fundamental math for physics

8/4/2019 Fundamental Math for Physics

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Fundamental Math for

PhysicsTrigonometry and Vectors


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Trigonometry

What is sine, cosine, tangent?

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Trigonometry functions

Trigonometry functions are defined forany angle by constructing a right angletriangle.

sin =opp/hypcos = adj/hyp

tan =opp/adj

or sin /cos

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Hypothenuse (hyp)opposite (opp)

adjacent (adj)


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Vectors vs Scalars

Some measurements has only aquantity, like length of an object in m,distance travelled in m, mass in kg,

temperature in K, speed in ms-1. For instance, speed is telling us how

fast something is moving, but no

mention of direction. These quantities are scalar quantities.

What about speed with a direction?

velocity 4


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Vectors

Quantities like velocity is a vectorquantity

It has both magnitude (size) and

direction For example, a car is travelling at 50

km/h (speed) heading north (direction)

50 km/h northwards is velocityAnother car may be travelling at 50km/h BUT southwards

They have the same speed but 5


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Vectors

Other examples of vectors are:

displacement, force, momentum andacceleration (covered later)

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Representing vectors inPhysics Useful to represent vectors using a

diagram

A vector is represented by arrows

An arrow is drawn so that it points inthe direction of the vector quantity it

represents50 km/h northwards

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Addition of scalars

Scalars can be simply added

E.g., Travel 5 m

travel another 3 m

Total distance covered = 8 m

E.g., Speed of car = 10 km/h

Increase in speed = 5 km/h

Final speed = 10 + 5 = 15 km/h

But vectors like velocity MUST beadded in a special way

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Displacement as example to addvectors Example:

Distance how far one has travelled

Displacement change in position of

an object

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Displacement as an example toadd vectors Below, the ball has travelled 3 m

horizontally and the 4 m vertically

Total distance travelled is 3 + 4 = 7 m

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Displacement as an example toadd vectors But what is the change in position

from original location?

Answer: (Pythagoras theorem)

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3 m

4 m


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Resultant

The red arrow is called the resultantdisplacement.

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3 m

4 m


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Components of a vector

Any displacement (red arrow) can beexpressed as the sum of two othervectors; green and blue

They are the components of theoriginal vector

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3 m

4 m


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Components of a vector

Note that the components chosen areperpendicular in directions, one horizontaland another vertical

The process of finding the components iscalled resolving the vector

into its components

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3 m

4 m


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Components of adisplacement A ball travels in the direction of the red

arrow.

The components of the displacement

are in green and blue; derived fromthe displacement

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3 m

4 m


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Find the resultant

We have seen that using Pythagorastheorem, we can find the resultantmagnitude, 5 m

What about the direction? tan = 4/3

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3 m

4 m


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Find the resultant

What about the direction?

tan = 4/3

This is the direction with reference to

the horizontal

vector found!

5 m at 53.1

to x axis

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3 m

4 m


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Find the components

Given the vector is 5 m at 53.1 to xaxis, can we find the components (xand y components)?

We have to use trigonometricfunctions

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3 m

4 m


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Find the components

Red: 5 m 53.1 from horizontal

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5 cos

5 sin


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Find the y component

sin = opp/hyp

opp = hyp x sin

=5 x sin 53.1

= 5 x 0.8 = 4

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4 m


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Find the x component

cos = adj/hyp

adj = hyp x cos

=5 x sin 53.1

= 5 x 0.6 = 3

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3 m


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Find the components(conclusion) Red: 5 m 53.1 from horizontal

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5 cos

5 sin


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Addition of vectors usingcomponents Two vectors in the same direction

Two vectors in opposite directions

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Addition of vectors usingcomponents Two vectors at different direction

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Addition of vectors usingcomponents Find the x and y components

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Addition of vectors usingcomponents Add them (simple addition

in x and y

directions)

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Resolving of vectors intocomponents Example:

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Magnitude: 5Direction: 60

Magnitude: 8Direction: 30

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