Electromagnetic Waves

Electromagnetic Waves



Electromagnetic waves are formed when an electric field (shown as blue arrows) couples with a magnetic field (shown as red arrows). The combination of oscillating electrical and magnetic fields of the electromagnetic waves are perpendicular to each other and to the direction of the wave. James Clerk Maxwell and Heinrich Hertz are two scientists who studied how electromagnetic waves are formed and how fast they travel.This is difficult to visualize, however the waveform has similar characteristics of other types of waves.

Although they seem different, radio waves, microwaves, x-rays, and even visible light are all electromagnetic waves are part of the electromagnetic spectrum, and each has a different range of wavelengths, which cause they waves to affect matter differently.The creation and detection of the wave depend much on the range of wavelengths.

Electromagnetic spectrum

The range of wavelengths for electromagnetic waves--from the very long to the very short--is called the Electromagnetic Spectrum:

* Radio and TV waves are the longest usable waves, having a wavelength of 1 mile
(1.5 kilometer) or more.
* Microwaves are used in telecommunication as well as for cooking food.
* Infrared waves are barely visible. They are the deep red rays you get from a heat lamp.
* Visible light waves are the radiation you can see with your eyes. Their wavelengths are in the range of 1/1000 centimeter.
* Ultraviolet rays are what give you sunburn and are used in "black lights" that make object glow.
* X-rays go through the body and are used for medical purposes.
* Gamma rays are dangerous rays coming from nuclear reactors and atomic bombs. They have the shortest wavelength in the electromagnetic spectrum of about 1/10,000,000 centimeter.






(Click on picture to get full view)

Waves in the electromagnetic spectrum vary in size from very long radio waves the size of buildings, to very short gamma-rays smaller than the size of the nucleus of an atom.

Did you know that electromagnetic waves can not only be described by their wavelength, but also by their energy and frequency? All three of these things are related to each other mathematically. This means that it is correct to talk about the energy of an X-ray or the wavelength of a microwave or the frequency of a radio wave. The electromagnetic spectrum includes, from longest wavelength to shortest: radio waves, microwaves, infrared, optical, ultraviolet, X-rays, and gamma-rays.

There are no gaps in between the electromagnetic spectrum.It is possible to find all possible frequencies that exist in this range.There is no boundary between one type of electromagnetic waves and the next.

(To Take Note:)
Gamma Rays - highest frequency, lowest wavelength
Radio waves - lowest frequency, longest wavelength
Visible light is just one of the 7 members of the family of electromagnetic waves
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Look at the picture of the electromagnetic spectrum. See if you can find answers to these questions:

1. What kind of electromagnetic radiation has the shortest wavelength? The longest?

2. What kind of electromagnetic radiation could be used to "see" molecules? A cold virus?

3. Why can't you use visible light to "see" molecules?

4. Some insects, like bees, can see light of shorter wavelengths than humans can see. What kind of radiation do you think a bee sees?
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Properties

1. Electromagnetic Waves are transverse waves.They are magnetic and electric fields of an electromagnetic wave which are perpendicular to each other and to the direction of the wave.



2.They transfer energy from one place to another.

3.They can travel through vacuum.They don't require any medium to travel from one point to another.

4.They can travel at the speed of light, 3.0 x 10*8 m/s but, will slow down when travelling through water or glass.

5.Equation of wave velocity also applies for electromagnetic waves,
Speed = Wavelength • Frequency

6.They obey Snell's laws of Reflection and Refraction.

7.They carry no electric charges as they are neither positively charged nor negatively charged.

8.The frequencies do not change when they travel from one medium to another as their frequency depends only on the source of the wave.Only their speeds and wavelengths change from one medium to another.

Wave Production and the ripple tank

The Ripple Tank



The ripple tank is a container that when filled with water permits the study of water waves.A concentrated light source positioned above the tank forms images of the waves on a screen beneath the tank.Wave crests and troughs project light and dark lines in the screen.




The crests act as converging lenses that focus light,producing the bright lines..The troughs act as diverging lenses that scatter light, producing the dark lines.
The depth at which the dipper is placed affects the amplitude of the waves, while the frequency of waves is determined by frequency of vibration of the dipper.

Refraction of waves and the depth of ripple tank

Refraction of waves involves a change in the direction of waves as they pass from one medium to another. Refraction is the bending of the path of the waves.It is accompanied by a change in speed and wavelength of the waves. It was mentioned that the speed of a wave is dependent upon the properties of the medium through which the waves travel. So if the medium (and its properties) are changed, the speed of the waves are changed.

The most significant property of water which would affect the speed of waves traveling on its surface is the depth of the water.

This boundary behavior of water waves can be observed in a ripple tank if the tank is partitioned into a deep and a shallow section. If a pane of glass is placed in the bottom of the tank, one part of the tank will be deep and the other part of the tank will be shallow. Waves traveling from the deep end to the shallow end can be seen to refract (i.e., bend), decrease wavelength (the wave fronts get closer together), and slow down (they take a longer time to travel the same distance). When traveling from deep water to shallow water, the waves are seen to bend in such a manner that they seem to be traveling more perpendicular to the surface.If traveling from shallow water to deep water, the waves bend in the opposite direction.



Water waves travel fastest when the medium is the deepest. Thus, if water waves are passing from deep water into shallow water, they will slow down and also the wavelength of the plane waves shorten.The frequency remains the same as it is determined by the dipper.Using the equation, v:f x L,the speed of the waves is therefore slower at the shallow water.



Refraction of waves can be demonstrated by placing the plastic sheet at an angle to the incoming waves .As observed earlier, the differene in the depth of water causes a change in speed of waves.Similar to light, when waves enter a region of shallow water at an angle, the waves refract.



Reflection of waves can be demonstrated by placing a straight barrier upright in the water causing the incoming incident waves to be reflected.The law of reflection is obeyed and the Angle of incidence is equal to the angle of reflection.



Reflection, refraction and diffraction are all boundary behaviors of waves associated with the bending of the path of a wave. The bending of the path is an observable behavior when the medium is a two- or three-dimensional medium. Reflection occurs when there is a bouncing off of a barrier. Reflection of waves off straight barriers follows the law of reflection. Reflection of waves off parabolic barriers results in the convergence of the waves at a focal point. Refraction is the change in direction of waves which occurs when waves travel from one medium to another. Refraction is always accompanied by a wavelength and speed change. Diffraction is the bending of waves around obstacles and openings. The amount of diffraction increases with increasing wavelength.

Properties of Wave motion

Describing Waves





(click picture to get full view)

Crest or peak: The highest point reached by a wave.

Trough: The lowest point reached by a wave.

Phase : Any Two sources of wave motion are said to be in phase if, at any moment, they have the same fractional displacement from the rest position and are moving in the same direction. If both conditions are not met the sources are out of phase.
Any two crests or troughs are always in phase

Wavelength : The shortest distance between any two points in a wave that are in phase.In a transverse wave, the points are two successive crests or troughs.In longitudinal wave, it is the distance between two successive compressions or rarefactions.S.I. Unit: Metre (m)

Amplitude : The maximum displacement (difference between an original position and a later position) of the material that is vibrating. Amplitude can be thought of visually as the highest and lowest points of a wave.It is the height of a crest or the depth of the trough measured from rest position.S.I. Unit: Metre(m)

In a longitudinal Wave,



(click picture to get full view)

Compression : A point on a medium through which a longitudinal wave is traveling which has the maximum density.It is a region where the coils are pressed together in a small amount of space.

Rarefaction : a point on a medium through which a longitudinal wave is traveling which has the minimum density.It is a region where the coils are spread apart, thus maximizing the distance between coils.



The wavelength of a wave is the length of one complete cycle of a wave. For a transverse wave, the wavelength is determined by measuring from crest to crest. A longitudinal wave does not have crest; so how can its wavelength be determined? The wavelength can always be determined by measuring the distance between any two corresponding points on adjacent waves. In the case of a longitudinal wave, a wavelength measurement is made by measuring the distance from a compression to the next compression or from a rarefaction to the next rarefaction

Questions:

Consider the diagram below in order to answer questions #1-2.



1. The wavelength of the wave in the diagram above is given by letter ______.

ANS

2. The amplitude of the wave in the diagram above is given by letter _____.

ANS

3. Indicate the interval which represents one full wavelength.



a. A to C

b. B to D

c. A to G

d. C to G

ANS

Displacement-Distance Graph

A wave:



Any points on graph above the rest position show positive displacements
and any points on graph below the rest position show negative displacement.

(click picture to get full view)

Displacement-Time Graph



If we freeze the wave motion at various times, we can observe the up-and-down movements of points on transverse wave.If we trace the movement of wave over one second ,we can obtain the displacement-time graph of wave.

Period (T): It is the time for a particle on a medium to make one complete vibrational cycle. Period, being a time, is measured in units of time such as seconds, hours, days or years.

Frequency(f) : It refers to how often the particles of the medium vibrate when a wave passes through the medium. S.I. unit: Hertz (Hz)where 1 Hz is equivalent to 1 cycle/second.

Frequency = Number of cycles/Time Interval

(The period (T) is the time required to complete one full cycle.)

Period and frequency exhibit a reciprocal relationship.



Since the symbol f is used for frequency and the symbol T is used for period, these equations are also expressed as:



Higher the frequency, more the no. of waves produced in one second,T is shorter.

Wave Speed (v) : In a time of one period, a crest on a transverse wave will have moved a distance of one wavelength.

Hence,

Equation for the Speed of a wave :

v : L/T

where ,
v- wave speed
L- wavelength
T-Period

Since f:1/T,

v:f x L



S.I. unit : m/s

Wave front: It is the imaginary line on a wave that joins all points that are in the same phase.It is usually drawn by joining all the wave crests.Depending on the formation of waves, wavefront can be concentric circles,plane straight lines or any shape.



Questions:

1. As the frequency of a wave increases, the period of the wave ___________.

a. decreases

b. increases

c. remains the same

ANSWER

2.The period of the sound wave produced by a 440 Hertz tuning fork is ___________.


Answer

3.A child in a swing makes one complete back and forth motion in 3.2 seconds. This statement provides information about the child's

a. speed

b. frequency

c. period

ANSWER

4. A period of 5.0 seconds corresponds to a frequency of ________ Hertz.

a. 0.2

b. 0.5

c. 0.02

d. 0.05

e. 0.002

Answer

Types of Waves

Types of Waves

Two types of waves exist: transverse and longitudinal.Both of these wave-types are traveling disturbances, but they are different because of the way that they travel. As a wave travels through a medium, the particles that make up the medium are disturbed from their resting, or “equilibrium” positions.After either type of wave passes through a medium, the particles return to their equilibrium positions. Thus, waves travel through a medium with no net displacement of the particles in the medium.

Transverse Waves




(A transverse wave. The particles move in a direction that is perpendicular to the direction of wave propagation.)

A transverse wave is one that causes the particles of the surrounding medium to vibrate in a direction at right angles to the direction of the wave. In a transverse wave, the particles are disturbed in a direction perpendicular to the direction of source(Vibration).A water wave is an example of a transverse wave. As water particles move up and down, the water wave itself appears to move to the right or left.Another example of these types of waves are light waves.



Longitudinal Waves


(A longitudinal wave, made up of compressions - areas where particles are close together - and rarefactions - areas where particles are spread out. The particles move in a direction that is parallel to the direction of wave propagation.)

In a longitudinal wave, the particles are disturbed in a direction parallel to the direction that the wave propagates. A longitudinal wave consists of “compressions” and “rarefactions” where particles are bunched together and spread out, respectively.
The direction of the wave motion is parallel to the direction to the direction of vibration.
Sound Waves are an example of longitudinal waves.


Comparison between transverse and longitudinal waves
----------------------------------------
Transverse waves

Definition:
The particles of the medium vibrate
at right angles to the direction of
wave motion

Longitudinal waves

Definition:
The particles of the medium vibrate
parallel to the direction of wave
motion.
----------------------------------------
Transverse waves

Movement:
The wave is propagated in the form
of crests and troughs

Longitudinal waves

Movement:
The wave is propagated in the form
of compressions and rarefactions.
-----------------------------------------
Transverse waves

Medium:
This type of wave motion is possible
in solids and on liquid surfaces

Longitudinal waves

Medium:
This type of wave motion is possible
in any medium (solid, liquid or gas)
-----------------------------------------
Transverse waves

Polarization:
These waves Can undergo polarization

Longitudinal waves

Polarization:
These waves do not undergo polarization
-----------------------------------------

Questions:
1. A transverse wave is transporting energy from east to west. The particles of the medium will move_____.

a. east to west only

b. both eastward and westward

c. north to south only

d. both northward and southward
(See Answer)

2.A wave is transporting energy from left to right. The particles of the medium are moving back and forth in a leftward and rightward direction. This type of wave is known as a ____.

a. mechanical


b. electromagnetic

c. transverse

d. longitudinal
(See Answer)

3. Describe how the fans in a stadium must move in order to produce a longitudinal stadium wave.

ANSWER

4. If you strike a horizontal rod vertically from above, what can be said about the waves created in the rod?

a. The particles vibrate horizontally along the direction of the rod.

b. The particles vibrate vertically, perpendicular to the direction of the rod.

c. The particles vibrate in circles, perpendicular to the direction of the rod.

d. The particles travel along the rod from the point of impact to its end.

See Answer

Wave Formation

HOW ARE WAVES FORMED?

Wave motion

Wave motion is defined as the movement of a distortion of a material or medium, where the individual parts or elements of the material only move or propogate back-and-forth, up-and-down, or in a cyclical pattern.It is just the distortion moving, where one part influences the next.



Example:



Probably the most familiar example of wave motion is the action of water waves. A boat at rest on the ocean moves up and down as water waves pass beneath it. The waves appear to be moving toward the shore. But the water particles that make up the wave are actually moving in a vertical direction. The boat itself does not move toward the shore or, if it does, it's at a much slower rate than that of the water waves themselves.

The energy carried by a water wave is obvious to anyone who has watched a wave hit the shore. Even small waves have enough energy to move bits of sand. Much larger waves can, of course, tear apart the shore and wash away homes.

Other Examples:

Wave motion on rope



A rope is fixed with one end to a wall and moving the other end up and down.Up and Down movements produce a vibration and oscillation.The rope waves produced move towards the wall while traveling up and down.In this case, the rope is the medium through which the waves move.The kinetic energy from the up-and-down movement is transferred by the wave without the rope itself moving from one end to another.

Waves in a ripple tank







In a ripple tank, a small dipper moves up and down the water surface.As a result, the water particles at the surface that are in contact with the dipper are made to move up and down.This up and down motion spreads to other parts of the water surface in the tank in the form of ripples.The kinetic energy from the up and down movement of the dipper is transferred to the water molecules in the surface.These water molecules then transfer the energy to the surrounding water molecules and so on.(Note: on the energy is transferred from the dipper to the water, not the water itself)



In a Nutshell...

We know that:
1.The source of a wave is a vibration or oscillation (Basically a disturbance to the medium)
2.Waves transfer energy from one point to another.
3.In waves,energy is transferred without the medium being transferred

What is a Wave?

Waves

What is a Wave?

A wave can be described as a disturbance that travels through a medium from one location to another location.It is made up of periodic motion,which is a motion repeated at regular intervals.One example of a periodic motion is the pendulum bob moving left to right and back to left.

It is thought of as a traveling disturbance.It travels energy from one place to another place transferring energy in the process.The source of any wave is a vibration or oscillation.



For example:

When a pebble is dropped in a pond, a few circular ripples, the disturbances in this case,move outward on the surface of the water.






When the slinky is stretched from end to end and is held at rest, it assumes a natural position known as the equilibrium or rest position.

To introduce a wave into the slinky, the first particle is displaced or moved from its equilibrium or rest position. The particle might be moved upwards or downwards, forwards or backwards; but once moved, it is returned to its original equilibrium or rest position.

The act of moving the first coil of the slinky in a given direction and then returning it to its equilibrium position creates a disturbance in the slinky.

We can then observe this disturbance moving through the slinky from one end to the other. If the first coil of the slinky is given a single back-and-forth vibration, then we call the observed motion of the disturbance through the slinky a slinky pulse.

A pulse is a single disturbance moving through a medium from one location to another location. However, if the first coil of the slinky is continuously and periodically vibrated in a back-and-forth manner, we would observe a repeating disturbance moving within the slinky which endures over some prolonged period of time.

The repeating and periodic disturbance which moves through a medium from one location to another is referred to as a wave.

Newton's third law of motion

According to Newton, whenever objects A and B interact with each other, they exert forces upon each other. When you sit in your chair, your body exerts a downward force on the chair and the chair exerts an upward force on your body. There are two forces resulting from this interaction - a force on the chair and a force on your body. These two forces are called action and reaction forces and are the subject of Newton's third law of motion

Newton's third law of motion is:

For every action, there is an equal and opposite reaction.These forces act on mutually opposite bodies.

The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.


Characteristics of forces:

1.Forces always come in pairs - equal and opposite action-reaction force pairs.
2.Action and reaction forces are equal in magnitude.
3.Action and reaction forces act in opposite direction.
4.Action and reaction forces act on different bodies.

Example:



A fish uses its fins to push water backwards. But a push on the water will only serve to accelerate the water. Since forces result from mutual interactions, the water must also be pushing the fish forwards, propelling the fish through the water. The size of the force on the water equals the size of the force on the fish; the direction of the force on the water (backwards) is opposite the direction of the force on the fish (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction force. Action-reaction force pairs make it possible for fish to swim.

Another Example:



A bird flies by use of its wings. The wings of a bird push air downwards. Since forces result from mutual interactions, the air must also be pushing the bird upwards. The size of the force on the air equals the size of the force on the bird; the direction of the force on the air (downwards) is opposite the direction of the force on the bird (upwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for birds to fly.

Questions:



1.While driving down the road, a firefly strikes the windshield of a bus and makes a quite obvious mess in front of the face of the driver. This is a clear case of Newton's third law of motion. The firefly hit the bus and the bus hits the firefly. Which of the two forces is greater: the force on the firefly or the force on the bus?

Hint: Read Chapter again!

Answer

2. Many people are familiar with the fact that a rifle recoils when fired. This recoil is the result of action-reaction force pairs. A gunpowder explosion creates hot gases which expand outward allowing the rifle to push forward on the bullet. Consistent with Newton's third law of motion, the bullet pushes backwards upon the rifle. The acceleration of the recoiling rifle is ...

a. greater than the acceleration of the bullet.

b. smaller than the acceleration of the bullet.

c. the same size as the acceleration of the bullet.

Answer

Newton's second law of motion

Objects at equilibrium (the condition in which all forces balance) will not accelerate. According to Newton, an object will only accelerate if there is a net or unbalanced force acting upon it. The presence of an unbalanced force will accelerate an object - changing either its speed, its direction, or both its speed and direction.

Newton's second law of motion can be formally stated as follows:

When a net force acts on an object of constant mass, the object will accelerate and move in the direction of the resultant force.The product of the mass and acceleration of the object is equal to the resultant force.



Equation:

a = Fnet / m

Where,

a- acceleration
F- Force acting on object
m- mass

This equation can be rearranged to as follows:

Fnet = m * a

Fnet = m * g
(g - acceleration due to gravity)

(acceleration of free fall is due to Earth's gravity is 10(m/s)/s)

Hence,

W= m * g




The acceleration is directly proportional to the net force; the net force equals mass times acceleration; the acceleration in the same direction as the net force; an acceleration is produced by a net force. The NET FORCE. It is important to remember this distinction.


It is the net force which is related to acceleration.

*the net force is the vector sum of all the forces. If all the individual forces acting upon an object are known, then the net force can be determined.

Consistent with the above equation, a unit of force is equal to a unit of mass times a unit of acceleration. By substituting standard metric units for force, mass, and acceleration into the above equation, the following unit equivalency can be written.

Effect of forces on motion and Newton's First Law of motion

Effect of forces on motion:

1. a stationary objects to start moving (Movement)
2. Moving object to accelerate (Acceleration)
3. Moving object to decelerate (Deceleration)
4. Moving object to change direction(Direction)

Easy to remember:
Movement
Acceleration
Deceleration
Direction

*
Zero acceleration refers to objects that are stationary or moving with constant
velocity.In this case, different forces acting on it are balanced,add up to zero,
resultant or net force add up to zero.

Newton and his laws of motion

Isaac Newton (a 17th century scientist) put forth a variety of laws which explain why objects move (or don't move) as they do. These three laws have become known as Newton's three laws of motion.

Newton's First Law of motion(sometimes referred to as the law of inertia):

An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

There are two parts to this statement - one which predicts the behavior of stationary objects and the other which predicts the behavior of moving objects. The two parts are summarized in the following diagram.




Example:

Suppose that you filled a baking dish to the rim with water and walked around an oval track making an attempt to complete a lap in the least amount of time. The water would have a tendency to spill from the container during specific locations on the track. In general the water spilled when:


* the container was at rest and you attempted to move it

* the container was in motion and you attempted to stop it

* the container was moving in one direction and you attempted to change its
direction.

The water spills whenever the state of motion of the container is changed. The water
resisted this change in its own state of motion.The container was moved from rest to a high speed at the starting line; the water remained at rest and spilled onto the table. The container was stopped near the finish line; the water kept moving and spilled over container's leading edge. The container was forced to move in a different direction to make it around a curve; the water kept moving in the same direction and spilled over its edge. The behavior of the water during the lap around the track can be explained by Newton's first law of motion.(Water keeps on doing what its doing)

More Examples:

*
Blood rushes from your head to your feet while quickly stopping when
riding on a descending elevator.

*
The head of a hammer can be tightened onto the wooden handle by banging the
bottom of the handle against a hard surface.

*
A brick is painlessly broken over the hand of a physics teacher by slamming it
with a hammer. (CAUTION: do not attempt this at home!)

*
To dislodge ketchup from the bottom of a ketchup bottle, it is often turned
upside down and thrusted downward at high speeds and then abruptly halted.

*
Headrests are placed in cars to prevent whiplash injuries during rear-end
collisions.

*
While riding a skateboard (or wagon or bicycle), you fly forward off the board
when hitting a curb or rock or other object which abruptly halts the motion of
the skateboard.

Vector Diagrams

Vector Diagrams

Vector diagrams are diagrams which depict the direction and relative magnitude of a
vector quantity by a vector arrow. Vector diagrams can be used to describe the velocity of a moving object during its motion.

Vector diagrams can be used to represent any vector quantity.
For example, acceleration, force, and momentum.

How to draw a vector diagram?

A vector arrow is used to represent the direction and relative size of a quantity. It will become a very important representation of an object's motion.

*In a vector diagram, the magnitude of a vector quantity is represented by the size
of the vector arrow. If the size of the arrow in each consecutive frame of the vector diagram is the same, then the magnitude of that vector is constant.

A straght arrow represents a vector motion and its length represents its magnitude.
The direction of the force is represented by the direction in which the arrowhead points.

A scale is used to represent the magnitude of the vector
( Magnitude represented by the scale must be accurate)
The direction of the drawing must also be accurate.

Adding Vectors

The net force is the vector sum of all the forces which act upon an object. That is to say, the net force is the sum of all the forces, taking into account the fact that a force is a vector and two forces of equal magnitude and opposite direction will cancel each other out.

The net force experienced by an object is determined by computing the vector sum of all the individual forces acting upon that object. That is the net force is the result(or resultant) of adding up all the force vectors to find a single vector that will produce the same effect as the two vectors added together.



Observe in the diagram above that a downward vector will provide a partial or full cancellation of an upward vector. And a leftward vector will provide a partial or full cancellation of a rightward vector. The addition of force vectors can be done in the same manner in order to determine the net force (i.e., the vector sum of all the individual forces). Consider the three situations below in which the net force is determined by summing the individual force vectors which are acting upon the objects.



1. Free-body diagrams for four situations are shown below. For each situation, determine the net force acting upon the object. Click the buttons to view the answers.



Answer to situation A
The net force is zero Newtons. All the individual forces balance each other (i.e., cancel each other out).

Answer to situation B
The net force is 5 Newtons, left. The vertical forces balance each other (i.e., cancel each other out). The leftward force (friction) remains unbalanced.

Answer to situation C
The net force is zero Newtons. All the individual forces balance each other (i.e., cancel each other out).

Answer to situation D
The net force is 15 Newtons, up. The upward force of air resistance is only partially balanced by the downward force of gravity - 15 N of upward force remains unbalanced.

2. Free-body diagrams for four situations are shown below. The net force is known for each situation. However, the magnitudes of a few of the individual forces are not known. Analyze each situation individually and determine the magnitude of the unknown forces. Then click the button to view the answers.



See Answer

How to add non-parallel vectors?

There are two equivalent ways to add vectors graphically: the tip-to-tail method and the parallelogram method. Both will get you to the same result, but one or the other is more convenient depending on the circumstances.

Tip-to-Tail Method
We can add any two vectors, A and B, by placing the tail of B so that it meets the tip of A. The sum, A + B, is the vector from the tail of A to the tip of B.



Note that you’ll get the same vector if you place the tip of B against the tail of A. In other words, A + B and B + A are equivalent.

Parallelogram Method
To add A and B using the parallelogram method, place the tail of B so that it meets the tail of A. Take these two vectors to be the first two adjacent sides of a parallelogram, and draw in the remaining two sides. The vector sum, A + B, extends from the tails of A and B across the diagonal to the opposite corner of the parallelogram. If the vectors are perpendicular and unequal in magnitude, the parallelogram will be a rectangle. If the vectors are perpendicular and equal in magnitude, the parallelogram will be a square.



Adding Vector Magnitudes
Of course, knowing what the sum of two vectors looks like is often not enough. Sometimes you’ll need to know the magnitude of the resultant vector. This, of course, depends not only on the magnitude of the two vectors you’re adding, but also on the angle between the two vectors.

Adding Perpendicular Vectors
Suppose vector A has a magnitude of 8, and vector B is perpendicular to A with a magnitude of 6. What is the magnitude of A + B? Since vectors A and B are perpendicular, the triangle formed by A, B, and A + B is a right triangle. We can use the Pythagorean Theorem to calculate the magnitude of A + B, which is



Adding Vectors at Other Angles
When A and B are neither perpendicular nor parallel, it is more difficult to calculate the magnitude of A + B because we can no longer use the Pythagorean Theorem. It is possible to calculate this sum using trigonometry, but SAT II Physics will never ask you to do this. For the most part, SAT II Physics will want you to show graphically what the sum will look like, following the tip-to-tail or parallelogram methods. On the rare occasions that you need to calculate the sum of vectors that are not perpendicular, you will be able to use the component method of vector addition, explained later in this chapter.

Example

Vector A has a magnitude of 9 and points due north, vector B has a magnitude of 3 and points due north, and vector C has a magnitude of 5 and points due west. What is the magnitude of the resultant vector, A + B + C?
First, add the two parallel vectors, A and B. Because they are parallel, this is a simple matter of straightforward addition: 9 + 3 = 12. So the vector A + B has a magnitude of 12 and points due north. Next, add A + B to C. These two vectors are perpendicular, so apply the Pythagorean Theorem:
The sum of the three vectors has a magnitude of 13. Though a little more time-consuming, adding three vectors is just as simple as adding two.