Mechanics Tutorial 1 - Vectors and Scalars

Central to the study of mechanics is the idea of the vector quantity that not only has a value or magnitude, 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.

Unit

Displacement

Distance

Metres (m)

Velocity

Speed

Metres per second (m s-1)

Acceleration

Metres per second2 (m s-2)

Momentum

Newton seconds (N s)

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:

• 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, although m s-1 is much better.

• Some vectors have the same units as scalars.  For example, velocity has units of m s-1 and speed has units m s-1.

• Work is the product of two vectors (Work = Force × distance moved in direction of force) but it is a scalar

• You will sometimes be asked to find the magnitude of the vector. This means that you just put down the value.

• In some situations, direction is absolutely critical.

Remember:

• The product of a vector × scalar is a vector:

• Momentum = mass × velocity.

• Momentum is vector, mass is scalar, velocity is a vector.

• The product of two vectors is a scalar.

• Potential energy = mass × gravity × displacement

• Energy is a scalar, mass is a scalar, while gravity and displacement are vectors.

• The square of a vector is a scalar.

• Displacement squared (= area) is always positive.

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 = (3 N)2 + (4 N)2 = 9 N2 + 16 N2 = 25 N2.

\ Resultant = Ö(25 N2) = 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

 Maths Window You need to understand trigonometrical functions in order to resolve vectors, and to work out the angles that the resultants make. You will have done this in Maths at school but in case you weren’t listening at the time… Sines, cosines, and tangents are ratios between the sides of a triangle for a given angle.   The symbol  is “theta”, a Greek letter ‘th’, which is used to denote angles. • sin  = opposite               hypotenuse • cos = adjacent              hypotenuse • tan  = opposite                adjacent • tan  = sin                cos  There are all sorts of different mnemonics to help you understand these relationships. A rule of thumb is that if you get a sine or cosine that is greater than 1, you’ve done it wrong.

Vectors not at Right Angles

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.

 Maths Window The cosine rule can be used to work out the resultant of two vectors that are not at right angles.   a2=b2+c2- 2bccosA   F22=F12+FR2- 2F1FRcosq

At A-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.

It is easier to add vectors that are not at right angles using resolution of vectors.

 What are the resultants of these vectors? What are the angles that the resultants make to the vertical in the previous question?
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.

• Fx = F cos q

• Fy = F sin q

Any vector in any direction can be resolved into horizontal and vertical components.  These can be calculated by accurate drawing or trigonometry.

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-1.  We can resolve the vectors, remembering that the weight acting vertically is the resultant force.

The force vectors are arranged like this:

Note that:

• The weight vector forms the hypotenuse of the triangle.

• Simple geometry shows that the angle between the weight and the normal is o.

• Normal means at 90 degrees to.  The normal force is the force on the road.

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.

 It is depressing how many students write weight in kilograms.  Watch out for this bear trap! You cannot talk of vertical and horizontal components of a vertical vector. The components are perpendicular to each other.

 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. (Use g = 9.8 N kg-1)

Adding vectors by resolution
When vectors are not at right angles, it is much easier to resolve each vector into vertical and horizontal components.

The components add up like this:

Add the horizontal vectors together:

Add the vertical vectors together:

Finally we do the vector addition:

Resolving Vectors in 3-dimensions

It is useful to be aware of this, but it won’t be in the exam.

The resultant vector is the vector sum of the x, y, and z components: