2.3.1 scalars and vectors Lesson 2

Vector Addition by Scale Drawing

Scalars and Vectors : “A scalar is any physical quantity that has magnitude” (e.g. Mass, density, volume & energy) “A vector is any physical quantity that has direction as well as magnitude” (e.g. Force, velocity & acceleration)

Scalars and Vectors : But also think of displacement and distance: “By road there are an infinite number of routes you can take from home to school, some shorter and many longer. However, once you arrive at school your displacement will always be the same home → school. The distance travelled will vary for each potential route”

Scalars and Vectors : Now consider adding a further journey after school to a friend’s house: We now have two displacements: Home → School & School → Friend’s House Clearly the end result could be achieved by a single displacement of Home → Friend’s House

Scalars and Vectors : Vector addition for displacements School Friend’s house Home Vector addition for displacements Today via scale drawing

An example: Consider an oil tanker being towed by a pair of tugs as shown: 8000 kN 20° 20° 8000 kN What is the combined (resultant) effect of these two forces?

The Tail-to-Head method... 40°

The Tail-to-Head method: Rules for scale drawing... Choose a scale and include a reference direction. Choose any of the vectors to be summed and draw it as an arrow in the correct direction and of the correct length- remember to put an arrowhead on the end to denote its direction. Take the next vector and draw it as an arrow starting from the arrowhead of the first vector in the correct direction and of the correct length. Continue until you have drawn each vector- each time starting from the head of the previous vector. In this way, the vectors to be added are drawn one after the other tail-to-head. The resultant is then the vector drawn from the tail of the first vector to the head of the last. Its magnitude can be determined from the length of its arrow using the scale. Its direction too can be determined from the scale diagram.

There are many equivalent constructions:

Vectors at Right Angles

Vectors at Right Angles In addition to scale drawing we can add pairs of vectors which are at right angles, (perpendicular) using Pythagoras’ theorem Displacement Vectors: Consider walking 3m, turning through 90° and then walking 4m. The equivalent displacement can be found by....

θ Our combined resultant vector (c) Walk 3m (a) Turn through 90° and walk 4m (b) To find the magnitude of the resultant vector we use Pythagoras’ theorem: c2 = a2 + b2 c = √(32 + 42) c = 5m

To find the direction, (taking θ as the angle relative to the initial direction) tan θ = opposite adjacent tan θ = 4 3 θ = 53.1°

The same can be applied to perpendicular forces... Redraw our force diagram as a vector diagram.... What is the resultant force? θ 5N 12N 13N at 67.4° to the 1st force

Some common sense stuff... Vectors in the same line 4N Simply sum if in the same direction 6N Resultant 10N And find the difference if they are in opposite directions 4N 6N 2N But how can we do this for random collections of vectors acting in all sorts of directions?

Resolving vectors into perpendicular components Resolving means to break down into.... Maths example first, then explanation... Consider an arbitrary displacement of 5km at 53° NE North 5km East 53°

Using some trusty trigonometry.... After button pressing we can find a pair of equivalent perpendicular vectors 4km North followed by 3km East North 5km 5 sin 53° 53° East 5 cos 53°

But so what? Why do this? It allows an arbitrary collection of vectors to be broken down into a collection of only vertical and horizontal components... In turn the components can be added and/or subtracted as required This will yield a resulting pair of vertical and horizontal components which can them be resolved to a single vector as required

Summary : Pairs of vectors at right angles can be resolved to a single resultant vector using Pythagoras’ theorem and trigonometry Arbitrary vectors can be resolved to perpendicular components, (typically vertical and horizontal). Once this has been done complex collections of arbitrary vectors can then be easily rationalised.

Calculation Sheet – Using Scalars and Vectors Support Sheets – Resolving Vectors Practical – Coplanar Forces (2.5) Lesson 2