Physics class 11 notes Chapter 1 Measurement

measurement Physics notes for class 11 chapter no 1, Khyber Pakhtunkhwa textbook board class 11 physics short question, numerical problems, and long question.

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Physics class 11 notes Measurement, Numerical, short question, long question,

1.  Define the number π  and show that 2π radian = 360°.

“It is defined as the circumference of a circle (2πr) divided by its diameter (d=2r) and is given by,
                                                                    2π radian = 360o.”
As one radian is defined as the
plane angle subtended at the centre of a circle by an arc of length equal to the radius and is given by,


Where ‘l’ is length of arc and ‘r’ is radius. As for a complete revolution, ‘l’ will be the circumference of circle i.e. 2πr. So


So, we get

Q.2 Define the terms

  • a. Error,
  • b. Uncertainty,
  • c. precision and
  • d. accuracy
  • in measurement.

a. Error:
“It is a difference or disagreement between measured and accepted standard value.”
b. Uncertainty:
“Error arises due to natural imperfection of an experimenter, limitations of apparatus, and sudden change in the environment is known as uncertainty.”
c. Precision:
“It is associated with the least count of the measuring instrument. We can say, smaller the least count of the instrument better will be its precision.”
d. Accuracy:
“It is nearest measured value to actual value and is related to the fractional error of instrument, So, smaller the magnitude of fractional error of instrument better will be the accuracy.”

3. Explain several repetitive phenomena occurring naturally which could serve as reasonable time standards.


Any phenomena that repeat itself in regular equal intervals of time can be used as a time standard. Some naturally occurring phenomena for time standard are as follows:

  • Revolution of earth about its axis: Earth completes its one revolution about its axis in 24 hours, which can be taken as a time standard.
  • Revolution of the earth around the sun: Earth completes its one circle around the sun in 365 days, which serves as a time standard of 1 year.
  • Revolution of the moon around the earth.
  • Back and forth motion of the pendulum: If we neglect the air friction then the motion of the pendulum under the action of gravity is also a time standard as it completes its one vibration in the same intervals.
  • Read more: Gravitation Class 11 handwritten notes

Question No 4

Why do we find it useful to have two units for the amount of substance, kilogram, and mole?


There are two SI units to measure the amount of substance named as,
It is a unit to measure the mass, where we have to deal with large quantities.
Mole is the SI unit to measure the mass of a system, which contains, as many elementary entities as there are atoms in 0.012kg mass of C-12. It is used when we are interested in measuring the number of entities instead of mass and these elementary entities must be specified, may be atoms, molecules, ions, other particles, or other specified groups of such smaller and fundamental particles. 

5. Show that the famous “Einstein equation” E = mc2 is dimensionally consistent.


Einstein equation is given by,

                                                                E = mc2

As energy is equivalent to work so, we can write

                                                                E = W = F.d

Putting this value in the Einstein equation, we get

                                                                F.d = mc2

Putting dimensions of mass, acceleration, displacement, and speed of light, we get

                                                    [MLT-2][L] = [M][LT-1]2

                                                    [ML2T-2] = [[ML2T-2]

Above equation shows that the Einstein equation is dimensionally consistent.

6. Deduce the dimensions of the Gravitational constant.

As, gravitational force is given by,


Where ‘G’ is gravitational constant, ‘M’ is mass of earth, ‘m’ is mass of any object an ‘r’ is the distance between object and center of earth. So, we get,

2 1

Putting dimensions of force, distance and masses, we get.


So dimensions of gravitational constant will be,


7. Find the dimensions of kinetic energy.

As, K.E is given by,


Where ½ is a dimensionless value as it is a constant, so putting dimensions f mass and velocity we get
                                                                K.E = [M] [LT-1]2


8. Give the drawbacks to use the period of a pendulum as a time standard.

Time period of a pendulum depends upon:

  • Length of string “L”
  • value of gravitational acceleration “g”
  • Air resistance

             T = 2π √(l/g)

Length of string varies with surrounding temperature i.e. it contracts in winter and expands in summer. Similarly, value of “g” also varies with height and depth. Air resistance affects the back and forth motion of pendulum i.e. larger the air friction smaller will be the amplitude.

Due to these variations, time period of pendulum is not a perfect standard for time.

9. Are radians and steradians the base units of SI? Justify your answer.

No, radians and steradians are not the base units of SI. These are geometrical units i.e. plane angle (angle subtended by two radii of circle) and solid angle (angle subtended by two radii of a sphere), respectively. As their values vary with diameter of circle and sphere so the general conference on weights and measures has not yet classified these units under either base units of derived units. These are called supplementary units.

10. What does the word “micro” signify in the words “micro wave oven”?

                                                    micro = 10-6
This means that the waves used in oven have 10-6 times shorter wavelength than all other radio waves. Energy and wavelength are inversely proportional to each other so shorter the wavelength, higher the energy. Therefore, microwaves have higher energy than other electromagnetic waves and can produce more heating effect. The reason we call oven as microwave oven is, that we use these higher energy microwaves in it.

11. Density of air is 1.2kg m-3. Change it into gm cm-3.

                                                Density of air = 1.2kgm-3
We know that,
                                1kg = 1000g               and               1m = 100cm
Putting these values in density of air, we get
                    Density of air = 1.2×1000g×(100cm)-3
                    Density of air = 1.2×103g×10-6cm-3
                    Density of air = 1.2×10-3gcm-3
Which gives us,


12. An old saying is that “A chain is only as strong as its weakest link. What analogue statement can you make regarding measurement?

Answer :
Analogous statement for that old saying regarding measurement can be,
     “The result of an experimental data is as accurate as the measurement  used in                                                                computation to measure that data.”

13. Differentiate between the light year and year.

It is the unit of time in which earth completes its one revolution around the sun i.e. 365 days.
Light Year:
It is basically unit of length and is defined as the distance travelled by the light in vacuum per year.

Comprehensive Questions Notes for Physics 1st year

1. Define Physics. Explain the scope and importance of physics in today’s technological world.



“Physics is a branch of science, which deals with the study of matter, energy and they are mutual


Scope and Importance:

Scope and importance of physics are very clear in everyday life. It gives us detailed information regarding several occurring events in the universe. It is not only a study of the smallest subatomic particles like electron, proton, neutron etc. However, it also covers the huge heavenly bodies like galaxies etc. It explains nature in terms of some fundamental laws e.g. Newton’s law, Archimedes law principle, Coulomb’s law etc. It has brought a great revolution in information technology and communication fields.

The information media and fast means of communication have brought the whole world into close contact. Global positioning system and tracking system of vehicles are also the latest achievements. Video mobile communication is now in common use.  It helped us to get many answers to some very common questions like, what is the moon? What is the sun? How the universe formed?

Physics has also developed many useful instruments for precise and accurate measurements.

2. What does a unit system mean? Give names of three different systems of units. In System International, explain what is meant by base units, derived units and supplementary units?


Unit system is a system used to define and describe different physical phenomena by relating them to some selected standards.

There are three different systems of units named as,

  • Base units.
  • Derived units.
  • Supplementary units.

Base Units:

Base units are SI units, which are defined as arbitrarily and cannot be defined in terms of other units. There are seven base units namely length, mass, time temperature, electric current, luminous intensity and amount of substance.

Derived Units:

These are units, used to measure all other physical quantities and they are derived from seven base units. Some derived units are newton, joule, watt etc.

Supplementary Units:

There are some units which general conference on weights and measures have not yet either classified under base units or derived units. These units are supplementary units. This class contains only two units named as radian and steradian.

3. Define the terms error and uncertainty. Explain different types of errors?



“It is a difference or disagreement between measured and accepted standard value.”


“Error arises due to natural imperfection of an experimenter, limitations of apparatus and sudden change in the environment is known as uncertainty.”

There are three types of error:

  • a. Personal error
  • b. Systematic error
  • c. Random error

a. Personal Error:

“The error due to carelessness or improper knowledge about an instrument or incorrect reading of a scale by an experimenter is called personal error.”

We can remove these types of errors up to some extent if the experimenter experiments with more care and is himself proficient with the necessary knowledge.

b. Systematic Error:

“The error due to faulty apparatus and poor calibration of the instrument itself is a systematic error.”

These errors can also arise due to zero error of the instrument. Therefore, experimenters can remove these errors by either using a more accurate instrument or by adding or subtracting the zero error form calculated reading.

c. Random Error:

“The error due to repeated measurement of the same quantity under the same condition that results in different values of the same quantity is called a random error.”

These types of errors can arise due to change in temperature or humidity or some other unknown causes in an experimental environment. Maintaining strict control conditions in the laboratory, repeating the measurements several times, and taking measures of measured values can minimize this error.

4. Briefly explain how we indicate uncertainties? With the help of examples, explain how uncertainties are measured in the final result in different cases?

Indicating Uncertainty:

Uncertainty in a measurement can be indicated by adding or subtracting the error or uncertainty from the measured value.

This can be understood by following example:

Let’s consider a box that has a length of 22.5cm. As the least count of a meter rod is 0.1cm so, the total uncertainty in length of box will be ±0.1cm, it means that there is ±0.05cm of uncertainty at both ends of the meter rod. Thus if one end of meter rod coincide with 10.5cm mark  and the other with 33.0cm mark then total length with uncertainty is given by:

                            (33.0±0.05)cm – (10.5±0.05)cm = (22.5±0.1)cm

It means the length of the box lies between 22.4cm and 22.6cm.

Calculating Uncertainty in the Final Result:

Let x, y and z are three different physical quantities with uncertainties of Δx, Δy and Δz, respectively.

(a) Rule for addition and subtractions:

Let x be the sum of y and z i.e. x = y+z. Then, uncertainty in value of x is given by,

                                            Δx = ± (Δy+Δz)

(b) Product and quotient rule:

If, we have


Then percentage uncertainty in x will be the sum of percentage uncertainties of y and z.
Percentage uncertainty in x =
                                   ± (Percentage uncertainty in y + percentage uncertainty in z)
It can be understood by following example,
As Ohm’s law states, 


Percentage error in R will be given by,


(c)  Power of a quantity:
For power, the individual percentage uncertainty is multiplied with the power of that quantity in final result. For example, if we want to calculate the volume of a sphere along with uncertainty. As a volume of a sphere is given by,


 Let,                                               r = (2.25±0.01) cm
                             Then percentage uncertainty in r will b

13 1

Then total uncertainty in r3, is given by
                                                                0.4% × 3 = 1.2%
So we get,


5. What does the dimension of a physical quantity mean? Explain what are its applications and limitations.


 “The powers positive or negative to which the fundamental units of the first system must be raised to give the unit of a given physical quantity are called dimensions of that quantity.”

Dimensions of physical quantity:

In physics, a dimension represents the nature of a physical quantity. For example, different quantities such as length, breadth, wavelength etc. are all measured in meters and have the dimension [L], similarly, the dimension of time is [T] and that of mass is [M]. 

Dimensions of derived quantities are products of dimensions of base quantities. For example, the dimension of velocity is given as

                                                    v = [LT-1]

Also as acceleration, “a” is velocity per unit time square so its dimensions will be,

                                                    a = [LT-2]

Some terms and limitations with dimensions

a)    Dimensional variables: Those physical dimensional quantities that have dimensions but variable magnitudes are dimensional variable quantities. For example, force, energy, acceleration etc.

b)     Dimensional Constants: These physical dimensional quantities are constant in magnitude. For example, speed of light in vacuum, Planck’s constant, gravitational constant etc.

c)     Dimensionless variables: These physical quantities have no dimensions but variable magnitudes. Some examples are plane angle, solid angle, strain and coefficient of friction etc.

d)    Dimensionless constants: Physical quantities which have no dimensions but are constants are called dimensionless constants e.g. 1, 2, 3, 4, π etc.

Applications of dimensional analysis:

a)    We can check the homogeneity of an equation with dimensional analysis.

For example, we can check the homogeneity of the following equation,


Numerical Problems Physics Class 11 Notes Chapter 1 Measurement

1. Express the following quantities by using prefixes.
(a) 4.0 x 10-4 m
(b) 15.0 x 10-8 s
(c) 7.5 x l0-7g



2. The length and width of a rectangular plate are (15.6 ± 0.1) cm and (10.80 ± 0.01) cm respectively. Calculate area of the plate and uncertainty in it. 



3. The length of a pendulum is (100.0 ± 0.1) cm. If the acceleration of free fall is (9.8 ± 0.1) ms -2, calculate the percentage uncertainty in Time period of the pendulum.

(± 0.6%)

Given Data:
                                           Length of pendulum = L+ΔL = (100.0±0.1) cm
                                            Free fall acceleration = g+Δg = (9.8±0.1) cm
To Find:
                                            Percentage uncertainty in time period = ΔT =?
                As time period of the pendulum is given by,


4.Theory suggests that drag force depends upon the viscosity of the medium, average radius of the object and velocity of the object moving through the fluid. Derive a formula for dragging force of fluid by using dimensional analysis.

(Hint viscosity =ML-1 T-1)

Answer :

Given Data:
Drag force D depends upon viscosity of medium η, average radius of object r and velocity of object v.
To Find:
                        Formula for dragging force = D =?
                 Let D has a direct relation with η, r and v as follows


5. (a) Suppose that the displacement of an object is related to time according to the expression x = B t2 . What are the dimensions of B?

(b) A displacement is related to time as x = A sin (2 π f t), where A and f are constants. Find the dimension of A?

[(a) LT-2,(b) L]


(a) Given Data:
                                               x = Bt2
To Find:
                              Dimensions of B =?
                                            x = Bt2
Writing dimensions, we get
                                            [L] = B [T]2
So, we get
   B = [LT-2]             Answer
(b) We have,
                                    X = Asin(2πft)
In dimensions we can write it as,
                                            [L] = A [T-1T]
So, this gives
              A = [L]        Answer