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11.1.1 Scalars and vectors

Scalars are quantities that have magnitude only:

Eg: Length, speed, mass, density, energy, power, temperature, charge and pd

Vectors are quantities that have both direction and magnitude:

Eg:  Displacement, force, torque, velocity, acceleration, momentum, electric current and electric charge.

Addition and subtraction of vectors

O A
B

When we add two vectors their sum is called the resultant.

In the diagram OB = OA  +  AB

The two vectors can form 2 sides of a triangle

If the vector arrows follow on from one another (for addition) then the resultant is the third side of the triangle.

Parallelogram rule:  it can be easier to make OA and AB two touching corners of a parallelogram.  The resultant is the diagonal across the parallelogram.  If done to scale this can be a good way to allow us to measure the magnitude (length) and direction (angle) of a resultant vector.

O           30o                                  A     

 

 B  

Do a scale diagram to calculate the resultant vector from adding Vector OA = 5cm North to Vector AB = 10cm East.  Check your answer using trigonometry.

NB. Problems will only be given using vectors at right angles – therefore calculation methods using trigonometry and Pythagoras are a bit easier.

NB2.  To subtract vectors – simply swap the direction of the subtracting vector and do the same as above.

Questions on adding and subtracting vectors:

Q1.  Consider two tug boats pulling a cruise liner out of a harbour.  Tugboat A is pulling with a force of 200kN N45oE. Tugboat B is pulling with a force of 300kN S45oE. 

Q2.   Draw sketches to help add and then subtract the following vectors:

  Vector 1

  Vector 2

 Addition   resultant

 Subtraction   resultant

20N South

20N East

 

 

 

5cm North

 

7cm West

 

 

 

40 m/s N50oE

 

60m/s S40oE

 

 

 

 Q3   A ferry is crossing a river.  Its engines propel the boat with a velocity of 10m/s North.  The river is moving with a current of 4m/s West.  Calculate the resultant velocity of the boat.  Make sure you draw a sketch to help.

Q4 and Q5 are from Breithaupt fourth edition – P.15 Q 1.9 and 1.10

 


Resolving
Vectors

Forces are vector quantities; they have both size and direction.

Resolving a force into its components means replacing the original force with two other forces which would have added together to produce the same effect.  

F

F1

F2

There are an infinite number of combinations of F1 and F2 which are equivalent to F but if ...

F1 and F2 are said to be components of F 

They are chosen to be at right angles to each other so that:

            F12 + F22  = F2

In general :

if force= F and

angle=q

F1= F cos q
       F2  = F sin q

 

Conditions for equilibrium

See P. 42 of Breithaupt – 4th ed.

Forces which act on objects but which do not cause an acceleration are in equilibrium with each other. The study of forces which are in equilibrium is called statics. 

We need to be able to consider forces in two dimensions - either when an object is at rest or when it is moving at a constant speed. In both of these situations the forces are in equilibrium. To do this, we need to be able to resolve forces.

The condition for equilibrium for three vector forces acting at a point is:

F1  +  F2  +  F3  =  0

This means that if the above relationship is true for the forces – then we can say they are “in equilibrium”.

We can prove equilibrium in two ways – by scale diagrams (3 forces must make a closed triangle) – or – by resolving all the forces into horizontal and vertical components (they must all add to zero)

Statics is the study of objects in equilibrium.

  This picture is in equilibrium.
      T                          T
 

 


                Weight          

If the forces on the picture are redrawn nose to tail then it is possible to determine if there is a resultant force or if the picture is at equilibrium.

T W
T

  If the forces form a closed triangle (or polygon if more than 3 forces are present) then the object is in equilibrium. This method relies on accurate drawing so is completely useless! Therefore we prefer to use the mathematical model for resolving forces.

Examples of resolving and finding resultant forces

R

          3N

    7N

R2 = 32 + 72

R = Ö58

   = 7.6N

Angle between resultant and 7N force = q

tan q = 3

            7

q = 23°

 

Experiments: Conditions for equilibrium

P. 42 Breithaupt – 4th ed. – with either:

spring balances – nailed and attached with string

OR

2 Weights hanging over pulleys attached to another weight in between

Draw scale vector diagrams of the three forces and prove that they are in equilibrium – first by using the closed triangle method and then by resolving forces.

Notes on resolving forces method:

Imagine 3 forces see diagram -

 

10N                                 10N  symmetrical

 

                  120o

   10N

half of 120o is angle to vertical
Prove they are in equilibrium like this:

Vertical component of top two forces = 2  x   10 x cos 60o = 10N

Vertical component of lower force =  -10N  (assume convention up = positive)
10 + -10 = 0  therefore have vertical equilibrium
 

Horizontal component of right hand top force =  -10 x sin 60o

(assume a convention right is positive)

Horizontal component of bottom force =

Adding gives zero again – so horizontal equilibrium. QED

 

 

 


 

Problems on resolving and equilibrium

Breithaupt – 4th Ed – P. 52 – Qns 3.1, 3.2, 3.3, 3.4, 3.5 - link to answer sheet

 

Turning effects

The moment of a force is a measure of it’s “turning effect”.  These are required in situations when forces are rotating something – eg. Levers.

    Distance  = s              force = F                moment =  Fs

Couple:  these are when 2 equal but opposite forces are turning an object (ie when they act at different points on the object)

  F

                                          

                     s                  F                             couple = Fs

                     

 The moment of a couple about any point  =  force  x  shortest distance between their lines of action

Principle of moments: 

When a body is in equilibrium  - the sum of the clockwise moments about any pivot must equal the sum of the anticlockwise moments about that pivot.

 

 

Centre of mass

The centre of mass (or the centre of gravity) of a body is the point where its weight is considered to act.

You should be able to use the idea of the centre of mass of an object to decide if the object is stable or not.  (Eg. Will it topple over or not?)

Problems on turning forces moments and centre of mass

Breithaupt 4th Edn. P. 52  Qns  3.6 - 3.9 (density = mass / volume), 3.10, 3.12 – 3.20 

(see how far you can get – don’t labour it – move onto the next if TOO hard and come back if you have time – but ensure you leave PLENTY of time – some of these questions are VERY challenging and extend your skills to grade “A” and beyond)