Errors......calculating errors in measurment........
http://www.rit.edu/cos/uphysics/uncertainties/Uncertaintiespart1.html
Friday, October 23, 2009
Wednesday, October 21, 2009
Unit 1, Module 2…….. WAVES!!!!
At the end of this lecture, students should be able to:
· Define a wave
· Classify the types of waves
· Be able to use and understand the terms associated with wave motion
· Represent transverse and longitudinal waves graphically
When a stone is thrown into a quiet lake, circular ripples spread out from the point of impact and travel over the surface of the lake. The disturbance eventually reaches some distant part of the lake and produces physical effects there, even though there is no actual transfer of water from the point of impact of the stone to the distant point. If the wave passes over a floating piece of wood, the wood bobs up and down but stays in the vicinity of its original position and does not move forward with the advancing wave front. An important aspect of waves is that they provide a means whereby one part of the universe can influence another part without the bodily transfer of matter between the two parts. Consider, for example, how you might communicate with a friend on the opposite bank of a lake. One way would be to shoot over an arrow with a message attached to it. This would involve the bodily transfer of matter. A second way would be to throw a stone into the lake. When the ripples reached the float of his fishing line and made it bob up and down, this would be a prearranged signal to your friend that it was time to go to lunch.
Recall: A wave is a disturbance of a medium from its equilibrium position and the disturbance travels from one region of space to another. It therefore has the ability to carry energy from place to place i.e. a traveling disturbance.
A single disturbance is called a pulse but repetitive disturbances create a wave train/wave.
Types of Waves.
I. Progressive Waves- a progressive wave or traveling wave consists of a disturbance moving from a source to surrounding places as a result of which energy is transferred from one point to another. There are two types of progressive waves:
a. Transverse Waves
b. Longitudinal waves.
Transverse Waves are those in which the articles vibrate perpendicularly to the direction of propagation and longitudinal waves are those in which the particles travel parallel to the direction of propagation.
(a) Transverse Wave (b) Longitudinal Wave
II. Mechanical Waves- these waves are produced by a disturbance in a material and are transmitted by the particles of the medium oscillating to and fro. Examples include: water waves in a ripple tank, waves on a stretched string, sound waves, shock waves as in earthquakes.
III. Electromagnetic Waves- these waves consist of a disturbance in the form of varying electric and magnetic fields. No medium is necessary and they travel more easily in a vacuum than in matter. Examples include: light, radio waves, X rays, gamma rays etc.
( See Electromagnetic spectrum)
IV. Stationary Waves- to be discussed at a later time.
Wave Language
Describing the behavior of waves can be discussed using the following terms:
I. Displacement: - distance moved by a vibrating particle from its equilibrium position.
II. Amplitude: - maximum displacement of a vibrating particle from its rest position.
III. Period T :- time taken for one complete oscillation
IV. Frequency f: - no. of complete oscillations in one second.
Mathematically related to period by the formula f=1/T
V. Wave speed: - when the source makes a complete vibration, then one wave is generated and the disturbance spreads out a distance of λ from the source. Thus fro f vibrations per second, then in that one second, the wave travels a distance of f λ and from mechanics we know that Speed = distance/ time : Speed = f λ/ one second : Hence v or c = f λ
Alternatively, we can say speed= distance/ time
= distance x 1/T but distance = λ and 1/T = f
Therefore, speed v or c= f λ.
VI. Polarization: - As a result of the transverse nature of vibration, transverse waves have an additional property that is not possessed by longitudinal waves. With transverse waves, the oscillations take place in many different directions and the wave is then said to be unpolarized. Polarizations are the process by which a wave’s oscillations are made to occur in one plane only. This is known as a polarized wave. To polarize a wave, it is effectively passed through a grid of parallel bars. Only oscillations in a plane parallel to those bars can pass through and thus the wave is said to be plane polarized.
VII. Coherent sources: - these sources have a constant phase difference and thus for a constant phase difference to occur, the waves under consideration must have the same frequency which in most cases originate from the same source. Recall for Young’s double slit experiment the light sources were coherent. How was this achieved?
VIII Phase: - Phase is a measure of whether two oscillations are in step with each other or if one lags behind the other. Oscillations which are in step with each other are said to be in phase and oscillations which are in opposite directions are said to be in antiphase. Antiphase oscillations lag behind each other by exactly half of a cycle. For values of lag between in phase and antiphase we measure the lag as a fraction of a full cycle measured in radians, degrees or wavelengths. A complete cycle is 2Π radians, 360 degrees or λ. Waves which are separated by whole wavelengths of λ are said to be in phase whereas fractions of λ causes it to be out of phase.
Consider the figure below showing four oscillations A, B, C and D. Oscillation A is in phase with oscillation B which is in antiphase with wave C and out of phase with oscillation D by quarter cycle or Π/2 radians. So, the actual value of difference or lag/ lead of one wave relative is referred to as the phase difference. Therefore wave D has a phase difference of Π/2 radians behind wave A. It is important to note that phase differences to be constant, the waves must have the same frequency.
IX Path difference : - When considering the behavior of two coherent waves from sources close to each other certain relationships between must be taken into account. The difference in the distance between each source and the particular point is known as the path difference. It can be measured in metres or wavelengths.
Graphical Representation of waves.
Two kinds of graphs may be drawn:
§ Displacement-distance graphs
§ Displacement-time graphs
Displacement-distance graphs show the displacement of the vibrating particles of the transmitting medium at different distances from the source at a certain instant of time.
With this graph, values of wavelength and amplitude can be obtained.
Displacement-time graphs show the displacement of one particle at a particular distance from the source with time. With this graph, one can obtain values of amplitude, period or frequency.
http://www.antonine-education.co.uk/physics_a2/module_4/Topic_4/topic_4.htm
PS : OPEN HYPERLINK ABOVE FOR ADITTIONAL INFORMATION
At the end of this lecture, students should be able to:
· Define a wave
· Classify the types of waves
· Be able to use and understand the terms associated with wave motion
· Represent transverse and longitudinal waves graphically
When a stone is thrown into a quiet lake, circular ripples spread out from the point of impact and travel over the surface of the lake. The disturbance eventually reaches some distant part of the lake and produces physical effects there, even though there is no actual transfer of water from the point of impact of the stone to the distant point. If the wave passes over a floating piece of wood, the wood bobs up and down but stays in the vicinity of its original position and does not move forward with the advancing wave front. An important aspect of waves is that they provide a means whereby one part of the universe can influence another part without the bodily transfer of matter between the two parts. Consider, for example, how you might communicate with a friend on the opposite bank of a lake. One way would be to shoot over an arrow with a message attached to it. This would involve the bodily transfer of matter. A second way would be to throw a stone into the lake. When the ripples reached the float of his fishing line and made it bob up and down, this would be a prearranged signal to your friend that it was time to go to lunch.
Recall: A wave is a disturbance of a medium from its equilibrium position and the disturbance travels from one region of space to another. It therefore has the ability to carry energy from place to place i.e. a traveling disturbance.
A single disturbance is called a pulse but repetitive disturbances create a wave train/wave.
Types of Waves.
I. Progressive Waves- a progressive wave or traveling wave consists of a disturbance moving from a source to surrounding places as a result of which energy is transferred from one point to another. There are two types of progressive waves:
a. Transverse Waves
b. Longitudinal waves.
Transverse Waves are those in which the articles vibrate perpendicularly to the direction of propagation and longitudinal waves are those in which the particles travel parallel to the direction of propagation.
(a) Transverse Wave (b) Longitudinal Wave
II. Mechanical Waves- these waves are produced by a disturbance in a material and are transmitted by the particles of the medium oscillating to and fro. Examples include: water waves in a ripple tank, waves on a stretched string, sound waves, shock waves as in earthquakes.
III. Electromagnetic Waves- these waves consist of a disturbance in the form of varying electric and magnetic fields. No medium is necessary and they travel more easily in a vacuum than in matter. Examples include: light, radio waves, X rays, gamma rays etc.
( See Electromagnetic spectrum)
IV. Stationary Waves- to be discussed at a later time.
Wave Language
Describing the behavior of waves can be discussed using the following terms:
I. Displacement: - distance moved by a vibrating particle from its equilibrium position.
II. Amplitude: - maximum displacement of a vibrating particle from its rest position.
III. Period T :- time taken for one complete oscillation
IV. Frequency f: - no. of complete oscillations in one second.
Mathematically related to period by the formula f=1/T
V. Wave speed: - when the source makes a complete vibration, then one wave is generated and the disturbance spreads out a distance of λ from the source. Thus fro f vibrations per second, then in that one second, the wave travels a distance of f λ and from mechanics we know that Speed = distance/ time : Speed = f λ/ one second : Hence v or c = f λ
Alternatively, we can say speed= distance/ time
= distance x 1/T but distance = λ and 1/T = f
Therefore, speed v or c= f λ.
VI. Polarization: - As a result of the transverse nature of vibration, transverse waves have an additional property that is not possessed by longitudinal waves. With transverse waves, the oscillations take place in many different directions and the wave is then said to be unpolarized. Polarizations are the process by which a wave’s oscillations are made to occur in one plane only. This is known as a polarized wave. To polarize a wave, it is effectively passed through a grid of parallel bars. Only oscillations in a plane parallel to those bars can pass through and thus the wave is said to be plane polarized.
VII. Coherent sources: - these sources have a constant phase difference and thus for a constant phase difference to occur, the waves under consideration must have the same frequency which in most cases originate from the same source. Recall for Young’s double slit experiment the light sources were coherent. How was this achieved?
VIII Phase: - Phase is a measure of whether two oscillations are in step with each other or if one lags behind the other. Oscillations which are in step with each other are said to be in phase and oscillations which are in opposite directions are said to be in antiphase. Antiphase oscillations lag behind each other by exactly half of a cycle. For values of lag between in phase and antiphase we measure the lag as a fraction of a full cycle measured in radians, degrees or wavelengths. A complete cycle is 2Π radians, 360 degrees or λ. Waves which are separated by whole wavelengths of λ are said to be in phase whereas fractions of λ causes it to be out of phase.
Consider the figure below showing four oscillations A, B, C and D. Oscillation A is in phase with oscillation B which is in antiphase with wave C and out of phase with oscillation D by quarter cycle or Π/2 radians. So, the actual value of difference or lag/ lead of one wave relative is referred to as the phase difference. Therefore wave D has a phase difference of Π/2 radians behind wave A. It is important to note that phase differences to be constant, the waves must have the same frequency.
IX Path difference : - When considering the behavior of two coherent waves from sources close to each other certain relationships between must be taken into account. The difference in the distance between each source and the particular point is known as the path difference. It can be measured in metres or wavelengths.
Graphical Representation of waves.
Two kinds of graphs may be drawn:
§ Displacement-distance graphs
§ Displacement-time graphs
Displacement-distance graphs show the displacement of the vibrating particles of the transmitting medium at different distances from the source at a certain instant of time.
With this graph, values of wavelength and amplitude can be obtained.
Displacement-time graphs show the displacement of one particle at a particular distance from the source with time. With this graph, one can obtain values of amplitude, period or frequency.
http://www.antonine-education.co.uk/physics_a2/module_4/Topic_4/topic_4.htm
PS : OPEN HYPERLINK ABOVE FOR ADITTIONAL INFORMATION
Simple Harmonic Motion
Simple harmonic motion
Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic, as it repeats itself at standard intervals in a specific manner - described as being sinusoidal, with constant amplitude. It is characterized by its amplitude which is always positive and depends on how motion starts initially, its period which is the time for a single oscillation, and its phase which depends on displacement as well as velocity of the moving object.
One definition of simple harmonic motion is "motion in which the acceleration of the oscillator is proportional to, and opposite in direction to the displacement from its equilibrium position", or .
A general equation describing simple harmonic motion is , where x is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and γ is the phase of oscillation. If there is no displacement at time t = 0, the phase γ = 0. A motion with frequency f has period .
Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterisation of more complicated motions through the techniques of Fourier analysis.
//
] Mathematics
It can be shown, by differentiating, exactly how the acceleration varies with time. The displacement is given by the function
We then differentiate once to get an expression for the velocity at any time.
And once again to get the acceleration at a given time.
These results can of course be simplified, giving us an expression for acceleration in terms of displacement.
When and if total energy is constant and kinetic the formula applies for simple harmonic motion, where E is considered the total energy while all energy is in its kinetic form. A representing the mean displacement of the spring from its rest position in MKS units.
Realizations
Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples:
Mass on a spring: A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with
With ω representing angular frequency.
Alternately, if the other factors are known and the period is to be found, this equation can be used:
Uniform circular motion: Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius R centered at the origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude R and angular speed ω.
Mass on a pendulum: In the small-angle approximation, the motion of a pendulum is shown to approximate simple harmonic motion. The period of a mass attached to a string of length with gravitation acceleration g is given by
This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position.
With θ being small, and therefore the expression becomes
which makes angular acceleration directly proportional to θ, satisfying the definition of Simple Harmonic Motion
For an exact solution not relying on a small-angle approximation, see pendulum (mathematics).
PS. Double click on highlighted words for additional information.
Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic, as it repeats itself at standard intervals in a specific manner - described as being sinusoidal, with constant amplitude. It is characterized by its amplitude which is always positive and depends on how motion starts initially, its period which is the time for a single oscillation, and its phase which depends on displacement as well as velocity of the moving object.
One definition of simple harmonic motion is "motion in which the acceleration of the oscillator is proportional to, and opposite in direction to the displacement from its equilibrium position", or .
A general equation describing simple harmonic motion is , where x is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and γ is the phase of oscillation. If there is no displacement at time t = 0, the phase γ = 0. A motion with frequency f has period .
Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterisation of more complicated motions through the techniques of Fourier analysis.
//
] Mathematics
It can be shown, by differentiating, exactly how the acceleration varies with time. The displacement is given by the function
We then differentiate once to get an expression for the velocity at any time.
And once again to get the acceleration at a given time.
These results can of course be simplified, giving us an expression for acceleration in terms of displacement.
When and if total energy is constant and kinetic the formula applies for simple harmonic motion, where E is considered the total energy while all energy is in its kinetic form. A representing the mean displacement of the spring from its rest position in MKS units.
Realizations
Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples:
Mass on a spring: A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with
With ω representing angular frequency.
Alternately, if the other factors are known and the period is to be found, this equation can be used:
Uniform circular motion: Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius R centered at the origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude R and angular speed ω.
Mass on a pendulum: In the small-angle approximation, the motion of a pendulum is shown to approximate simple harmonic motion. The period of a mass attached to a string of length with gravitation acceleration g is given by
This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position.
With θ being small, and therefore the expression becomes
which makes angular acceleration directly proportional to θ, satisfying the definition of Simple Harmonic Motion
For an exact solution not relying on a small-angle approximation, see pendulum (mathematics).
PS. Double click on highlighted words for additional information.
Subscribe to:
Comments (Atom)