Physics and Physical Physics and Physical MeasurementMeasurement

Topic 1.3 Scalars and VectorsTopic 1.3 Scalars and Vectors

Scalars QuantitiesScalars Quantities Scalars can be completely Scalars can be completely

described by described by magnitude magnitude (size)(size) Scalars can be added algebraicallyScalars can be added algebraically They are expressed as positive or They are expressed as positive or

negative numbers and a unitnegative numbers and a unit examples include :- mass, electric examples include :- mass, electric

charge, distancecharge, distance, speed, energy, speed, energy

Vector QuantitiesVector Quantities Vectors need both a Vectors need both a magnitudemagnitude and a and a

directiondirection to describe them (also a point to describe them (also a point of application)of application)

When expressing vectors as a symbol, you When expressing vectors as a symbol, you need to adopt a recognized notationneed to adopt a recognized notation

e.g. e.g. They need to be added, subtracted and They need to be added, subtracted and

multiplied in a special waymultiplied in a special way Examples :- velocity, weight, acceleration, Examples :- velocity, weight, acceleration,

displacement, momentum, forcedisplacement, momentum, force

Addition and SubtractionAddition and Subtraction The The Resultant Resultant (Net) (Net) is the result is the result

vector that comes from adding or vector that comes from adding or subtracting a number of vectorssubtracting a number of vectors

If vectors have the same or opposite If vectors have the same or opposite directions the addition can be done directions the addition can be done simplysimply

same direction : addsame direction : add opposite direction : subtractopposite direction : subtract

Co-planar vectorsCo-planar vectors The addition of co-planar vectors that do not The addition of co-planar vectors that do not

have the same or opposite direction can be have the same or opposite direction can be solved by using scale drawings to get an solved by using scale drawings to get an accurate resultantaccurate resultant

Or if an estimation is required, they can be Or if an estimation is required, they can be drawn roughlydrawn roughly

or by Pythagoras’ theorem and trigonometryor by Pythagoras’ theorem and trigonometry Vectors can be represented by a straight line Vectors can be represented by a straight line

segment with an arrow at the endsegment with an arrow at the end

Triangle of VectorsTriangle of Vectors Two vectors are added by drawing Two vectors are added by drawing

to scale and with the correct to scale and with the correct direction the two vectors with the direction the two vectors with the tail of one at the tip of the other.tail of one at the tip of the other.

The resultant vector is the third The resultant vector is the third side of the triangle and the arrow side of the triangle and the arrow head points in the direction from head points in the direction from the ‘free’ tail to the ‘free’ tipthe ‘free’ tail to the ‘free’ tip

ExampleExample

a b+ =

R = a + b

Parallelogram of VectorsParallelogram of Vectors Place the two vectors tail to tail, to Place the two vectors tail to tail, to

scale and with the correct directionsscale and with the correct directions Then complete the parallelogramThen complete the parallelogram The diagonal starting where the two The diagonal starting where the two

tails meet and finishing where the tails meet and finishing where the two arrows meet becomes the two arrows meet becomes the resultant vectorresultant vector

ExampleExample

a b+ =R = a + b

More than 2More than 2 If there are more than 2 co-planar If there are more than 2 co-planar

vectors to be added, place them all vectors to be added, place them all head to tail to form polygon when head to tail to form polygon when the resultant is drawn from the the resultant is drawn from the ‘free’ tail to the ‘free’ tip.‘free’ tail to the ‘free’ tip.

Notice that the order doesn’t Notice that the order doesn’t matter!matter!

Subtraction of VectorsSubtraction of Vectors To subtract a vector, you reverse To subtract a vector, you reverse

the direction of that vector to get the direction of that vector to get the negative of itthe negative of it

Then you simply add that vectorThen you simply add that vector

ExampleExample

a b- =

R = a + (- b)-b

Multiplying ScalarsMultiplying Scalars Scalars are multiplied and divided Scalars are multiplied and divided

in the normal algebraic mannerin the normal algebraic manner

Do not forget units!Do not forget units! 5m / 2s = 2.5 ms5m / 2s = 2.5 ms-1-1

2kW x 3h = 6 kWh (kilowatt-hours)2kW x 3h = 6 kWh (kilowatt-hours)

Multiplying VectorsMultiplying Vectors A vector multiplied by a scalar gives a A vector multiplied by a scalar gives a

vector with the same direction as the vector with the same direction as the vector and magnitude equal to the product vector and magnitude equal to the product of the scalar and a vector magnitudeof the scalar and a vector magnitude

A vector divided by a scalar gives a vector A vector divided by a scalar gives a vector with same direction as the vector and with same direction as the vector and magnitude equal to the vector magnitude magnitude equal to the vector magnitude divided by the scalardivided by the scalar

You don’t need to be able to multiply a You don’t need to be able to multiply a vector by another vectorvector by another vector

Resolving VectorsResolving Vectors The process of finding the The process of finding the

Components Components of vectors is called of vectors is called Resolving Resolving vectorsvectors

Just as 2 vectors can be added to Just as 2 vectors can be added to give a resultant, a single vector can give a resultant, a single vector can be split into 2 components or partsbe split into 2 components or parts

The RuleThe Rule A vector can be split into two A vector can be split into two

perpendicular componentsperpendicular components These could be the vertical and These could be the vertical and

horizontal componentshorizontal components

Vertical component

Horizontal component

Or parallel to and perpendicular to Or parallel to and perpendicular to an inclined planean inclined plane

These vertical and horizontal These vertical and horizontal components could be the vertical components could be the vertical and horizontal components of and horizontal components of velocity for projectile motionvelocity for projectile motion

Or the forces perpendicular to and Or the forces perpendicular to and along an inclined planealong an inclined plane

Doing the TrigonometryDoing the Trigonometry

Sin = opp/hyp = y/V

Cos = adj/hyp = x/V

V

y

x

Therefore y = Vsin In this case this is the vertical component

Therefore x = Vcos In this case this is the horizontal component

V cos

V sin

Quick WayQuick Way If you resolve through the angle it If you resolve through the angle it

is is coscos

If you resolve ‘not’ through the If you resolve ‘not’ through the angle it isangle it is

sinsin

Adding 2 or More Adding 2 or More Vectors by ComponentsVectors by Components First resolve into components (making First resolve into components (making

sure that all are in the same 2 directions)sure that all are in the same 2 directions) Then add the components in each of the Then add the components in each of the

2 directions2 directions Recombine them into a Recombine them into a resultant resultant vectorvector This will involve using Pythagoras´ This will involve using Pythagoras´

theoremtheorem

QuestionQuestion Three strings are attached to a Three strings are attached to a

small metal ring. 2 of the strings small metal ring. 2 of the strings make an angle of 70make an angle of 70oo and each is and each is pulled with a force of 7N.pulled with a force of 7N.

What force must be applied to the What force must be applied to the 3rd string to keep the ring 3rd string to keep the ring stationary?stationary?

AnswerAnswer Draw a diagramDraw a diagram

7N 7N

F

70o

7 sin 35o7 sin 35o

7 cos 35o + 7 cos 35o

HorizontallyHorizontally 7 sin 357 sin 35o o - 7 sin 35- 7 sin 35o o = 0= 0

VerticallyVertically 7 cos 357 cos 35o o + 7 cos 35+ 7 cos 35o o = F= F F = 11.5NF = 11.5N And at what angle?And at what angle? 145145o o to one of the strings.to one of the strings.