Unit 1: Introduction

Units of Measurements

A unit is a standard, agreed-upon quantity by which we measure other physical quantities. A measurement consists of a numerical value and a unit (e.g., "10 metres").

There are two main systems of units:

1. SI (Système International d'Unités): The modern, internationally accepted metric system. It is a coherent system built on seven base units. It is also known as the MKS system (Metre-Kilogram-Second).
2. CGS (Centimetre-Gram-Second): An older metric system, still used in some fields of physics (like electromagnetism).

Base Units in SI and CGS

Quantity	SI Unit (MKS)	SI Symbol	CGS Unit	CGS Symbol
Length	Metre	m	Centimetre	cm
Mass	Kilogram	kg	Gram	g
Time	Second	s	Second	s

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Conversion to SI and CGS

Converting between systems involves using the fundamental relationships between their base units. This is essential for ensuring calculations are consistent.

Fundamental Conversions

• 1 metre (m) = 100 centimetres (cm)
• 1 kilogram (kg) = 1000 grams (g)
• 1 second (s) = 1 second (s)

Examples of Derived Unit Conversions

To convert a derived unit, substitute the base conversions.

1. Force (Unit: Newton (N) in SI, Dyne (dyn) in CGS)

• Definition: Force = mass × acceleration (F = ma)
• SI Unit: 1 N = 1 kg × m/s²
• CGS Unit: 1 dyn = 1 g × cm/s²
• Conversion:
  1 N = 1 kg × m/s²
  1 N = (1000 g) × (100 cm)/s²
  1 N = 100,000 g·cm/s²
  1 N = 10⁵ dyn

2. Energy/Work (Unit: Joule (J) in SI, Erg (erg) in CGS)

• Definition: Work = Force × distance (W = Fd)
• SI Unit: 1 J = 1 N × m
• CGS Unit: 1 erg = 1 dyn × cm
• Conversion:
  1 J = 1 N × m
  1 J = (10⁵ dyn) × (100 cm)
  1 J = 10,000,000 dyn·cm
  1 J = 10⁷ erg

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Familiarization with Measuring Instruments

These are common precision instruments used in a workshop to measure length.

1. Meter Scale (Metre Rule)

• Description: A simple ruler, typically 1 metre long.
• Least Count: The smallest measurement it can accurately read. For a standard metre rule, this is 1 millimetre (mm) or 0.1 centimetre (cm).
• Use: For approximate measurements of length where high precision is not needed.
• Common Error: Parallax error. This occurs if you don't read the scale from a position directly perpendicular to the mark.

2. Vernier Caliper

• Description: A more precise instrument with two scales: a fixed Main Scale (like a ruler) and a sliding Vernier Scale. It has "jaws" to grip objects.
• Principle: The Vernier scale is built such that 9 divisions on the main scale (MSD) are equal in length to 10 divisions on the Vernier scale (VSD).
• Least Count (LC): The smallest value it can measure.
  LC = 1 Main Scale Division (MSD) - 1 Vernier Scale Division (VSD)
  LC = 1 mm - 0.9 mm = 0.1 mm or 0.01 cm (This is the standard value).
• Reading:
  Total Reading = Main Scale Reading (MSR) + (Vernier Scale Coincidence (VSC) × Least Count (LC))
  1. MSR: Read the main scale mark just *before* the zero mark of the Vernier scale.
  2. VSC: Find the one mark on the sliding Vernier scale that aligns *perfectly* with any mark on the main scale.
• Zero Error: If the zero of the main scale and the zero of the Vernier scale do not coincide when the jaws are closed, there is a zero error that must be added or subtracted.

3. Screw Gauge (Micrometer)

• Description: An even more precise instrument used for small diameters and thicknesses. It has a U-frame, a main "sleeve" scale, and a rotating "thimble" scale.
• Principle: Based on the principle of a screw. The pitch is the distance the spindle moves forward in one complete rotation of the thimble (usually 0.5 mm or 1.0 mm).
• Least Count (LC):
  LC = Pitch / (Total divisions on Thimble Scale)
  Standard LC = 0.5 mm / 50 = 0.01 mm or 0.001 cm. (10 times more precise than a Vernier caliper).
• Reading:
  Total Reading = Main Scale Reading + (Thimble Scale Coincidence × Least Count)
  1. MSR: Read the last full millimetre mark visible on the sleeve (and check if the 0.5 mm mark after it is also visible).
  2. TSC: Read the division on the rotating thimble scale that aligns with the horizontal datum line of the sleeve.
• Zero Error: Also has zero errors (positive or negative) if the zero on the thimble does not align with the datum line when the jaws are closed.

4. Sextant

• Description: An instrument used to measure the angle between two visible objects.
• Principle: Based on the law of reflection. It uses a system of two mirrors (an index mirror and a horizon mirror) to bring the images of two objects (e.g., the horizon and the Sun/star) into coincidence. The angle is then read off a graduated arc.
• Use: Classically used in celestial navigation to find one's latitude. In this course, it's used for finding the height of distant objects.

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Using Instruments for Measurement

The correct instrument should be chosen for the job, based on the required precision.

• To measure length of a bench: Use a Meter Scale. Precision is low.
• To measure diameter of a cylinder (e.g., a beaker): Use a Vernier Caliper. The outer jaws are used to grip the cylinder.
• To measure diameter of a thin wire: Use a Screw Gauge. A Vernier is not precise enough and its jaws apply too much pressure, which can deform the wire.
• To measure thickness of a metal sheet: Use a Screw Gauge. This provides the necessary high precision (e.g., 0.52 mm).

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Use of Sextant

A sextant measures angles. We can use it with simple trigonometry to find the height of buildings or mountains.

To measure the height (H) of a building:

1. Stand at a known, measurable distance `D` from the base of the building. Measure `D` with a tape measure.
2. Use the sextant to measure the angle of elevation (θ) from your eye level to the top of the building.
3. Let `h` be the height of the building *above* your eye level. From trigonometry:
   tan(θ) = Opposite / Adjacent = h / D
   So, h = D × tan(θ)
4. The total height of the building `H` is `h` plus your own eye-level height (`h_eye`).

   H = (D × tan(θ)) + h_eye

To measure the height (H) of a mountain:

We cannot measure the distance `D` to the center of the mountain. So, we must take two readings.

1. Stand at point A. Use the sextant to measure the angle of elevation θ₁.
2. Walk a known distance `d` straight towards the mountain, to point B.
3. At point B, use the sextant again to measure the new, larger angle of elevation θ₂.
4. Let the distance from B to the base be `x`. The height `H` is:
   From triangle 1 (at A): tan(θ₁) = H / (d + x) → H = (d + x) tan(θ₁)
   From triangle 2 (at B): tan(θ₂) = H / x → H = x tan(θ₂)
5. We have two equations. Set them equal to find `x`:
   (d + x) tan(θ₁) = x tan(θ₂)
   d tan(θ₁) + x tan(θ₁) = x tan(θ₂)
   d tan(θ₁) = x (tan(θ₂) - tan(θ₁))
   x = d tan(θ₁) / (tan(θ₂) - tan(θ₁))
6. Now substitute this `x` back into the simpler equation:

   H = [d tan(θ₁) tan(θ₂)] / [tan(θ₂) - tan(θ₁)]