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### TOPIC 2

Physical Quantities: Scalar and Vector Quantities

Lecture Outline

- Scalar Quantities
- Vector Quantities
- Addition of Vector Quantities
- Application of Vector Quantities

Lecture Objectives

- After this lecture, students will be able:
- To define scalar quantities
- To define vector quantities
- To analyze vector addition using graphical approach
- To analyze vector addition using vector components

Scalars

Scalar quantities are those which are described solely by their magnitudeSome examples are:

Mass e.g. 14 [kg], 36 [lbs], …

Time e.g. 10 seconds, 40 minutes, …

Volume e.g. 1000 cm3, 4 litres, 12 gallons

Temperature e.g 14 oF , 25 oC, …

Voltage e.g. 9 Volts, etc

Vectors

Vector quantities are those which need to be described by BOTH magnitude and direction

Some of the most common examples which we will encounter are:

Velocity e.g. 100 [mi/hr] NORTH

Acceleration e.g. 10 [m/sec2] at 35o with respect to EAST

Force e.g. 980 [Newtons] straight down (270o)

Momentum e.g. 200 [kg m/sec] at 90o.

Vectors recall

- Quantities with magnitude and direction
- Magnitude
- Size or quantity in unit of measurement
- 10 meters, 5 Newtons

- Direction
- Compass direction: North, Southeast
- Angle: 25o from the positive x axis

- Examples
- Distance—5 miles North
- Acceleration—10m/s2 down
- Force—100 Newton at an angle of 30o above the horizontal

- an arrow drawn to scale used to represent a vector quantity
- vector notation

Ways to add vectors

- Graphically
- Tip to tail method
- Parallelogram

- Using components

Graphical addition of vectors

Arrange tail of vector B on tip of vector A. The vector sum C, called the resultant, is drawn from the tail of vector A to the tip of vector B. The order of addition does not matter.

B

A

C

C = A + B

Parallelogram Method

A

C

B

B (parallel)

Note: B has direction and size, but starting point is arbitrary

Subtraction and Multiplication of Vector

1. The negative of a vector is a vector of the same magnitude but in the opposite direction. Thus vector v = 5 m/s due east, then –v = 5 m/s due west.

2. In order to subtract one vector from another, rewrite the problem s that the rules of vector addition can be applied.

Example :

A – B can be written as A + (-B)

Vector components

y

A vector R which lies at some angle q to an axis has two perpendicular components, Rxand Ry which lie on the axes.

If you draw a straight line from the tip of vector R to the axis, this distance along the axis is the magnitude of the component.

R

Ry

q

x

Rx

Resolving a vector into components

y

Applying trigonometry, the components of the vector R can be defined.

Rx = Rcos(q)

Ry = Rsin(q)

tan q = Ry / Rx

R = √(Rx2 + Ry2)

R

Ry

q

x

Rx

Analytical addition of vectors

- To add vectors analytically, add their components
- Resolve individual vectors to be added into x and y or North and East components using trig
- Add all of the x components
- Add all of the y components
- Find the magnitude of the Resultant (the vector sum) by taking the square root of the sum of the squares of the x and y components
- Find the angle of the Resultant using trig

Summary

- In nature, there are two types of quantities, scalars and vectors
- Scalars have only magnitude, whereas vectors have both magnitude and direction.
- The vectors we learned about are distance, velocity, acceleration, force, and momentum
- The scalars we learned about are time, and Energy.

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