Mechanics Tutorial 1 - 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:

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

Vector

Scalar

Unit

Displacement

Distance

Metres (m)

Velocity

Speed

Metres per second (m/s)

Acceleration

 

Metres per second2 (m/s2)

Momentum

 

Newton seconds (Ns)

Force

 

Newtons (N)

 

Work , Energy

Joules (J)

 

Voltage

Volts (V)

 

Temperature

Degrees Celsius (oC)

 

Frequency

Hertz (Hz)

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

Note:

 

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 90o 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  Ž q = tan-1(0.75) = 36.9o

  We can also add vectors that are not at right angles:

In the picture above we can see the resultant of two forces that are not at right angles.  We can show that they make a vector triangle by moving Force 1:

Alternatively we can use a parallelogram of forces as shown below:

The resultant can be worked out by accurate drawing.  Or you can use the cosine rule:

At AS level, you would only have to add vectors at 90 degrees to each other.  If there were vectors as shown above, then you would normally be expected to use accurate drawing.  The question would tell you to do accurate drawing, although I am sure that if you got the right answer from the cosine rule, you would be awarded full credit.

Question 1

What are the resultants of these vectors?

Answer
Question 2 What are the angles that the resultants make to the vertical in the previous question?  Answer
 
 
Resolution of Vectors

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

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 9.8 m s-2 and the force of gravity is 9.8 N/kg.  We can resolve the vectors, remembering that the weight acting vertically is the resultant force.

Remember:

 

It is depressing how many students write weight in kilograms.  Watch out for this bear trap!

Question 3

The car has a mass of 1100 kg, and the angle is 10o

Calculate:

(a) the weight of the car,

(b) the force on the road,

(c) the downhill force.                            

Answer