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Lecture 14
An Introductory Physics Course with Peter Eyland
Lecture 13 (Vectors and Scalars)
In this lecture the following are introduced:
• Descartes and Cartesian coordinates
• Vectors and scalars
• Polar coordinates
• Adding vectors sequentially and simultaneously
• Subtracting vectors
• Vector components and adding by components
• Moment of force
• Mass and weight
• Equilibrium and centre of gravity
Where am I?
This question has either many answers or none; because it is incomplete.
In spatial terms it probably asks: "where am I, with reference to my home, or my destination?"
Cartesian Coordinates
Rene Descartes (1596 to 1650) said, we need a
reference point and reference direction and
from this we can uniquely identify every point in a plane (or in space). He continued with this approach and created what is known as Coordinate Geometry. 
The Cartesian coordinate system has an origin and two (or three) perpendicular axes, "x" and "y" (and "z").
A point is space is specified by coordinates along each axis. 
Distance and displacement
Once we can uniquely define where we are, we have to distinguish between distance and displacement.
The distance you measure between two points depends
on the actual path you take between the points (and there are an infinite number of paths you can take!). 
The displacement has the length and direction of the minimum straightline between the two points. 
Vectors and Scalars
Because displacement has both length and direction, two numbers are needed to describe it.
It is called a "vector" quantity. Distance only needs one number. Quantities that are described by one number are called "scalars".
Examples of scalars are mass, angle and time.
To distinguish between vectors and scalars, textbooks use boldface type for vectors and plainface for scalars.
You need to distinguish between displacement (written as r) and distance (written as r).
In Cartesian coordinates:  
A vector of length one unit (a "unit vector") along the xaxis is written as
i and a unit vector along the yaxis is written as j.
Hence r_{x} = r_{x}i and r_{y} = r_{y}j. 
Polar coordinates
There is another system called the Polar coordinate system, where you specify a displacement by its straightline length and its angle from the reference direction. In the Polar coordinate system, 
Example: an angle which is 22^{O} towards the East of South.
Angle conventions

Circular measure

The angle between two vectors
This can be tricky at times, so it is best to put the two "tails" of the arrows together and then measure the smaller angle. 
Changing between Cartesian and Polar
r = (400m, 300m) in Cartesian notation is shown in the diagram below. 
For Polar notation with maths angle convention. so, r = (500m, 37^{0}) 
Adding displacements
Displacements add in a different way from familiar quantities like mass, angle and time. One way to show how they add is to draw a scale diagram. Here are two displacements, 3000 km Sydney to Darwin then 2600 km Darwin to Perth. They are equal to a 3200 km displacement Sydney to Perth. 
This triangle diagram is clear when the displacements are
sequential.
In the triangle method you place the tail of the second arrow on the head of the first arrow and then join start to finish. When things happen simultaneously, a parallelogram diagram is appropriate (though in practice, either can be used for either situation). 
Subtracting vectors
To subtract vectors you simply add the negative. The negative of a vector has the same size but is in the opposite direction to its positive direction. 
Formal definition of Vector quantities
Vectors are quantities which have
• size (or magnitude),
• direction, and
• add like displacements.
Force
A force is a physical push or pull. It is vector quantity because both size and direction matter. The size is measured in a unit called the Newton. When a force is seen to have effects at right angles, the parts of the force in those directions are called "components".
This can be seen in the case of a yacht where the wind force (at right angle to the sail area) pushes the yacht forward but also tilts it sideways.
We can express this in a diagram this as follows 
Component Notation
If North is vertically up the page, then θ is an azimuthal angle and in polar coordinates, the force F = (F , θ).
It would be written in Cartesian notation (with x as the East direction) as (F·Sinθ , F·Cosθ)
But note carefully:
If the maths convention is to be used, then the angle would be the complement of θ (i.e. 90^{O}  θ) and so the force
would be written in Cartesian notation as (F·Cosθ , F·Sinθ)
Adding by components
Instead of scale diagrams it is often more accurate to add vectors by using the method of components.
For example, adding two vectors (with the maths convention for angles). Let the vectors be A=(A , α) and B=(B , β)
Taking components, i.e. expressing in Cartesian form, A = (A·Cosα , A·Sinα ) and
B = (B·Cosβ , B·Sinβ )
The magnitude of the sum is written as: 
and the angle is 
The sum is thus, (A+B,θ).
Vector addition example 1
Add the vectors (3,13) and (4,7) then subtract (2,8) and express the result in polar form.
Vector addition example 2
Add the vectors (2,30^{O}) and (3,45^{O}) then subtract (1,60^{O}), (where the angles follow the maths convention). Express the result in Cartesian form.
Moments of force
The moment of a force gives the turning or rotating effect of the force. The moment about the turning point is defined as the magnitude of the force times the perpendicular distance from the turning point to the line of action of the force.
Take a seesaw, where there is a plank which can rotate about a point
(called a fulcrum), shown as a triangle Δ.
The angle θ, between the force, F, and the position vector, r, is the obtuse angle shown,
because it is the angle between the vectors when their tails are placed together.
The distance from the turning point to the line of action of the force is,
r·cos(θ90) = r·sinθ.
The size of the moment about the turning point (Δ) is M = F(r·sinθ).
The moment can also be thought of this way. It is the displacement r of the force from the fulcrum
times the component of the force at right angles to that displacement, i.e. (F·sinθ)r.
This emphasises that the component at right angles is the part of the force that actually causes turning.
Since F(r·sinθ) = (F·sinθ)r = Fr·sinθ, either way is right.
Note that the unit for Moment of force will be Newton·metre.
Moments will cause rotation about an axis through the object, and the rotation about this axis can be in one of two directions 
clockwise or anticlockwise. Hence, moments have direction and so are vectors.
Mass and weight
Mass is a scalar quantity, giving the amount of matter that a body has. It's unit is the kilogram.
Weight is a vector quantity, giving the gravitational force that acts downwards on a mass. Accordingly, it has the unit of force which is the Newton.
Weight is the product of mass (m) and the acceleration of gravity (g = 9.8 m.s^{2}). W = m·g
In space (where there is little or no gravity) you lose your weight, but not your mass.
Example
A child, of mass 20kg, is sitting on the left side of a seesaw. The child is 2.4m along the plank from the fulcrum.
The seesaw is initially inclined upwards at an angle of 25^{O} to the horizontal. Find the moment of the child's weight.
Stable, unstable and neutral equilibrium.
Equilibrium means "equally balanced", so the object has balanced forces on it and it will remain how it is placed.
With stable equilibrium, a small displacement will result in the object returning
to the initial balance.
With unstable equilibrium, a small displacement will result in the object moving further away
from balance.
With neutral equilibrium, a small displacement dosen't change the balance.
Centre of gravity
The centre of gravity of an object is the point where all the gravitational force can be considered to act through. If you fix the object at that point there will be no rotation by gravity. If you tilt an object, it will fall over only when the center of gravity lies outside the supporting base of the object.
If you suspend an object so that its center of gravity lies below the point of suspension, it will be unusually stable.
It may oscillate, but it won't fall over.
An object thrown through the air may spin and rotate, but its center of gravity will follow a smooth trajectory.
As the following sketches show, the centre of gravity does not have be within the object.
Summarising:
The Cartesian coordinate system has an origin and perpendicular axes. A point is space is specified by coordinates along each axis.
The distance between two points depends on the actual path you take.
The displacement has the length and direction of the minimum straightline between the two points.
Polar coordinates have a straightline length in a specified direction.
Vectors can be added by arrowed lines in scale diagrams. The arrowed lines may be positioned sequentially or simultaneously.
Vector subtraction is done by adding the negative.
Vector components are vectors at right angles which add to give the original vector.
Vectors may be added without scale diagrams by using components.
The moment about the turning point is defined as the magnitude of the force, times, the perpendicular distance from the turning point to the line of action of the force.
Mass measures the amount of matter that a body has, and weight is the gravitational force on a mass.
Equilibrium means balanced forces, and centre of gravity is the point where all the gravitational force can be considered to act through.
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