Scalars & Vectors

Name: ________________

Class: _________________

Index: ________________

Learning objectives

At the end of this unit you should be able to :1. State what is meant by scalars and vectors quantities and give common examples of each;

2. Add two vectors to determine a resultant (graphical method);

3. Solve problem for a static point mass under the action of 3 forces for 2 dimensional cases (graphical method).

Scalar quantities are quantities that have magnitude only.

Vector quantities are quantities that have both magnitude and direction.

Scalars VectorsDistance Displacement

Speed Velocity

Mass Weight

Time Acceleration

Pressure Force

Energy Momentum

Volume

Density

Addition of Scalar Quantities

The addition of scalar quantities is very simple as it involves addition or subtraction using simple arithmetic. An example is 4 kg plus 6 kg always gives the answer 10 kg.

Addition of Vector Quantities

The addition of vector quantities is not as straight forward as that of scalar quantities as the direction of vector quantities must be considered.

Adding parallel vectors

4 N

6 N

Resultant force on block = 4 + 6 = 10 N to the right

Resultant force on block = 6 - 4 = 2 N to the left

4 N

6 N

Observations: Vectors can be represented graphically by arrows. The length of the arrow represents the magnitude of the vector. The direction of the arrow represents the direction of the vector.

Vector Addition (Method 1 – Tip to tail)• The resultant (force) of two vectors is a single vector which

produces the same effect as given two vectors.• To add vectors P and Q, the starting point of vector P is

placed at the ending point of vector Q. The vector sum or the resultant vector of P and Q is represented by vector R, which is the vector joining the starting point of vector P to the ending point of vector Q. The same resultant vector is obtained irrespective of whichever order the vectors are added.

R

P

QQ

P

• Vectors acting at an angle to each other can be added graphically using the parallelogram law.

• The parallelogram law of vector addition states that if two vectors acting at a point are represented by the sides of a parallelogram drawn from that point, their resultant is represented by the diagonal which passes through that point of the parallelogram.

Vector Addition (Method 2 - Parallelogram)(Method 2 - Parallelogram)

P

Q

P

Q

R

ExampleA force of 3 N acts at 90o to a force of 4 N. find the magnitude and direction of their resultant, R.

Solution

R2 = 32 + 42 = 25 R = 5 N tan = 3/4 = 37o

3N

4N

R

Scale : 1 unit rep. 1 N

Example

A force of 10 N acts at 45o to a force of 4 N. find the magnitude and direction of their resultant, R.

Solution Scale : 1 unit rep. 2 N

From the diagram, R = 6.2 units = 12.4 N ( =13° counter-clockwise from 10 N force)

450

4N

10 N

R

Figure A below shows a catapult used to project an object. Force F pulls back the object, creating tension in the rubber cords. The tension force in each rubber cord is 20N and the two cords are at 60° to each other. Fig. B shows the direction of the two tension forces acting on the object. By making a scale drawing of Fig. B, find the resultant of these two tension forces acting on the object. State the scale of that you use.

Application

Solution (Parallelogram Method):

Scale: Let 1cm rep. 4 N

60°5 cm 5 cm

Since 1cm rep 4 N, the resultant force is 8.6 x 4 = 34.4 N

8.6 cm

Application

Figure below shows the top view of the wooden block. Two forces, magnitude of 30 N and 20 N, act on the wooden block (not drawn to scale). By means of a scale diagram, determine the magnitude and direction of the resultant force acting on the wooden block.

30 N

20 N

40

15

Solution (Tip to tail Method):

Scale: Let 1cm rep. 10 N

Since 1cm rep 10 N, the resultant force is 4.5 x 10 = 45 N at an angle of 25° clockwise from the 30N force

3 cm

2 cm

4.5 cm

55°

Tip to tail method

References

http://www.jaunted.com/tag/Singapore%20Flyer