Measurement of Length, Mass and Time

This topic supplies the "big and small" numbers boards love in 1-mark fill-in and MCQ items — parallax for stars, light year, parsec, atomic mass unit. It also grounds the order-of-magnitude estimation skill tested in reasoning questions.

Why it matters

No single instrument spans all scales (10⁻¹⁵ m nucleus to 10²⁶ m observable universe). Physicists use direct methods (metre scale, vernier, screw gauge) for ordinary sizes and indirect methods (parallax, echo/RADAR, spectroscopy) for the very large and very small.

Measuring large distances — parallax

View a distant object against a far background from two points a basis b apart; the object shifts by angle θ (the parallax). Then distance D = b / θ (θ in radians). Used to find the distance of the Moon and nearby stars.

Key units and standards

Quantity	Special unit	Value in SI
Length (astronomy)	1 light year	9.46 × 10¹⁵ m
Length (astronomy)	1 parsec	3.08 × 10¹⁶ m
Length (astronomy)	1 AU	1.496 × 10¹¹ m
Length (atomic)	1 angstrom (Å)	10⁻¹⁰ m
Length (nuclear)	1 fermi	10⁻¹⁵ m
Mass (atomic)	1 u (amu)	1.66 × 10⁻²⁷ kg

Time is now kept by the caesium atomic clock: 1 second = 9,192,631,770 vibrations of the caesium-133 atom. Ordinary masses → common balance; charged particles → mass spectrograph.

Exam Tricks & Tips

• 🎯 Parsec > light year > AU in size — remember 1 parsec ≈ 3.26 light years.
• 🎯 In D = b/θ, θ must be in radians; convert from arc-second/degree first (1° = π/180 rad).
• 🎯 Light year and parsec are units of distance, not time — a favourite trap MCQ.
• 🎯 Angstrom (10⁻¹⁰ m) for atoms, fermi (10⁻¹⁵ m) for nuclei — don't swap them.
• 🎯 For nearby-star parallax the basis is the diameter of Earth's orbit (2 AU), taken 6 months apart.
• ❌ Common mistake: using θ in degrees in the parallax formula — gives an answer ~57× too small.

Expected exam pattern

1-mark: define parsec/light year/amu, or state the caesium-clock definition of the second. 2–3 mark numerical: parallax distance of Moon/star, or convert between Å, nm and m.

Quick recap

Direct vs indirect measurement across scales. Parallax: D = b/θ (θ in rad). Know light year, parsec, AU, angstrom, fermi, amu and the caesium definition of the second.

Cover the answer, recall, then check. 11 cards on measuring length, mass and time.

Q1. State the parallax formula for distance.
A1. D = b/θ, where b is the basis (separation of observation points) and θ is the parallax angle in radians.

Q2. Must the parallax angle be in degrees or radians?
A2. Radians — always convert first.

Q3. How is one second defined using the caesium clock?
A3. The time for 9,192,631,770 vibrations of the caesium-133 atom.

Q4. What is 1 light year in metres?
A4. 9.46 × 10¹⁵ m — the distance light travels in one year.

Q5. What is 1 parsec and how does it compare to a light year?
A5. 3.08 × 10¹⁶ m ≈ 3.26 light years.

Q6. What is 1 astronomical unit (AU)?
A6. The mean Earth–Sun distance, 1.496 × 10¹¹ m.

Q7. Define the angstrom and the fermi.
A7. 1 Å = 10⁻¹⁰ m (atomic sizes); 1 fermi = 10⁻¹⁵ m (nuclear sizes).

Q8. What is 1 unified atomic mass unit (u) in kg?
A8. 1 u = 1.66 × 10⁻²⁷ kg (1/12 the mass of a carbon-12 atom).

Q9. Which instrument measures the mass of charged particles?
A9. The mass spectrograph.

Q10. For measuring a nearby star's distance, what basis is used?
A10. The diameter of Earth's orbit (2 AU), observations taken 6 months apart.

Q11. Name one indirect method to measure large distances.
A11. Parallax (or RADAR/echo methods).

From the size of an atom to the distance of a galaxy, physics spans some 41 orders of magnitude — and one toolkit of clever methods measures it all.

Definition / core idea

Measurement assigns a number and unit to a quantity by comparing it with a standard. Different ranges need different techniques: direct instruments for the human scale, indirect (parallax, echo, decay) methods for the very large or very small.

Deep explanation

Beginner — direct length measurement

• Metre scale: 1 mm resolution.
• Vernier callipers: least count = 1 main-scale division − 1 vernier division, typically 0.1 mm = 0.01 cm.
• Screw gauge (micrometer): least count = pitch / number of circular divisions, typically 0.01 mm.
  Reading = main-scale reading + (coincident division × least count).

Intermediate — large distances by parallax

To measure the distance D of a far object, view it from two points a basis b apart; it shifts by angle θ (parallax). Then
D = b / θ (θ in radians).
The astronomical unit and star distances use Earth's orbit as the basis.

Advanced — very small and very large

• Size of a molecule: spread a known volume of oleic acid into a thin film; thickness t = Volume/Area gives the molecular diameter (~10⁻⁹ m).
• Distances by echo/RADAR/LASER: send a pulse, measure round-trip time t; distance = c·t/2.
• Time: modern atomic (caesium) clocks are accurate to 1 part in 10¹³ (lose 1 s in 3 lakh years). Time ranges from nuclear (10⁻²⁴ s) to the age of the universe (10¹⁷ s).

Orders of magnitude to remember

Size of nucleus ~10⁻¹⁴ m, atom ~10⁻¹⁰ m, human ~1 m, Earth's radius ~6.4×10⁶ m, distance to Sun ~1.5×10¹¹ m.

Worked example

A screw gauge has pitch 1 mm and 100 divisions on the circular scale. A wire reads main scale 2 mm and 45th circular division coincides.
Least count = 1 mm/100 = 0.01 mm.
Diameter = 2 mm + 45 × 0.01 mm = 2 + 0.45 = 2.45 mm.

Real-world application

Parallax scaled up gives astronomers stellar distances; scaled down, the same geometry lets your two eyes judge how far away a cricket ball is.

Exam tricks & shortcuts

• Least count of screw gauge = pitch ÷ circular divisions. Vernier LC = 1 MSD − 1 VSD.
• For parallax always convert θ to radians (multiply arc-seconds by 4.85×10⁻⁶).
• 1 light year = 9.46×10¹⁵ m; 1 parsec = 3.08×10¹⁶ m; 1 AU = 1.5×10¹¹ m.

Key formulas

• Parallax method (distance of a star): D = b/θ (b = baseline separation, θ = parallax angle in rad, D = distance).
• Angular size: d = D·α (d = actual size, D = distance, α = angular diameter in rad).
• Range of lengths: ~10⁻¹⁴ m (nucleus) to 10²⁶ m (universe).
• Range of masses: ~10⁻³⁰ kg (electron) to 10⁵⁵ kg (universe); 1 u = 1.66 × 10⁻²⁷ kg.
• Range of times: ~10⁻²² s to 10¹⁷ s.
• Reflection/echo (radar, sonar): distance = ½·v·t (v = wave speed, t = round-trip time).

Use the metre scale, vernier callipers (least count 0.1 mm) and screw gauge (least count 0.01 mm) for progressively smaller lengths.

Worked Example

Problem: A screw gauge has a pitch of 1 mm and 100 divisions on its circular scale. While measuring the diameter of a wire, the main scale reads 2 mm and the 47th circular-scale division coincides with the reference line. There is no zero error. Find the diameter of the wire.

Solution:
Step 1 — Find the least count (LC). The least count of a screw gauge is the pitch divided by the number of circular-scale divisions:
LC = pitch / number of divisions = 1 mm / 100 = 0.01 mm.

Step 2 — Read the main scale (MSR):
MSR = 2 mm.

Step 3 — Read the circular scale contribution:
Circular scale reading = coinciding division × LC = 47 × 0.01 mm = 0.47 mm.

Step 4 — Add the two readings (zero error is nil, so no correction):
Diameter = MSR + (circular scale × LC) = 2 mm + 0.47 mm = 2.47 mm.

Step 5 — Express in SI units if required:
2.47 mm = 2.47 × 10⁻³ m.

Answer: The diameter of the wire is 2.47 mm (2.47 × 10⁻³ m).