Vectors and Scalars

Scalars and Vectors

Central to the study of mechanics is the idea of the vector quantity that not only has a value, but direction as well.  Examples include:

• acceleration, 

• force, 

• velocity.

  Any quantity that does not specify a direction is a scalar, examples of which include:

• energy

• temperature.

Notice that the units are the same, regardless of whether they are vectors or scalars.

Note:

• Some vectors can be used as scalar quantities.

• There are some scalar quantities, e.g. temperature, that have no vector equivalent.

• The shorthand for metres per second is either written ms^-1 or m/s.  Either is acceptable.

• Work is the product of two vectors (Work = Force × distance moved in direction of force) but it is a scalar.  Click on the link below for an animation to show why.  You will need Windows Media Player (or similar).

Adding Vectors

If the force vectors of 3N and 4N are in the same direction, they simply add together.

The heavy arrow indicates the resultant force.

If the vectors are in opposite directions, we subtract.

We can see that the resultant is now just 1 N.

If the two vectors are at 90^o use Pythagoras’ Theorem.

Resultant² = 3² + 4² = 9 + 16 = 25.     

\ Resultant = Ö(25) = 5 N

To work out the angle we use the tan function:

                        tan q = ¾ = 0.75  Þ q = tan^-1(0.75) = 36.9^o

Resolution of Vectors

We can resolve any vector into two components at 90^o to each other.  They are called the vertical and the horizontal components.

• F_x = F cos q

• F_y = F sin q

Consider a car going up a hill. 

The angle of the hill is q degrees.  We must note that the weight (given by the mass in kilograms × acceleration due to gravity) is always pointing vertically down.  Acceleration due to gravity can be taken as 10 m s^-2 and the force of gravity is 10 N/kg.  We can resolve the vectors, remembering that the weight acting vertically is the resultant force.

Remember:

• Mass is how much material there is in an object.  It is measured in kilograms (kg).
• Weight is a force measured in newtons (N).  It is the product of mass and the acceleration due to gravity.  It is the force exerted by a mass due to gravity.

Vector calculations can be done using scale drawing.

• Use a sharp pencil

• Use graph paper

• Choose a scale for both vectors and stick to it

• Measure the angles with a protractor.

Equilibrium for Coplanar Forces

Forces in equilibrium mean that they are balanced.  Coplanar forces act in the same plane.  Two balanced forces are equal in magnitude but opposite in direction to the other.

We can see easily from the free body diagram that the resultant force is zero. 

If we are considering three coplanar forces in equilibrium, use the triangle of forces rule:

 If 3 forces acting at a point can be represented in size or direction by the sides of a closed triangle, then the forces are in equilibrium, provided their directions can form a closed triangle.

This means that the forces can follow each other round a triangle

Notice how

• The forces form into a closed triangle.

• The directions of the forces go round the triangle.

In statics, remember that all forces add up to zero.  That does not mean that there are no forces;  the forces balance each other out.

At AS level, you are most likely to encounter symmetrical systems like this.  If the system is NOT symmetrical, don't panic.  Remember:

• The weight (downwards force) can be split into two force vectors.  These add up to the weight.

• The horizontal force vectors add up to zero.

• If you are completely stuck, use accurate drawing!  Whatever you do, don't leave a blank!