Scalars and Vectors: Chapter 3, X Physics,Physics tutors Karachi

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Physical Quantities:

All those quantities which can be measured are called physical quantities. Physical

Quantities can be measured by means of magnitude and units.

Magnitude:

A symbol which gives us the quantity of substance is called magnitude.

Unit:

A symbol which gives its relation is called unit. e.g.

5 KG 5: Magnitude Kg: Unit

10 N 10: Magnitude N : Unit

Type of Physical Quantities:

Physical quantities are of two types.

(i) Scalar quantities (ii) Vector Quantities

(i) Scalar Quantities:

Definition:

“All those physical quantities which are completely specified when only magnitude with

suitable units are given are called scalar quantities”.

OR

“Quantities which can be specified by number having appropriate units are scalar

quantities”.

Example:

Mass, Distance, Time, Speed, Energy, Temperature, Work, Volume, density, etc.

(ii) Vector Quantities:

“All those physical quantities which are completely specified only when both

magnitude with suitable units as well as direction are called vector quantity”.

OR

“Quantities having both magnitude and direction with appropriate units are called

vector quantities”.

Example:

Displacement, velocity, acceleration, force, weight, momentum, torque, etc.

Vector Representation:

A vector is represented graphically by a directed line segment or an arrow head

segment. The length of arrow represents the magnitude while arrowhead represents

the direction. Representation of vector can be performed in three steps.

Step I: “Suitable Scale”

In this step we choose suitable scale in order to represent the given vector.

A

Step II: “ Reference axis”

In this step we represent the direction of given vector in reference axis. N

W E

S

Step III: “Representation”

In this step we represent the given vector by the help of above choosen scale and

indicated direction.

Negative of a vector:

Negative vector is defined as:

“A vector just equal in magnitude but exactly opposite in direction is called negative of

a vector.”

OR

“A vector having the same magnitude as that of a given vector but opposite in

direction is called negative of a vector.”

Example:

Negative of a A

Vector A is –A - A

Addition of Vectors by Head to Tail Rule:

Consider two vector A and B represented by lines OP and OQ respectively as shown in

figure we can and these two vectors by placing the tail of second vector to the head of

first vector. Now join the tail of first vector to the head of last vector, it gives us the

resultant vector as shown in Fig.

B

B P

P

B A

If more than two vectors are to be added than same method will be adopted for that.

First draw the first vector than place the second vector so that its tail is at the head of

first vector. Now add all the vectors just by joining the head of first vector to the tail of

next vector and in last join the tail of first vector to the head of last vector it gives us

resultant vector as shown in figure.

D

C

A

B

Resultant Vector:

“A vector which joins the tail of first vector to the head of last vector is called resultant

vector.”

OR

“ A single vector which gives the combined effect of all the vectors which are to be

added is called resultant vectors.”

It can be represented by the following equation:

R = A + B + C + ………

Where R is the resultant vector, while A, B, C, are vectors to be added.

Subtraction of Vector:

Vectors cannot be subtracted directly. They are subtracted by means of addition. To

subtract vector from another vector sign of the vector is changed and then added to

the other vector. For example if a vector B is to be subtracted from a Vecto: A then A-B

is found by adding vector A and –B. Subtraction of vectors can be illustrated as follows:

A

B -B - B

A

Trigonometry:

Trigonometry is an important branch of mathematics and is used to solve various

problems in Physics. Considering the right angle triangle ABC,

angle and

In this triangle. C

0

A B

Base: (AB) Side adjacent to the angle 0 is called Base.

Perpandicular: (BC) Side opposite to the angle 0 is called Perpendicular.

Hypotaneous: (AC) Largest side or the side opposite to the right angle is called

Hypotaneous

The ratios between any two sides of the right angled triangle is represented by

different names. Some of important ratios are as follows:

Sin 0 = Perp / Hyp Cosec 0 = Hyp / Perp

Cos 0 = Base / Hyp Sec 0 = Hyp / Base

Tan 0 = Perp / Base Cot 0 = Base / Perp

The values of trigonometric ratios are changed if 0 is changed.

Resolution of Vector:

The process of splitting a vector into its parts (components) is called resolution of a

vector. Generally a vector is reloved into two components at right angle to each other

Such components are called rectangular components.

Horizontal Comp: The component which is along horizontal direction is called

Horizontal component.

Vertical Comp: The Component which is along vertical direction is called vertical

Component.

Consider a vector F, which shows the representative line AB making angle 0 with x-axis

from B draw perpendicular BC on x-axis. Suppose AB and AC are represented by two

vectors. Vector BC is parallel to y-axis and y

vector AC is along x-axis. Hence we denote

vector AC by F_x and vector AB by F_y by

applying head to tail rule of vector addition,

the sum of vectors F_x and F_y is equal to F.

Therefore, F_x and F_y are rectangular components B

of Vector F.

F

The magnitude of these components can be F_y

determined by using trigonometric ratios.

Now considering right angle triangle ABC. A x

F_x C

HOR Comp / X Comp: VER Comp / Y Comp:

Cos 0 = Base / Hyp Sin 0 = Perp / Hyp

Cos 0 = AC / AB Cos 0 = BC / AB

But But

AC = F_x1 AB = F BC = F_y1 AB = F

Cos 0 = F_x / F Sin0 = F_y / F

F_x = FCos0 F_y = Fsin0

Addition of Rectangular Components of Vectors:

OR

Composition of Vector:

Rectangular components of vector (components that are perpendicular to each other)

can be join together to form resultant vector or original vector.

Considering right angle triangle ABC. C

Where:

F_x = AB = Base F F_y

F_y = BC = Perp

F = AC = Hyp

For magnitude of vector using pythagorons theorem. 0

(H)² = (B)² + (P)² A B

(AC)² = (AB)² + (BC)² F_x

¾¾¾¾¾¾¾

AC = Ö (AB)² + (BC)²

OR

¾¾¾¾¾

ÖF = F_x² + F_y²

For direction of vector using trigonometric ratio:

Tan 0 = Perp / Base

Tan 0 = BC / AC

Tan 0 = F_y / F_x

0 = Tan^-1 (F_y / F_x)

Measurement: Chapter 2, X Physics

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Q1. What is meant by Measurement?

Ans. MEASUREMENT

The following information about a body or an event is called measurement.

i. Size and nature of a body is described with a scale.

ii. A clock describes an event.

iii. Hence the reading will give the scale or clock about a body or an event is known

as measurement.

Q2. Define the term UNIT?

Ans. UNIT:

Such quantities that are used to express physical quantities are called UNIT.

Examples:

Unit of mass is known is Kilogram.

Unit of weight is Newton.

Unit of distance is Meter.

In short, without mentioning unit we cannot described any physical quantity.

Q3. What do you mean by fundamental Unit?

Ans. FUNDAMENTAL UNIT:

Units used to express fundamental quantities are called Fundamental Units.

OR

The unit of quantity of matter, Length and time are called Fundamental units. Because

these are very essential for any experiment. We cannot define any body, event or any

experiment without these fundamental units.

Q4. What do you mean by derived units?

Ans. DERIVED UNITS:

The units, which are derived from fundamental units, are called Derived Units.

Example:

Unit of Area is derived from unit of distance or length. Similarly the division of distance

and time derives the unit of velocity, unit of volume is also derived from distance or

length.

Q5. What is meant by System of units?

Ans. SYSTEM OF UNITS:

The fundamental units and derived fundamental units are called System Units.

Q6. How many systems of units in World?

Ans. There are four system of unit in the world, which are as follows:

1. M.K.S System (Meter – Kilogram – Second)

2. C.G.S System (Centimetre – Gram – Second)

3. F.P.S. or BE System (Foot – Pound – Second) or British Engineering System

4. S.I.System (System International)

Q7. Give the unit of mass, distance and time in different system of Units.

Ans.

Name of system	Unit of Length Or Distance	Unit of Mass	Unit of Time
M.K.S System C.G.S.System F.P.S or BE System S.I System	Meter (m) Centimetre (cm) Foot (ft) Meter (m)	Kilogram (Kg) Gram (g) Slug Kilogram (kg)	Second Second Second Second

Q8. What is meant by System of International? Also write the basic units of S.I.

System.

Ans. SYSTEM OF INTERNATIONAL (S.I. System):

For two centuries a large number of conversion factors had to be memorized to

convert basic units into practical units and vice versa. This difficulty was removed in

MKSA (meter-kilogram-second-ampere) system, in which ampere was adopted as a

fundamental electrical unit. This MKSA system had been internationally adopted and

called S.I. System.

Fundamental Unit of S.I. system

Units of length or distance meter (m) Unit of Mass Kilogram (kg) Unit of time Second (Sec) Unit of Current Ampere (Amp)

Q9. What is difference between M.K.S. and S.I. System?

S.No.	M.K.S. System	S.I. System
1. 2. 3.	It has three fundamental units. It stand for the unit of mass, distance and time. The unit of mass, distance and time is Kg, and Sec respectively.	It has four fundamental units. It stands for the units of mass,Distance time & current. The unit of mass, distance time and current are Kg, Meter, Sec and Amp respectively.

Ans.

Q10. Write short note on following.

i. Meter ii. Kilograms iii. Second iv. Ampere

Ans. METRE:

It is the unit of length in 1983, the general conference in S.I. system of weight and

measures defined “one meter length as the length of the path traveled by light in a

vacuum during a time interval of 1/29979.2458 of a second.

KILOGRAM:

It is the unit of mass in S.I. System. The standard kilogram is the mass of certain

cylindrical piece of platinum iridium alloy kept at Sevres France. The cylinder is 3.9 cm

in diameter and 3.9cm in length.

SECOND:

It is the unit of time in all system. It is defined as the duration of 9192631778 cycles of

the radiation corresponding to the transition between two levels of the ground state of

the Caesium atom. It is denoted by “sec”.

AMPERE:

It is the unit of current in S.I.system. Its symbol is “A”. It is defined as the current which

is flowing in two straight parallel wires of infinite length placed one meter apart in a

vacuum, will produce on each of the wire with a force 2x10⁷ Newton per meter length.

Q11. Write the important fundamental and derived units of S.I.system?

Ans. DERIVED UNITS:

Angle Density Speed Force Pressure Energy-Work-Heat Power Charge Electric Potential Resistance Conductance Capacitance Inductance Magnetic Flux Magnetic Field

Radians Kilogram per cubic metre Metre per second Newton Pascal Joule Watts Volts Coulomb Ohm Seimens Farad Henry Webre Teals

Kg/m3 m/second H Pascal Joule Watt volts coulomb ohm seimens farad henry webre teals

IMPORTANT UNITS

	Physical quantities	Units
1. 2. 3.	Temperature Light Intensity Amount of matter	Kelvin (K) Candila (Cd) Mole (n)

Q12. Give the rule of Notation in SI. System.

Ans. The following are the important rule for notations

i. Full stop should not use in any unit e.g., J.S is wrong, JS is correct. Cm is wrong

but cm is correct.

ii. Similarly cms is wrong, whereas cm is correct.

iii. We often write Kg/m³ m/s, but correct is Kgm³, ms¹,

Q13. Write short note of following:

i. Physical Balance ii. Vernier Calliper iii. Stop Watch

iv. Micrometre Screw Gauge v. Measuring Cylinder

Ans. PHYSICAL BALANCE:

Construction:

i. It is the first hand of lever in which length of both arms is equal.

ii. The beam between two arms is made-up

of Aluminium The beam A is provided with

two identical pans P & P ’ suspended by knife

edges K and K ‘.

iii. When we rotate the screw B, the knife lifts

the beam up from the support T and R is a

pillar. The pans are lifted up from the base

Working:

i. The beam is made horizontal with the help

of screw F and F ‘.

ii. The body whose mass is to be determined is

put on the left pan and known mass I put on

the right pan.

iii. Maintaining the pointer at zero position,

the mass is determined.

iv. It should be noted that the physical balance measures mass, certainly not the

weight and its is quite independent of the value of “g”.

VERNIER CALLIPER:

Construction:

i. A vernier caliper is consists of a rectangular steel bar whose one side is

graduated in cms.

ii. It consists of two scales, one is called Main scale and other is called Vernier

Scale.

iii. The vernier scale freely moves on the main scale.

iv. Vernier scale is 9mm long and is divided into 10 equal parts.

v. The difference between main scale and vernier scale is 0.1mm or 0.01cm, which

called Vernier constant or “Lease Count”.

vi. Least count is used to find the fractional part.

vii. With help of this device we can accurately make measurement upto one tenth

(1/10) of a millimeter or one hundredth (1/100) of a centimeter.

Use:

i. The vernier slides move on main scale until its jaw just touch the ends of the

object being measured.

ii. Suppose a reading of 5.34cm is shown in figure.

iii. This is the reading of main scale.

iv. When a nearest perpendicular line of a vernier scale is coincides with a main

scale marks. The reading obtained in division is called Vernier scale reading.

Suppose it is 4 division as shown in the above figure.

MICROMETER SCREW GUAGE:

Construction:

i. It has two scales, one is called main scale and other is called Circular scale.

ii. It is used to measure the diameter and thickness of small objects.

iii. Spindle is an important part in Screw guage as shown in figure.

iv. Spindle is fitted with a graduated thimble.

v. To protect the screwed portion it kept into enclosed cylinder.

vi. Spindle moves through 0.5mm or 0.05cm for each complete rotation. This

distance is called pitch of the screw gauge.

Use:

i. The circular scale moves on main scale, which is in millimeter.

ii. When the object is being measured, we rotate circular scale clockwise till spindle

just touches the object. Now we may take reading.

STOP WATCH:

Construction:

i. We use a stopwatch for measurement of time,

in our laboratories.

ii. It has two hands M and S.

iii. ‘M’ is for minutes and ‘S’ is for seconds.

iv. Normally both M and S coincide with position

o2f zero.

v. The button ‘B’ is pressed and released for

recording the time interval between the start

and stop of the event.

vi. By pressing and releasing button ‘B’ the needle,

of stopwatch returned at zero and ready for next

fresh reading.

MEASURING CYLINDER:

Construction:

i. It is made-up of glass.

ii. A scale in cubic centimeter or millimeter printed on it.

iii. It is used to find volume of liquids.

iv. When we pour liquid into the cylinder, the level of liquid in the cylinder is noted.

v. We should keep the eye in level with the bottom of the meniscus of the liquid in

order to read the volume correctly.

vi. The liquid surface and the cylinder must be on a horizontal table.

Q14. Define the following:

i. Directly Proportional

ii. Inversely Proportional

iii. Proportional Constant

Ans. DIRECTLY PROPORTIONAL

Such relation between two physical quantities in which one is increased other is also

increased, and similarly when one is decreased, the other is decreased, is called Directly

Proportional. The Graph of such relation is always in straight line. Graph of Directly

Proportional.

Example: V a T

This equation shows the relation of directly proportional between Volume and

Absolute Temperature.

INVERSLY PROPORTIONAL:

If one physical quantity is increasing, then the other is decreasing , and if one were

decreasing, the other would be increasing then this relation is termed INVERSLY

PROPORTIONAL relation.

Example:

If volume and pressure are two physical quantities. Then the relation between them is

V a 1/P

Graph of Inversely Proportional:

The graph of inversely proportional relation between two physical quantities is always

forms a curve shape.

PROPORTIONAL CONSTANT:

To change the symbol of inversely or directly proportional between two physical

quantities. We must use a symbol (K), which is called Proportional Constant.

Example:

V a T V = KT

Where a is a symbol of directly proportional

in this case K is proportional constant.

Q15. What is a Graph?

Ans. It the way of showing the nature of relation between two physical quantities either it is

straight or in curve path.

Q16: How is Graph Construct?

Ans. METHOD OF CONSTRUCTING GRAPH:

i. It is always construct on paper on which

there are many straight horizontal and

vertical lines.

ii. First, we draw two lines, which are

perpendicular to each other as shown in

figure.

iii. The vertical line is called Y-axis and the

horizontal line is called X-axis.

iv. Where both x and y-axis intersect each

other is called origin.

v. The value of Y-axis is always positive from

origin to upward direction and negative

from origin to downward.

vi. The value x-axis is always negative from origin

to left side and positive from origin to right.

vii. All positive and negative value constructed on graph in small points. Then these

points are joined with each other and get the graph.

Q17. Write the advantages of Graph.

Ans. ADVANTAGES OF GRAPH:

i. It is denoted the reactor between two physical quantities.

ii. It also shows the change of relation between two physical quantities.

iii. A doctor diagnosis his patient, by a graph which is drawn between the patient ‘s

condition and medicines to be recommended.

iv. Businessmen can also the state of his business by drawing a graph between investment made and profit earned.

Q18. What is meant by Error?

Ans. ERROR:

The difference between the measured and the actual value of any quantity is called

Error.

Q19. How many types of Errors are. Define each.

Ans. There are three types of errors, which are as follows:

i. Personal Error.

ii. Systematic Error.

iii. Random Error.

i PERSONAL ERROR:

This error occurs when the instruments are used improperly.

ii SYSTEMATIC ERROR:

If difference between actual and experimental reading is due to the fault of the

measuring instrument then it is called Systematic Error.

iii. RANDOM ERROR:

If difference between actual and experimental reading is due to change in physical

state then it is called Random Error.

Q20. What is meant by Correction?

Ans. CORRECTION:

If difference between actual and experiment reading is more and more minimized,

then it is called Correctness. This can be done only when we use the instrument

properly and remove their fault, stop the change of physical state.