Measurements and Experimentation

1. System of Units

Physics is a quantitative science. To understand phenomena, we must measure physical quantities like length, mass, time, temperature, etc.

Concept Measurement: It is the process of comparing a given physical quantity with a known standard quantity of the same nature.

Unit: The standard quantity used for comparison is called a unit. A measurement is always expressed as:
Physical Quantity = Numerical Value × Unit.

Fundamental and Derived Units

Fundamental (Base) Units: The units which are independent and cannot be derived from other units. (e.g., Mass, Length, Time).
Derived Units: The units which can be expressed in terms of fundamental units. (e.g., Area = length × breadth = m², Velocity = distance/time = m/s).

Systems of Unit

Before the international standardization, three major systems were used:

C.G.S. System: Centimetre (Length), Gram (Mass), Second (Time).
F.P.S. System: Foot (Length), Pound (Mass), Second (Time). (British System)
M.K.S. System: Metre (Length), Kilogram (Mass), Second (Time).

Today, we use the S.I. System (Système International d'Unités), which is an expanded and modified version of the M.K.S. system. It includes 7 fundamental units (Metre, Kilogram, Second, Kelvin, Ampere, Candela, Mole).

2. Measurement of Length

The standard S.I. unit of length is the metre (m).

Fact Bigger Units of Length (For astronomy):

Astronomical Unit (A.U.): The mean distance between the earth and the sun. 1 A.U. = 1.496 × 10¹¹ m
Light Year (ly): The distance travelled by light in a vacuum in one year. 1 ly = 9.46 × 10¹⁵ m
Parsec: The distance at which an arc of length 1 A.U. subtends an angle of 1 second of arc. 1 Parsec = 3.26 ly = 3.08 × 10¹⁶ m

Smaller Units of Length

Micron (μm): 10^-6 m (Used to express the size of bacteria).
Nanometre (nm): 10^-9 m (Used to express the wavelength of light).
Angstrom (Å): 10^-10 m (Used to express the size of an atom).
Fermi (f): 10^-15 m (Used to express the size of a nucleus).

3. Measuring Instruments: Vernier Callipers

A standard metre scale can only measure accurately up to 1 mm (0.1 cm). To measure lengths smaller than 1 mm, we use a Vernier Callipers.

Concept Principle of Vernier: It works on the principle of using two scales—a main scale and a sliding vernier scale. The graduations on the vernier scale are slightly smaller than the graduations on the main scale.

Least Count (Vernier Constant)

The Least Count (L.C.) of an instrument is the smallest measurement that can be taken accurately with it.

L.C. = Value of 1 Main Scale Division (x) - Value of 1 Vernier Scale Division (y)

Alternatively: L.C. = Value of 1 Main Scale Division / Total number of divisions on Vernier Scale

For a standard Vernier Callipers: L.C. = 1 mm / 10 = 0.1 mm = 0.01 cm.

Important Zero Error in Vernier Callipers

If the zero mark of the vernier scale does not perfectly align with the zero mark of the main scale when the jaws are closed, the instrument has a Zero Error.

Positive Zero Error: The vernier zero is to the right of the main scale zero. (Correction is always negative).
Negative Zero Error: The vernier zero is to the left of the main scale zero. (Correction is always positive).

True Reading = Observed Reading - (Zero Error with sign)

4. Measuring Instruments: Micrometer Screw Gauge

A screw gauge is used to measure even smaller lengths, like the thickness of a wire or a glass plate, accurately up to 0.001 cm (or 0.01 mm).

Principle and Pitch

It works on the principle of a screw in a nut. The linear distance moved by the screw when it is given one complete rotation is called its Pitch.

Pitch = Distance moved by the screw / Number of complete rotations (Standard pitch = 1 mm)

Least Count of Screw Gauge

L.C. = Pitch / Total number of divisions on the circular scale

For a standard screw gauge (Pitch=1mm, Circular divisions=100): L.C. = 1 mm / 100 = 0.01 mm = 0.001 cm.

5. Measurement of Mass and Time

Mass

The S.I. unit of mass is the kilogram (kg). For sub-atomic particles (protons, neutrons, electrons), we use the atomic mass unit (a.m.u.) or unified mass (u).

1 a.m.u. = 1/12th the mass of a Carbon-12 atom = 1.66 × 10^-27 kg.

Time

The S.I. unit of time is the second (s). Other units include:

Solar Day: The time taken by the Earth to complete one rotation on its axis with respect to the sun.
Tropical Year: The year in which a total solar eclipse occurs.
Leap Year: A year containing 366 days (the month of February has 29 days).

6. Simple Pendulum

A simple pendulum consists of a heavy point mass (called a bob) suspended by a weightless, inextensible, and perfectly flexible string from a rigid support.

Concept Important Terms:

Oscillation: One complete to-and-fro motion of the bob.
Time Period (T): The time taken to complete one oscillation.
Frequency (f): The number of oscillations made in one second. (f = 1/T)
Amplitude (a): The maximum displacement of the bob from its mean position on either side.
Effective Length (l): The distance from the point of suspension to the center of gravity of the bob.

Factors Affecting Time Period of a Simple Pendulum

The time period $T$ of a simple pendulum is given by the formula:

T = 2π √(l / g)

$T$ is directly proportional to the square root of its effective length ($\sqrt{l}$).
$T$ is inversely proportional to the square root of acceleration due to gravity ($\sqrt{g}$).
$T$ does NOT depend on the mass or material of the bob.
$T$ does NOT depend on the amplitude (provided the swing is small).

Fact Second's Pendulum: A pendulum whose time period is exactly 2 seconds is called a Second's pendulum. The effective length of a second's pendulum on the surface of the earth is approximately 1 metre (100 cm).

7. Practice Questions & Assignments

Test your understanding with these exam-oriented questions. Click on the questions to reveal the step-by-step answers.

Q1. A vernier calliper has a main scale marked in mm. Its vernier scale has 20 divisions which coincide with 19 divisions of the main scale. Calculate its Least Count.

Value of 1 Main Scale Division (MSD) = 1 mm

Total divisions on vernier scale (n) = 20

Least Count (L.C.) = 1 MSD / n
L.C. = 1 mm / 20 = 0.05 mm = 0.005 cm.

Q2. Define a Light Year. Is it a unit of time or distance?

A Light Year is the distance travelled by light in a vacuum in one year. Despite the word "year", it is a unit of distance (not time) used in astronomy.

1 Light Year = 9.46 × 10¹⁵ metres.

Q3. Compare the time periods of two pendulums of lengths 1m and 4m.

We know that Time Period $T$ is directly proportional to the square root of the effective length ($\sqrt{l}$).

Therefore, T₁ / T₂ = √(l₁ / l₂)

T₁ / T₂ = √(1 / 4) = 1 / 2

The ratio of their time periods is 1:2.

Q4. A screw gauge has a pitch of 1 mm and 50 divisions on its circular scale. Calculate its least count.

Pitch = 1 mm

Total circular divisions = 50

Least Count = Pitch / Total circular divisions
L.C. = 1 mm / 50 = 0.02 mm = 0.002 cm.

Q5. Why is the bob of a simple pendulum usually made of a heavy, dense material like brass or iron?

A bob made of a heavy, dense material ensures that its center of gravity remains localized at a point. Secondly, air resistance on a heavy bob is negligible compared to a lighter bob of the same volume, allowing it to oscillate for a longer time.