Graphical Vector Addition - METNET



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In 3-Dimentional space the point P is represented as

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The coordinate system shown above is known as a right-handed co-ordinate system, because it is possible, using the right hand, to point the index finger in the positive direction of the x-axis, the middle finger in the positive direction of the y-axis, and the thumb in the positive direction of the z-axis, as in Figure below. An equivalent way of defining a right-handed system is if you can point your thumb upwards in the positive z-axis direction while using the remaining four fingers to rotate the x-axis towards the y-axis. Doing the same thing with the left hand is what defines a left-handed coordinate system

A (nonzero) vector is a directed line segment drawn from a point P (called its initial point) to a point Q (called its terminal point), with P and Q being distinct points. The vector is denoted by PQ. Its magnitude is the length of the line segment, denoted by |PQ|, and its direction is the same as that of the directed line segment. The zero vector is just a point, and it is denoted by 0.

To indicate the direction of a vector, we draw an arrow from its initial point to its terminal point. We will often denote a vector by a single bold-faced letter (e.g. v) and use the terms “magnitude” and “length” interchangeably. Note that our definition could apply to systems with any number of dimensions see Figure below,

Note: Two nonzero vectors are said to be equal if they have the same magnitude and the same direction. Any vector with zero magnitude is equal to the zero vector.

Unless otherwise indicated, when speaking of “the vector” with a given magnitude

and direction, we will mean the one whose initial point is at the origin of the

coordinate system.

Example:

Graphical Vector Addition

Adding two vectors a and b graphically can be visualized like two successive walks, with the vector sum being the vector distance from the beginning to the end point. Representing the vectors by arrows drawn to scale, the beginning of vector b is placed at the end of vector a. The vector sum c can be drawn as the vector from the beginning to the end point. The process can be done mathematically by finding the components of A and B, combining to form the components of R, and then converting to polar form.

Example of Vector Components

Finding the components of vectors for vector addition involves forming a right triangle from each vector and using the standard triangle trigonometry. The vector sum can be found by combining these components

Polar Form Example

After finding the components for the vectors A and B, and combining them to find the components of the resultant vector R, the result can be put in polar form by some caution should be exercised in evaluating the angle with a calculator because of ambiguities in the arctangent on calculators.

Combining Vector Components

After finding the components for the vectors A and B, these components may be just simply added to find the components of the resultant vector R.

The components fully specify the resultant of the vector addition, but it is often desirable to put the resultant in polar form.

Magnitude and Direction from Components

If the components of a vector are known, then its magnitude and direction can be calculated with the use of the Pythagorean relationship and triangle trig. This is called the polar form of the vector.

Vector Addition, Two Vectors

Vector addition involves finding vector components, adding them and finding the polar form of the resultant.

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Note : i.i = j.j = k.k = 1 and i.j = j.k = k.i = 0 where I, j , k are unit vectors perpendicular to each other.

Cross product or Vector Product

In the above section we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vectors that does result in another vector which is perpendicular to the two vectors. This product, called the cross product or Vector Product.

If C =AxB then, C can be calculated by

Find i X j

In general

Note: For Cross Product the following rules are applicable

Position

Specifying the position of an object is essential in describing motion. In one dimension some typical ways are

In two dimensions, either cartesian or polar coordinates may be used, and the use of unit vectors is common. A position vector r may be expressed in terms of the unit vectors.

In three dimensions, Cartesian or spherical polar coordinates are used, as well as other coordinate systems for specific geometries.

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The vector change in position associated with a motion is called the displacement.

Displacement:

The figure shows a displacement vector representing the difference between two points in the x-y plane. (For now, the examples are in two dimensional space; three dimensional space will include z coordinate also).

• point A: x=2, y=1.   A (2, 1)

• point B: x=7, y=3.   B (7, 3)

Computing the displacement from A to B can be done as two separate problems:

• The x displacement is the difference in the X values: 7-2 = 5

• The y displacement is the difference in the Y values: 3-1 = 2

The displacement vector expressed [pic] is: = [pic](5, 2)

Displacement vectors are often visualized as an arrow connecting two points. In the diagram point A is the tail of the vector and point B is the tip of the vector. (However, recall that vectors do not have a position, so this is just a convenient place to draw it, not where it is.)

When the points are visited in the opposite order, the displacement vector points in the opposite direction. The displacement from A to B is different from the displacement from B to A. Think of displacement as "directions on how to walk from one point to another." So, if you are standing on point A and wish to get to point B, the displacement (5, 2) says "walk 5 units in the positive X direction, then walk 2 unit in the positive Y direction." Of course, to get from point B to point A you need different directions: the displacement (-5, -2) says "walk 5 units in the negative X direction, then walk 2 unit in the negative Y direction," which puts you back on point A.

The displacement from point Start to point Finish : displacement = (Finish x - Start x , Finish y - Start y)

Velocity

The average speed of an object is defined as the distance traveled divided by the time elapsed. Velocity is a vector quantity, and average velocity can be defined as the displacement divided by the time. For the special case of straight line motion in the x direction, the average velocity takes the form:

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The units for velocity can be implied from the definition to be meters/second or in general any distance unit over any time unit.

You can approach an expression for the instantaneous velocity at any point on the path by taking the limit as the time interval gets smaller and smaller. Such a limiting process is called a derivative and the instantaneous velocity can be defined as

Average Velocity, General

The average speed of an object is defined as the distance traveled divided by the time elapsed. Velocity is a vector quantity, and average velocity can be defined as the displacement divided by the time. For general cases involving non-constant acceleration, this definition must be applied directly because the straight line average velocity expressions do not work.

Warning! The average velocity is not given by

Since the velocities are vector in different directions and the acceleration is not constant.

If the positions of the initial and final points are known, then the distance relationship can be used to find the displacement

Acceleration

Acceleration is defined as the rate of change of velocity. Acceleration is inherently a vector quantity, and an object will have non-zero acceleration if its speed and/or direction is changing. The average acceleration is given by

where the small arrows indicate the vector quantities. The operation of subtracting the initial from the final velocity must be done by vector addition since they are inherently vectors.

The units for acceleration can be implied from the definition to be meters/second divided by seconds, usually written m/s2.

The instantaneous acceleration at any time may be obtained by taking the limit of the average acceleration as the time interval approaches zero. This is the derivative of the velocity with respect to time:

Circular Motion

Θ = angular displacement

Ω = ω = dΘ/dt = angular velocity

α= dω/dt = angular acceleration[pic]

What are Dynamics? And why dynamics?

Dynamics is a branch of mechanics which deals with the description and explanation of the atmospheric and oceanic motion with emphasis on forces and physical laws that govern such motion.

“A study of forces on moving objects is considered in dynamics”.

Let F is the net force acting on an object of mass m moving with velocity ‘V’ then

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Where F is the net force action on the body (parcel) and F1, F2 etc., are various forces acting on the body on different directions. If m is constant then the above equation becomes [pic] where [pic] is acceleration of the object.

On the other hand Kinematics deals only with geometrical description of the motion of the particles along the curves such as Galileo’s laws of falling bodies or Kepler’s laws of planetary motion.

(Instructor has to give a brief introduction of various forces such as Gravitational, electostatic force, nuclear force, magnetic force etc.,. the presence of acceleration term in the equation of motions. Distinguish between “Dynamics and Kinematics” with examples)

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(Instructor may ask the various laws and refresh the basics concepts with examples and probing questions, units and dimensions etc.,)

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(Explain local derivative and total derivative with some examples and Temperature, moisture advection, advection fog etc.,_)

The Vector Differential Operator (DEL)

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[pic]is a vector differential operator possesses properties analogous to those of ordinary vectors. It is useful in defining three quantities which arise in practical application such as the gradient, the divergence and the curl. The operator [pic]is also known as ‘nabla’

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Isobar spacing and the magnitude of the pressure gradient

• The magnitude of the pressure gradient can be assessed by noting the spacing of the isobars....

• If the isobars are close together, the pressure gradient is large

• If the isobars are far apart, the pressure gradient is small

• The pressure gradient force, like any other force, has a magnitude and a direction:

• Direction - the pressure gradient force direction is ALWAYS directed from high to low pressure and is ALWAYS perpendicular to the isobars

• Magnitude - is determined by computing the pressure gradient

• Q:  What is the direction of the pressure gradient force surrounding an area of low pressure?

• Q:  What is the direction of the pressure gradient force surrounding an area of high pressure?

• Q:  Is the magnitude of the pressure gradient larger surrounding the high or the low? 

Variation of ‘g’ with depth

What will happen as go inside the earth? Let ρ be the density of the earth and R be the radius, then mass M of the earth is

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Let [pic] be the gravity at the surface (A in fig) of the earth and [pic]the gravity at the depth d from the surface (B in fig) or which is at a distance r from the center of the earth. Then

Where,[pic].

From the above equation we find that,

Thus g’ is linearly proportional to ‘r’ inside the earth and approaches zero as r tends to zero.

(Explain the term gravity and how it varies with examples)

Tidal Effect

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Tides are caused by the effects of gravity in the earth-moon-sun system, and the movement of those three bodies within the system. If you imagine that the earth is completely covered in water, there are two bulges of water - one towards the moon and another on the opposite side. The rise and fall in sea-level is caused by the earth rotating on its axis underneath these bulges of water. There are two tides a day because it passes under two bulges for each rotation (24 hours). This is called the lunar tide.

Two bulges of water are also caused by the sun, called the solar tide - and these can either reinforce or partially cancel out the lunar tide to give spring and neap tides.

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Do you know why forecasters are very much concerned about tidal effect at the time of cyclone crossing the land and Tsunami?

(Instructor may recall the 1998 Orisa cyclone and how the storm surge is related to tidal effect)

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Question of thoughts: Do you know the earth is rotating in a viscous medium such as atmosphere and water which drags the rotation of the earth. In spite of the viscous drags, how earth continuously rotating without changing its angular velocity?

Do you thing that geostationary satellites and polar orbiting satellites spending energy while revolving around the earth?

(Instructor may introduce the term Frictional force and its importance in meteorology with examples)

Coriolis force On Nargis

Effect of Coriolis force on Global Wind and its direction

(Instructor clearly defines the difference between real and pseudo forces with various examples (gravity, pressure gradient force, Columb force, magnetic force etc., so that the trainees feels why we call Coriolis and centrifugal forces are false force)

The above equation can also be called equation of motion of the atmosphere or momentum equation in vector form or momentum equation in Cartesian form.

(Instructor can ask the trainees to write the above equation in terms of its components and equate to the respective terms)

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Geostrophic Approximation and Geostrophic Wind

For midlatitude synoptic scale disturbances, the Coriolis force and pressure gradient force are in approximate balance and leads to the Geostrophic approximation.

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The geostrophic balance is a diagnostic expression that gives the approximate relationship between the pressure field and horizontal velocity in large-scale extra tropical systems. The above approximation contains no reference to time and therefore cannot be used to predict the evolution of the velocity field. It is for this reason that the geostrophic relationship is called a diagnostic relationship.

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The Rossby number: Rossby number is a measure of the validity of the Geostrophic approximation and it is ratio of magnitude of the acceleration to the Coriolis force. This ratio is a non-dimensional number given by

Ro = U / (f0 L)

The smallness of the Rossby number ( ................
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