y mu mv mv u m1 1 + 2 2 = 1 1 + 2 2Kinetic energy is the energy associated with motion.
The kinetic energy of an object with mass m and speed v is given by 2 2 1
mv KE=
Potential energy is the energy associated with an object’s position or chemical composition. Potential energy in a spring
The potential energy stored in a spring is given by 2 2 1
kx PE=
Where k is the spring constant and x, the extension of the spring Gravitational potential energy
The energy of an object due to its position relative to the surface of the earth.
The gravitational potential energy of an object of mass, m and height h, relative to an arbitraryly set zero is PE =mgh
Unit of Energy
The unit of energy is joule (J). 1 joule = 1 N m
1 joule = 1 kg m2s-2 Power is the rate of doing work
Power is the rate of energy used Power is the rate of energy supplied P = Power
W = work, E = Energy used/supplied t = time taken for work done
Conservation of mechanical Energy
Mechanical energy describes the potential energy and kinetic energy present in the components of a mechanical system.
The mechanical energy of the system is conserved. The energy changes form from kinetic energy to potential energy and vice versa.
Note: Energy is always conserved, but in a mechanical system the energy is assumed not to change into chemical, nuclear or electrical.
Work-Energy theorem
When a net external force does work, W on an object, the kinetic energy of the object changes from its initial value KEo to a final value KEf, the difference between the two being
Rotational motion (3 hrs)
Motion of a body rotating/ spinning about an axis.
When a body is rotating/spinning about an axis then each points on the body are moving in circular paths centered at the axis of rotation
If a line is drawn from each points to the axis of rotation, the radial lines will have all have the same angular speed about the axis of rotation.
The radial lines moves at the same angular speed. The motion of the radial lines are common to the body.
Thus the rotating body can be described using the motion of a radial line on the body. Angular position, angular displacement and angular velocity
Angular position is defined by r ∠θ, where r is the distance from the axis of rotation and
θ the angle the radial line makes with the positive x- axis
Angular displacement – the change in angular position. θ∆ = θf - θI
Angular speed
Average angular speed
0 1 0 1 t t t − − = ∆ ∆ = θ θ θ ω
Instantaneous angular speed
dt d t t
θ
θ
ω
= ∆ ∆ =lim∆→0Note: If moving CCW, angular velocity is positive, moving CW, angular velocity is negative Angular acceleration
Average Angular acceleration
0 1 0 1 t t t − − = ∆ ∆ = ω ω ω α
Instantaneous Angular acceleration
dt d t t
ω
ω
α
= ∆ ∆ =lim∆→0Constant angular acceleration in rotational motion
When a body is moving with a constant angular acceleration in rotational motion, a set of equations to describe the motion is obtained.
θ α ω ω α ω θ ω ω θ ω ω ω α θ θ θ ϖ ∆ + = + = ∆ + = ∆ ∆ ∆ = − − = ∆ ∆ = − − = 2 2 1 2 ) ( 2 0 2 1 2 0 1 0 0 1 0 1 0 1 0 1 t t t t t t t t t
The equations of motion for rotational motion are similar to that of linear motion. Relation Between Linear and Rotational Quantities
The distance along the circular path is give as
ω θ θ θ r dt d r dt r d dt ds r s = = = = ) ( But vT dt ds = ω r vT = ∴
Differentiating the velocity with respect to time gives the tangential acceleration
T a = ω ω rα dt d r dt r d dt dv = = = ( ) α r aT = ∴
Static equilibrium (3 hrs)
Equilibrium of a particle
An particle is in equilibrium if the sum of forces acting on it is zero.
∑
F =mar=0r
Definition of Torque Torque/Moment is a vector.
It is a result of a vector multiplication called the cross product.
θ τ τ sin rF F r = × =r r r
( vector definition of torque / magnitude of torque by vector resolution and directon by observation)
If the torque/moment causes a rotation in the clockwise direction, it’s direction is said to be negative.
If the toque/moment causes a rotation in the counter clockwise direction, it’s direction is said to be positive.
Equilibrium of rigid body
An rigid body is in equilibrium if the sum of forces acting on it is zero, and the sum of moments on the body is zero.
Physically, a body in equilibrium has zero translational / linear acceleration and zero rotational acceleration.
Translational Equilibrium
∑
F =mar=0r
Acceleration is zero, velocity is constant or zero. Rotational Equilibrium
Sum of moment is zero
∑
=∑
r×F =0r r r τ
Oscillatory motion (3 hrs)
The pendulum and mass – spring system are oscillation system where the plum bob or mass oscillates about an equilibrium point.
In both cases the oscillation follow the simple harmonic motion. Thus the two system are referred to as simple harmonic oscillators.
Periodic motion
A motion is periodic, if it repeats itself at standard intervals in a specific manner. Simple harmonic motion (SHM)
Definition of Simple Harmonic Motion
An object is said to be moving in a simple harmonic motion if its motion is restricted to a path, passing through an equilibrium point, with its acceleration proportional to the displacement from the equilibrium point and always pointing towards the equilibrium point.
Using Newton’s 2nd Law, the previous definition of S.H.M. imply that the force acting on the moving mass is proportional to the displacement form the
equilibrium point and is pointing towards the equilibrium point.
kx dt x d m or x dt x d m then dt x d m dt dv m ma F but x F − = − ∝ = = = − ∝ 2 2 2 2 2 2 .... .... ..
F is opposite in direction to the displacement, x. Both are vectors. k is a constant of proportionality
The differential equation kx dt x d − = 2 2
has a solution of the form
φ ω φ ω + = + = t A t x or t A t x sin ) ( cos ) (
The SHM Equation & Solution
x(t) = A cos ωt+φ : displacement from equilibrium dx(t)/dt = -ω A sin ωt+φ : velocity d2x(t)/dt2 = -ω2 A cos ωt+φ : acceleration d2x(t)/dt2 = - kx(t) - ω2 A cos ωt+φ = - kA cos ωt+φ ω2 = k Simple Pendulum
The weight has a component along the direction of displacement but opposite in direction. This acts as the force in the SHM.
θ θ θ θ θ θ θ θ θ θ θ θ θ l g dt d l g dt d g dt d l mg dt d ml dt d l dt s d dt d l dt ds l s mg dt s d m mg ma − = − = − = − = = = = − = − = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 sin sin sin sin sin The final equation has the SHM form.
Its solution is for the angular displacement θ, where θ(t) = A sin ωt +φ Where, l g f l g l g = = = π ω ω 2 2
Conservation of Energy in SHM
Free vibration : The SHM system is left to oscillate on its own. The oscillating mass changes kinetic energy into potential energy and vice versa during its motion. If no energy is lost due to friction or air resistance, then the amplitude of the oscillation remains a constant.
Total energy remains a constant.
Total energy = ½ mv2 + ½ kx2 (spring in SHM)
E = ½ m(- ω A cos (ωt + φ))2 + ½ k(A sin (ωt + φ))2
E = ½ m (k/m) A2 cos2 (ωt + φ) + ½ k A2 sin2 (ωt + φ) E = ½ k A2
In free vibration, the system vibrates /oscillates with the natural frequency of the system. The frequency of SHM / SH oscillators are the natural frequency. Energy is conserved in free vibration.
In forced vibration, the system is forced to oscillates / vibrates at a frequency which is not the natural frequency of the system – i.e.. A source is needed in forced vibration, whereas in free oscillation a source of vibration is not needed. Energy is not conserved in forced vibration.
Mechanics of solids (2 hrs)
Elasticity
An object is elastic if it returns to its original dimension when the stress acting upon it is removed.
The ratio of stress to strain is linear to a certain limit, called the proportionality limit. Beyond the proportionality limit, the relation is non linear.
The object will return to its original dimension until the elastic limit is reached. Beyond the elastic limit the object will not return to its original dimension.
Its behavior is said to be plastic – large deformation for a small increase in applied force. The ratio of tensile stress to strain is the elastic modules.
The linear proportionality of stress and strain is called Hooke’s Law.
Young’s Modulus
Tensile stress is defined as the ratio of the applied force perpendicular to the surface to the cross section area A;
Tensile stress =
A F⊥
Scalar, SI Unit : Pascal or Nm-2 Tensile strain = o o o l l l l l ∆ = −
Young’s Modulus = Tensile stress / Tensile strain =
A F⊥ / o l l ∆ Shear modulus Shear strain o l x ∆ =
The corresponding elastic modulus is called the shear modulus. The shear modulus
o l x A F S ∆ = // Bulk modulus
When the stress is a uniform pressure on all sides, and the resulting deformation is a change in volume, the stress is called bulk (volume) stress and the relative change in volume, bulk (volume strain).
Bulk (volume) strain =
o V
V
∆
The ratio of stress to strain is called the Bulk Modulus. Bulk Modulus, o V V p B / ∆ ∆ − =
where the change in pressure is positive, while the change in volume is negative. This gives Bulk Modulus a positive value.
The unit of bulk modulus is Pa.
The reciprocal of the bulk modulus is the compressibility and is denoted as
p V V B k o ∆ ∆ − = = 1 / k, where
Mechanics of fluids (2 hrs)
Density and relative density Density and specific weight
The density of a fluid is defined as its mass divided by its volume.
ρ = m / V
The SI unit of density is kg/m3.
The density of water is 1000 kg/m3 (1 g/cm3).
The specific weight of a fluid is its density divided by the density of water. The specific weight has no unit.
Pressure
Static pressure
The static pressure is the pressure in a non moving fluids. By definition Pressure = Force / Area
P = F /A,
The SI unit of pressure is N/m2 or Pascal (Pa)
Therefore the pressure due to a static fluid is the force exerted by the fluid divided by the area on which the force acts.
Suppose a column of fluid has a cross sectional area A and height h. Its volume is then given as V = A x h.
The mass of the fluid, m is given as the product of its density, ρ and its volume ,
V.
Its weight then is W = F = mg = ρVg
The pressure on top of the fluid column is P1.
The pressure exerted by the column is Pl = F/A = ρVg /A = ρAhg /A = ρhg Then the pressure at the bottom of the fluid column is P2 = P1 + ρhg
The pressure inside a fluid is
P = P0 + ρhg, where P0 is the atmospheric pressure, ρ the fluid’s density, g acceleration
due to gravity and h the depth inside a fluid. Fluid static – Pascal’s Law
Pascal’s Law states that the pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluids and the walls of the containing vessel. This is the principle used in a hydraulic lift.
The pressure on both piston are the same. The pressure on the smaller piston is P = F1/A1.
The pressure on the larger piston is P = F2/A2.
Thus P = F1/A1 = F2/A2
or F1/F2 = A1/A2 or F2 = F1 A2 /A1
Thus a smaller force can be amplified Buoyancy.
The upward force acting on the body is F2 – F1 =A(P atm +ρgh2 – (P atm +ρgh1))
= A(ρgh2 –ρgh1) = Aρg (h2 - h1)
= ρg A (h2 - h1)
= ρg V, where V = A (h2 - h1) is the volume of the fluid displaced by body
This is the buoyant force acting on the body.
For a body which floats it weight W = mg =buoyant force. (Static equilibrium) For a body which sinks W > buoyant force.
Vibrations and waves (2 hrs)
Introduction
Waves are transmission of disturbances through a medium, outwards from a source producing the disturbances.
Waves are produced by vibration of a source. Type of waves
Transverse wave
In a transverse wave, the disturbances causes the particle of the medium to oscillates in a direction perpendicular to the direction of travel of the wave.
Longitudinal waves
In a longitudinal wave, the disturbances causes the particles of the medium to oscillates along the direction of travel of the wave.
(In both the above examples the disturbances are produced at the source and changes sinusoidal, producing the characteristic sine wave form.)
Characteristic of wave motion Wave parameters
The classic wave (transverse and longitudinal) is produced by a source which oscillates in simple harmonic motion. In a transverse wave the oscillation of the source is perpendicular to
The source oscillates in SHM with the displacement equation y (t) = A sin (ωt + δ) , where A
is the maximum displacement (amplitude) , ω the angular frequency of the SHM, and t the
time of the oscillation and δ the phase shift/constant/difference. The quantity (ωt + δ) is called the phase of the oscillation.
The relationship of the SHM and the waveform propagated can be shown as below.
amplitude Wavelength, λ Wavelength, λ x y Occurs at t = 0 T 2T t y T 2T displacement-position graph displacement time graph
Period = the time taken for one oscillation of the SHM = the time taken to make one waveform = T
Frequency = the number of oscillation in one second = the number of waveform in one second
To form N wave form will take a time of NT . Thus the number of wave in a time NT is N, the the frequency is, f =
T 1 NT
N
=
wavelength = the size of one waveform in the direction of propagation = the distance a wave travel in time T , wavelength, λ = vT where v is the speed of propagation of the wave. A complete oscillation (or one wave form) is made in time T, the angular displacement of the SHM in this time is 2π. Therefore angular displacement θ = 2π = ωT =
f
ω
Thus ω = 2πf
Thus the oscillation can be written as y (t) = A sin (2πf t + δ)
Now if the source is at x = 0, and the equation of the SHM is y (t) = A sin (2πf t ), then the oscillations of the particles along the positive x axis all lags behind the source. Ie, y (x, t) = A sin (2πf t - δ(x))
δ(x) ∝ x , where is the distance the wave travel in time t, thus x = vt But wavelength, λ = 2π rad
Thus λ ∝ π δ x 2 λ = π δ x 2 , then δ = kx , where k = λ π 2
The propagation of Waves
The propagation of wave is the movement of the waveform from the source outwards. For mechanical wave, the propagation will follows the type of medium. i.e. in a string the wave will travel along the string, water waves may spread radially if it is from a point source. Spherical waves or planar waves can be produced depending on the source.
The speed at which the waveform travels is called the wave velocity or wave speed. As a complete wave length moves a distanceλ in a time T, thus the speed at which it moves is v =
λ/T = f.λ
Wave equations
Harmonic wave equation
Note : A phase constant φ is included in the general form in case the source of oscillation has an initial displacement (ie y(o) = A sin (φ)
x y y (x, t) = A sin (2πf t + kx +φ), Direction of propagation Later in time Earlier in time
Temperature, Thermal expansion and the Ideal Gas Law (4 hrs)
Temperature and thermometers
Temperature – the measure of the degree of “hotness” of a body. Temperature scale.
– Celsius scale (most commonly used) • 0oC - Freezing Point of water • 100oC – Boiling point of water
– Kelvin Scale (Absolute temperature scale)(SI unit)
• 0K – absolute zero, lowest temperature, all molecules stops vibrating equals -273oC , • 273K equals 0oC, • 373K equals 100oC • Change of 1K = change of 1oC • θ(K) = (θ(oC) +273) (K) – Fahrenheit Scale • 32oF equals 0oC • 212oF equals 100oC • θ(oF) = (9/5)θ(oC) +32) (oF) • Types of thermometers – Mercury in glass
• Uses the expansion/contraction of mercury due to absorption/loss of heat. The level of mercury rises/fall against a calibrated scale. • Ex : Room thermometer, Clinical thermometer
– Resistance thermometer
• Uses the change in the resistance of a wire as a measure of temperature change
– Thermocouple
• EMF generated between the two wire junctions kept at different temperature is measured. The e.m.f. is proportional to the difference in the junctions temperature.
– Constant Volume gas
• The temperature is proportional to the change in pressure of the gas. Calibration of a temperature scale
A measurable physical change with temperature is obtained. The change should be linear in nature.
The increase in length when a rod or tube is heated is called linear expansion. coefficient of linear
The coefficient of linear expansion is defined as the ratio of the change in length per unit change in temperature to its original length.
α = θ ∆ ∆ o l l unit : K –1 or oC-1
From the definition loα∆θ = ∆l = change in length
Thus new length l = lo + ∆l
= lo + loα∆θ
= lo ( 1 + α∆θ ) = the new length
Area expansion (solid)
The change in area due to a change in temperature The new area A is given by A = Ao (1 +β∆θ ) Where β is the coefficient of area expansion, β =
θ ∆ ∆ o A A
Relationship between α, coefficient of linear expansion and β, coefficient of areal expansion If a lamina has dimension Xo x Yo, then after a change in temperature,
X = Xo ( 1 + α∆θ ) and Y = Yo ( 1 + α∆θ )
The new area XY is then (Xo ( 1 + α∆θ )).(Yo ( 1 + α∆θ ))
A = Xo Yo( 1 + α∆θ ). ( 1 + α∆θ ) = Xo Yo ( 1 + 1(α∆θ) + 1(α∆θ) +(α∆θ)(α∆θ)) = Xo Yo ( 1 + 2(α∆θ) + (α∆θ)(α∆θ)) if we disregard (α∆θ)(α∆θ) as α∆θ = o l l ∆
is less then 1, the ( o l l ∆ )2 << 1 we’ll have A = Xo Yo ( 1 + 2(α∆θ) ) = Ao (1 + 2α∆θ) = Ao (1 + β∆θ), where β = 2α
Volume expansion (solid)
The change in volume due to a change in temperature. The new volume V is given by Vo (1 +γ∆θ )
Where γ is the coefficient of volume expansion, γ = θ ∆ ∆ o V V
Relationship between α, coefficient of linear expansion and γ, coefficient of volume expansion
If a cuboid has dimension Xo x Yox Zo, then after a change in temperature
X = Xo ( 1 + α∆θ ) , Y = Yo ( 1 + α∆θ ) and Z = Zo ( 1 + α∆θ )
The new volume XYZ can be shown as XYZ = Xo Yo Zo ( 1 + 3(α∆θ) ) after disregarding
smaller terms,
Thus γ , the coefficient of volume expansion is three times α, coefficient of linear expansion.
The gas laws and Absolute temperature Mole
The mole is defined as the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon 12.
Avogadro Number
The number of molecule in one mole or gram molecular weight of a substance is 6.022045 x 1023.
Mol volume
The volume occupied by a mol or a gram molecular weight of any gas at standard conditions is 22.414 l
Boyle’s Law
At a constant temperature the volume of a given quantity of any gas varies inversely as the pressure to which the gas is subjected. For a perfect gas, changing from pressure P and volume V to pressure P’ and volume V’ without change of temperature
PV = P’V’ or PV = constant then P ∝ 1/V
Charles Law (Charles – Gay – Lussac Law)
An empirical generalization that in a gaseous system at constant pressure, the temperature increase and the relative volume increase stand in approximately the same proportion for all so-called perfect gases.
∆T /∆v is a constant
or ∆T ∝∆V
If the absolute temperature scale is used (i.e. Kelvin scale) Then V ∝ T V P 1/V P V
Pressure Law
For a gas at constant mass and volume, its pressure P is proportional to its temperature T P ∝ T
The ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834.
The state of an amount of gas is determined by its pressure, volume, and temperature according to the equation:
where
T P
"R" has a different value for each different unit of pressure used. The values are... R = 8.314472 (pascals/kPa) R = .0821 (atms) R = 62.4 (torr/mmHg) R = 1.2 (psi)
The ideal gas law is the most accurate for monoatomic gases at high temperatures and low pressures. This follows because the law neglects the size of the gas molecules and the intermolecular attractions. Obviously the neglect of molecular size becomes less important for larger volumes, i.e., for lower pressures. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy 3kT/2, i.e., with increasing
temperatures. The more accurate Van der Waals equation takes into consideration molecular size and attraction. The ideal gas law mathematically follows from statistical mechanics of primitive identical particles (point particles without internal structure) which do not interact, but exchange momentum (and hence kinetic energy) in elastic collisions.
Heat and The First Law of Thermodynamics (4 hrs)
Heat as energy transfer
Heat, symbolized by Q, is energy transferred from one body or system to another due to a
difference in temperature. Specific heat
• Heat capacity and specific heat capacity.
– Heat capacity - the amount of heat transferred to/from a substance for every unit change in temperature.
– Specific heat capacity – the amount of heat transferred to/from a unit mass of substance for every unit change in temperature.
Latent heat of fusion and evaporation
– Latent heat – the amount of heat needed to change a substance from one state of matter to another. (i.e. from solid to liquid vice versa or from liquid to gas vice versa)
– Specific Latent heat - the amount of heat needed to change a unit mass of substance from one state of matter to another
The first law of thermodynamics
The first law of thermodynamics is an expression of the universal law of conservation of energy, and identifies heat transfer as a form of energy transfer. The most common form of the first law of thermodynamics is:
The increase in the internal energy of a thermodynamic system is equal to the amount of heat energy added to the system minus the work done by the system on the surroundings
θ ∆ = H C θ ∆ = m H c m H L=
