VECTORS - Diocesan College



VECTORS

VECTORS AND SCALARS

A scalar is a physical quantity that has magnitude only. Examples: time, distance, speed, mass, energy, work and power.

A vector is a physical quantity that has both magnitude and direction. Examples: displacement, velocity, acceleration, weight and momentum.

Illustration of the difference between a vector and a scalar:

A girl walks a distance of 800 m along a winding path from school to home. The straight-line distance of her home from the school is 500 m and the direction is north-east. Write down the values for the distance and displacement of her home from the school.

Distance = 800 m (Magnitude only)

Displacement = 500 m, NE (Magnitude and direction)

Distance is the magnitude of the actual distance covered by a moving object from the starting point to the finishing point.

Displacement is the magnitude and direction of the straight-line distance from the starting point to the end point of a motion.

REPRESENTATION OF DIRECTION

There are two methods of representing direction:

1. The use of the points of the compass as a reference from which to measure angles:

Example: 30o east of north or N 30o E.

2. The use of bearings, i.e. the clockwise measurement of angles from north or the vertical.

Example: Bearing 210o

REPRESENTATION OF VECTORS

Graphical representation:

A vector is represented graphically by an arrow drawn to scale. The magnitude of the vector is represented by the length of the arrow and the direction of the vector by the direction of the arrow.

When drawing a vector, the following must be shown:

(1) the reference direction, e.g. north;

(2) the scale used;

(3) an arrow head to show direction.

Symbolic representation:

The symbol of a vector is distinguished from the symbol of a scalar by means of an arrow written above the symbol of a vector.

Example: Distance (scalar): s Speed (scalar): v

Displacement (vector): s Velocity (vector): v

ADDITION OF VECTORS

Vectors can be added by algebraic addition if they act along the same straight line or by accurate geometric construction or trigonometric calculation if they operate at an angle to each other. The answer to a vector addition is called the resultant which is the vector from the tail of the first vector to the head of the last vector.

A resultant of a number of vectors is the single vector that will have the same effect as all the original vectors acting together.

1. Addition of vectors acting along the same straight line by algebraic summation

A vector is considered negative with respect to another if it acts in the opposite direction. One therefore has to state which direction is positive.

Example:

Two vectors of 12 N and 5 N act in opposite directions along the same straight line. Calculate the resultant.

By calculation:

Let the direction of the 12 N vector be positive.

Resultant = +12 N – 5 N = + 7 N, i.e. 7 N in the direction of the 12 N vector.

2 N

The same thing would apply to two vectors acting vertically.

Resultant

Example: 5 N 7 N

Two vectors of 2 N and 5 N act in the same direction vertically.

2.2 Addition of two vectors by trigonometric calculation.

This method will be limited to right angled triangles.

In this method, a rough diagram is drawn and the magnitude of the resultant is calculated using Pythagoras’s Theorem and the angle using trigonometric ratios.

Example:

A girl walks due west for a distance of 50 m and then 30 m due south. Calculate her resultant displacement.

R2 = 502 + 302 (Pythagoras) tan ( = 30/50 = 0,6

= 2500 + 900 = 3 400 ( ( = 30,96o

( R = 58,31 m

Resultant = 58,31 m, W 30,96o S or 58,31 m, 239,04o

VECTOR COMPONENTS

The process of resolving a vector into two components is the reverse operation of adding two vectors to obtain a resultant. When resolving a vector into components, two vectors are determined that will given a resultant equal to the single vector.

1. Components at right angles to each other

Vertical component: [pic]

Horizontal component: [pic]

Example:

A roller of mass 200 kg is pulled across the lawn at a constant speed by a gardener exerting a force of 300 N at an angle of 30o to the horizontal. Calculate:

a. the magnitude and direction of the force of resistance to motion experienced by the roller in a horizontal direction.

b. the resultant vertical force exerted by the roller on the lawn.

Answer:

a. [pic], to the right

Force of friction = 259,81 to the left.

b. [pic]

W = mg = 200 ( 9,8 = 1 960 N, downwards

Resultant vertical force = 1960 – 150 = 1810 N, downwards

2. Components on a slope

Component down the slope: [pic]

Component perpendicular to the slope: [pic]

Example:

A block of mass 2 kg remains stationery on a plane inclined at 30o to the horizontal.

a. Calculate the magnitudes of the components of the weight of the block parallel and perpendicular to the slope.

b. What is the magnitude and direction of the force of friction between the block and the plane?

Answer:

a. W = mg = 2 ( 9,8 = 19,6 N

[pic], down the slope.

[pic], perpendicular to the slope

b. Force of friction = 9,8 N up the slope.

EXTENTION WORK

1. Addition of more than two vectors by accurate geometric construction

In the polygon method, vectors are constructed head to tail and the length and direction of the line drawn from the starting point to the finishing point represents the resultant.

Example: Three vectors of 3 N, 60o, 4 N, 90o and 5 N, 230o act at a point. Determine the resultant by accurate construction and measurement.

Note: The larger the scale, the smaller the margin of error.

It is best to draw a rough sketch before starting your construction in order to determine the position of the starting point so that the construction does not run off the page.

2. Addition of two vectors at an angle to each other

2.1 By geometric construction

The following two methods can be used:

2.1.1 In the triangular method, the vectors are constructed head to tail and the third side of the triangle is completed by drawing an arrow from the starting point to the finishing point. The magnitude and direction of the third side as taken from the starting point is the resultant.

Example: A girl walks a distance of 100 m due east before turning and walking a distance of 50 m in a direction E 60o S. Determine her resultant displacement by accurate construction and measurement.

(The above is a computer generated sketch diagram and is therefore not an accurate construction.)

2. In the parallelogram method, the vectors are constructed tail to tail, the parallelogram is completed and the resultant is determined by measuring the magnitude and direction of the diagonal from the starting point.

FORCES IN EQUILIBRIUM

The Triangle Law states that if three forces are in equilibrium, they can be represented in both magnitude and direction by the three sides of a triangle taken in order.

The equilibrant of a number of forces is the single force that keeps all the other forces acting on a body in equilibrium. The equilibrant is equal in magnitude but opposite in direction to the resultant of the forces.

Example:

An object of mass 2 kg is suspended from a light string. A second string is attached to the first at a point P. A force F is applied horizontally until the first string makes an angle of 30o with the horizontal.

1 Write down an expression for the vertical component of force T.

2 Calculate the magnitude of force T.

3 Calculate the magnitude of force F.

Answer:

1. Ty = mg

2. Ty = mg = 2 x 9,8 = 19,6 N

[pic]

3. [pic]

EXERCISES

1. Which of the following are both vector quantities?

A time and distance C velocity and momentum

B speed and energy D momentum and time

2. Which of the following quantities is not a vector?

A force C velocity

B displacement D energy

3. The resultant of two forces acting on a body

A acts at the same point but in the opposite direction.

B keeps the body in equilibrium.

C produces the same effect as the two forces together.

D has the opposite effect of the two forces together.

4. Two cyclists travelled from town X to town Y. Their average velocities between the two towns were the same, but their average speeds differed. This is possible if

A they travelled on different roads.

B they travelled at different times.

C their distances and displacements were equal in magnitude.

D they travelled on the same road, but took different times to travel from X to Y.

5. If a body is displaced 24 m horizontally and then 32 m vertically, the magnitude of the resultant displacement, in metres, is

A 1,3. B 8.

C 40. D 56.

6. In which of the following vector diagrams is the resultant of the three vectors zero?

A B

C D

7. Two forces act at the same point on a body. When the angle between the two forces increases from 40o to 140o, the magnitude of the resultant of the two forces will

A increase.

B decrease.

C first decrease and then increase.

D first increase and then decrease.

8. In the diagram, A and B are vectors at an angle ( to each other. For which value of ( will the resultant vector have the smallest magnitude?

A 0o

B 90o

C 120o

D 180o

9. A and B are vectors of equal magnitude. B is a vector of which the direction can be changed. For which value of ( will the resultant of A and B have the same magnitude as A or B?

A 30o

B 60o

C 120o

D 150o

10. The accompanying diagram represents three forces x, y and z. Which one of the following vector equations describes the situation?

A x + y = z

B x = y + z

C x + y + z = 0

D y = x + z

11. Two forces X and Y can be replaced by a single force of magnitude 5 N. If the magnitude of X is 2 N, which one of the following can be the magnitude of force Y?

A 1 N

B 2 N

C 6 N

D 10 N

12. A ship is sailing slowly due west. In which direction should a passenger walk on the deck at P, at a suitable speed, to move due south?

A along PD

B along PB

C along PC

D along PA

13. Two ships, X and Y, are travelling in different directions at equal speeds. The actual direction of X is due north, but to an observer on Y the apparent direction of motion of X is north-east. The actual direction of Y must be

A west.

B east.

C south-west.

D south-east.

14. Two forces of 10 N each act at the same point and the magnitude of their resultant is 10 N. Which one of the following statements is true?

A The forces are perpendicular to each other.

B The angle between the forces is 60o.

C The angle between the forces is 120o.

D The angle between the forces is 30o.

15. If three vectors with magnitudes 30, 40 and 50 are added together, which one of the following cannot represent the magnitude of the resultant?

A 0 N

B 28 N

C 55 N

D 130 N

16. The vector that best represents the resultant of the forces F1 and F2 shown acting on point P in the diagram is

A B C D

17. A force is represented by AB in the diagram. This force is the resultant of a force represented by AD and a force F which is not shown. Force F is direction from

A D to B.

B B to D.

C A to C.

D C to A.

18. A man travels 50 km due west, then 40 km due south and finally 20 km due east. The magnitude of his displacement, in km, is

A 30.

B 50.

C 60.

D 90.

19. A person walks 5 km north-east and then 5 km south-east. Her displacement is

A 10 km.

B 7,07 km.

C 10 km east.

D 7,07 km east.

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