Units, Measurement & Dimensions PYQs – Physics, SSC & Railway

Answer: Option 1 — Newton
Force = mass × acceleration, so its SI unit is the newton (N), the force that gives a mass of 1 kg an acceleration of 1 m/s².
In base units 1 N = 1 kg·m/s².
Revision note — SI units named after scientists
• Newton (N) — force; 1 N = 1 kg·m/s²; the CGS unit is the dyne, and 1 N = 10⁵ dyne.
• Joule (J) — work and energy; 1 J = 1 N·m, the work done when a force of 1 N moves a body 1 m; the CGS unit is the erg (1 J = 10⁷ erg).
• Watt (W) — power; 1 W = 1 J/s; 1 horsepower = 746 W.
• Pascal (Pa) — pressure; 1 Pa = 1 N/m²; 1 atmosphere = 1.013 × 10⁵ Pa = 760 mm of Hg.
• Kilogram-force (kgf) is a gravitational unit: 1 kgf = 9.8 N, the weight of a 1 kg mass on Earth.
• Other named derived units: hertz (frequency), coulomb (charge), volt (potential), ohm (resistance), tesla (magnetic field), weber (magnetic flux).
Do not mix these up
• Newton vs newton-metre — the newton measures force; the newton-metre is the joule, a unit of work (and of torque).
• Newton vs kilogram — the kilogram is a unit of mass; the newton is a unit of force or weight (weight = mass × g).
Also asked from this topic: the combination of base units equivalent to the newton (kg·m/s²); the SI unit of pressure (pascal, N/m²); the SI unit of work (joule).

Correct Answer:
Correct Answer: Option 1 – pascal
Explanation:
Pascal (Pa = N/m²) is the SI unit of pressure, not work. Joule, watt-second, and newton-metre are all valid, equivalent units of work/energy (1 J = 1 N·m = 1 W·s).

Correct Answer:
Correct Answer: Option 4 – It makes the constant k in the formula "F = k ma" equal to 1
Explanation:
From Newton's second law, F ∝ ma, so F = k·ma. Thenewtonis defined as the force that produces an acceleration of 1 m/s² in a mass of 1 kg, which makes the proportionality constant k equal to1, simplifying the equation to F = ma.
Important Facts:
• 1 Newton = 1 kg·m/s².
• This choice of unit avoids the need for a numerical constant in Newton's second law equation.

Correct Answer:
Correct Answer: Option 2 – Kilogram meter per second squared
Explanation:
By Newton's second law, Force = mass × acceleration. In SI units, mass is in kilograms (kg) and acceleration is in meters per second squared (m/s²). Therefore, 1 newton = 1kg·m/s².
Important Facts:
• 1 N = 1 kg·m·s⁻², a derived SI unit.
• This is a standard dimensional relationship taught in mechanics.

Answer: Option 4 – Joule (J).
• Bothworkandkinetic energyare forms of energy and are measured injoules (J).

Answer: Option 3 — Joule (J)
Energy (including PE) is measured injoules→ Option 3.

Answer: Option 1 — meter per second
The SI unit of velocity ismetre per second (m/s)→ Option 1.

Density = Mass/Volume
Dimensional Formula: [M L⁻³]
SI base unit of Mass = kilogram (kg)
SI base unit of Length = metre (m)
SI base unit of Volume = m³
Density in SI base units = kg/m³
Note: Litre, gram, milligram are NOT SI base units
Answer: Option 2 → Kilogram per metre cubed

Topic:Work and Energy - SI Unit and Dimensions
Work/Energy Definition:Work = Force × Displacement
SI Unit:Joule (J) = Newton × meter = N·m
Dimensional Formula:
Force = M L T⁻² and Displacement = L
Work = [M L T⁻²] × [L] =M L² T⁻²
Other Options:
Pascal → Pressure unit (M L⁻¹ T⁻²)
Watt → Power unit (M L² T⁻³)
Newton → Force unit (M L T⁻²)
Answer: Option 4 → joule and M L² T⁻²

Fundamental vs Derived Quantities
Fundamental Quantities (7 in SI):
Length, Mass, Time, Electric Current, Temperature, Amount of Substance, Luminous Intensity
Derived Quantities:
Formed by combination of fundamental quantities
Analysis:
Time → Fundamental quantity ✓
Electric Current → Fundamental quantity ✓
Luminous Intensity → Fundamental quantity ✓
Velocity → Speed = Distance/Time = Derived quantity ✗
Answer: Option 4 → Velocity

Force is defined by Newton's Second Law:
Force = Mass × Acceleration
Dimension of Mass = [M]
Dimension of Acceleration = [LT⁻²]
Therefore, Dimension of Force:
[F] = [M] × [LT⁻²] = [MLT⁻²]
This represents: mass × length × time⁻²
Answer: Option 4 → mass × length × time⁻²

Unit Analysis:
m/s² = meters per second squared
This representsrate of change of velocity
Velocity = m/s
Acceleration = change in velocity/time = (m/s)/s = m/s²
Force → kg·m/s² (Newton)
Energy → kg·m²/s² (Joule)
Answer: Option 3 → acceleration

Topic: SI Units - Temperature
SI System (International System of Units)
Base unit for temperature =Kelvin (K)
Other units:
• Joule → Energy unit
• Fahrenheit → Non-SI temperature scale
• Celsius → Derived unit, not base SI unit
Answer: Option 2 → Kelvin

Topic:SI Units - Mass Conversion
1 tonne = 1 metric ton
Conversion:
1 tonne = 1000 kg = 10³ kg
This is the standard SI definition.
Answer: Option 1 → 10³ kg

Topic:Dimensional Formula of Energy
Energy (Work) = Force × Distance
Force = Mass × Acceleration = M × LT⁻²
Dimensional formula of Force =[M L T⁻²]
Energy = [M L T⁻²] × [L] =[M L² T⁻²]
This represents kinetic energy (½mv²), potential energy (mgh),
and all forms of energy in SI system.
Answer: Option 3 → [M L² T⁻²]

SI Prefix Multiplicative Factors:
Nano (n)= 10⁻⁹
Centi (c)= 10⁻²
Milli (m)= 10⁻³
Micro (μ)= 10⁻⁶
The prefix with factor 10⁻⁹ isNano.
Answer: Option 1 → Nano

Pressure Definition:
Pressure = Force / Area
SI Units:
Force = Newton (N)
Area = square meter (m²)
Pressure = N/m² (Pascal)
Answer: Option 4 → N/m²

Dimensional Analysis of Electric Charge
Electric charge Q is defined by the relation:
Q = I × t
Where:
I = Electric current (Ampere, A) - SI base unit
t = Time (Second, s) - SI base unit
Dimension:[Q] = [A][T]
Unit: 1 Coulomb = 1 Ampere × 1 Second
Answer: Option 1 → current and time

Correct Answer: 7.6 × 10⁶ s — Option (a)
Calculation: 88 Earth days × 24 hours/day × 3600 seconds/hour
= 88 × 24 × 3600
= 88 × 86,400
= 7,603,200 seconds
≈ 7.6 × 10⁶ seconds
This corresponds to Mercury, the closest planet to the Sun, which has an orbital period of approximately 88 Earth days (87.97 days to be precise). The planet's short orbital period reflects its proximity to the Sun and high orbital speed (~47.87 km/s).
Option (b) 6.8 × 10⁶ s — Incorrect: 6.8 × 10⁶ s ≈ 78.7 days, which is less than 88 days.
Option (c) 7.1 × 10⁶ s — Incorrect: 7.1 × 10⁶ s ≈ 82.2 days, not 88 days.
Option (d) 7.9 × 10⁶ s — Incorrect: 7.9 × 10⁶ s ≈ 91.4 days, which exceeds 88 days.

Given: Force (F) unit = Newton (N)
Formula:F = m × a
Where m = mass (kg), a = acceleration (m/s²)
Unit of Force = kg × m/s²
1 Newton = 1 kg⋅m/s²
Answer: Option 2 → kg⋅m/s²

Topic: Units of Measurement
Quintal is a unit used to measureweight or mass.
1 quintal = 100 kilograms
Commonly used in agriculture and trade for measuring crops, grains, and other heavy commodities.
Answer: Option 2 → Weight

SI Unit of Thermodynamic Temperature
The International System of Units (SI) definesKelvin (K)as the base unit for thermodynamic temperature.
Other temperature scales:
• Celsius (°C) - derived unit, not SI base unit
• Fahrenheit (°F) - imperial system
• Rankine (°R) - absolute scale in imperial system
Kelvin scale starts from absolute zero (0 K = -273.15°C).
Answer: Option 2 → Kelvin

Given: 0.00200
Rule for Significant Figures:
• Leading zeros are NOT significant
• Trailing zeros after decimal ARE significant
In 0.00200:
• 0.00 (leading zeros) → NOT significant
• 2, 0, 0 (after first non-zero digit) → significant
Significant figures =2, 0, 0 = 3 figures
Answer: Option 1 → 3

(The final speed of the train is given as 40 km)/(h.)
(To convert km)/(h to m/s, multiply by (1000 m / 1 km) and divide by (3600 s / 1 hour).)
(Speed in m)/(s) = 40 × (1000/3600) = 40 × (5/18) = (200)/(18) = (11.11 m)/(s.)

The SI unit of force is derived from Newton's Second Law of Motion: F = m × a.
Here, m is mass in kilograms (kg) and a is acceleration in metres per second squared (m/s²).
Therefore, the unit of force is kg·m/s², called a Newton (N).
Option 4 'kg m/s²' correctly represents the SI unit of force.

The SI (International System of Units) unit of work is the joule (J). Work is defined as the product of force and displacement in the direction of force: W = F × d. One joule equals the work done when a force of one newton moves an object one metre in the direction of force.
Why other options are incorrect:
- Weber (Wb): The SI unit of magnetic flux, not work.
- Tesla (T): The SI unit of magnetic field strength (magnetic flux density), not work.
- Coulomb (C): The SI unit of electric charge, not work.

Correct Answer: 3 (1000)
The prefix 'kilo' (k) means 1000. Therefore, 1 kilojoule (kJ) = 1000 joules (J). This is a standard SI unit conversion: 1 kJ = 10^3 J = 1000 J.
Option 1 (10) is incorrect - that would be the conversion for 'deca' prefix.
Option 2 (100) is incorrect - that would be the conversion for 'hecto' prefix.
Option 4 (10000) is incorrect.

The correct answer is 10⁷ ergs (Option 1).
1 joule (SI unit of energy) is equivalent to 10⁷ ergs (CGS unit of energy). This conversion arises because: 1 J = 1 kg·m²/s² = 1000 g × (100 cm)² / s² = 10⁷ g·cm²/s² = 10⁷ ergs.
Option 2 - 10⁸ ergs: This is 10 times larger than the correct value; 1 joule = 10⁷ ergs, not 10⁸.
Option 3 - 10⁹ ergs: This is 100 times larger than the correct value.
Option 4 - 10⁶ ergs: This is 10 times smaller than the correct value.

The correct answer is g·cm²/s² (Option 2).
The erg is the unit of energy in the CGS (centimeter-gram-second) system. One erg equals 1 g·cm²/s² and is equivalent to 10⁻⁷ joules. It is named from the Greek word for 'work.'
Option 1 - kg·m²/s²: This is the joule, the unit of energy in the SI (MKS) system.
Option 3 - kg·cm²/s²: This mixes SI (kg) and CGS (cm) units and is not a standard unit of energy.
Option 4 - g·m²/s²: This also mixes CGS (g) and SI (m) units and is not a standard unit of energy.

The correct answer is Distance can be negative (Option 3) - this statement is incorrect.
Distance is a scalar quantity that measures the total path length traveled by an object. It is always non-negative (zero or positive) because it only has magnitude and no direction. Therefore, the statement 'Distance can be negative' is incorrect.
Option 1 - Speed is a scalar quantity: This is correct. Speed only has magnitude and no direction, making it a scalar.
Option 2 - Displacement is a vector quantity: This is correct. Displacement has both magnitude and direction (shortest path from start to end point).
Option 4 - Force is a vector quantity: This is correct. Force has both magnitude and direction, making it a vector quantity.

1 Newton = 10^5 dynes. This follows from the definitions: 1 Newton (SI unit) = 1 kg·m /s², and 1 dyne (CGS unit) = 1 g·cm /s². Converting: 1 kg = 1000 g = 10^3 g, and 1 m = 100 cm = 10^2 cm. Therefore, 1 N = 10^3 g × 10^2 cm/s² = 10^5 dyne. Option 2 is correct.
Option 1 is wrong: 10^4 dyne is not equal to 1 Newton; this would correspond to 0.1 N, which is 100 times smaller than the correct value of 10^5 dyne.
Option 3 is wrong: 10^6 dyne would be equal to 10 Newtons, not 1 Newton; it is 10 times more than the correct conversion factor.
Option 4 is wrong: 10^7 dyne would equal 100 Newtons, not 1 Newton; this is 100 times larger than the correct conversion of 10^5 dyne per Newton.

Correct Answer: Option 3 – Trigonometric and logarithmic functions are valid only when applied to quantities that are dimensionless
In dimensional analysis, mathematical functions like trigonometric (sin, cos, tan), logarithmic (log, ln), exponential ( e^x ), and other transcendental functions are only mathematically valid when their arguments are dimensionless (pure numbers). This is because these functions are defined by infinite series (e.g., sin x = x − x³/3! + …) — adding terms with different dimensions would be meaningless. Therefore, quantities inside sin, log, e^x must always be dimensionless.
Option 1 (Dimensionally consistent = always physically correct): FALSE. A dimensionally correct equation may still be physically wrong. For example, it could have wrong numerical constants or missing terms. Dimensional consistency is necessary but not sufficient for physical correctness.
Option 2 (Verifies magnitude and physical validity): FALSE. Dimensional analysis cannot determine numerical constants (like 1/2 in KE = ½mv²) or verify physical validity — it only checks dimensional consistency.
Option 4 (Refractive index has dimensions): FALSE. Refractive index (n = c/v) is a ratio of two speeds — it is dimensionless. All pure ratios of like quantities are dimensionless.

Correct Answer: Option 1 — m s⁻²
📖 Given
• The physical quantity is acceleration (a).
• Acceleration is defined as the rate of change of velocity.
✍️ Formula
a = change in velocity / time = (v − u) / t
🔢 Solution
• Unit of velocity = m/s = m s⁻¹
• Unit of time = s
• Unit of acceleration = (m s⁻¹) / s = m s⁻²
So acceleration is measured in metre per second squared (m s⁻²) — it tells how much the velocity changes in every one second. Acceleration is a vector quantity; if it is negative it is called retardation or deceleration.
❌ Why the other options are incorrect
✗ m s² — this comes from multiplying by time instead of dividing by it; it is not the unit of any common quantity.
✗ m s⁻¹ — this is the SI unit of velocity or speed, one step before acceleration.
✗ m s — metre × second; it is not the unit of acceleration at all.
📌 SSC & Railway Exam Facts
• Acceleration due to gravity, g = 9.8 m s⁻², also written as 9.8 m/s².
• Equations of motion: v = u + at, s = ut + ½at², v² = u² + 2as.
• SI unit of force = newton (N) and 1 N = 1 kg m s⁻².
• In uniform circular motion the speed is constant but the body is still accelerated, because the direction keeps changing.
• The slope of a velocity–time graph gives the acceleration; its area gives the displacement.

Correct Answer: Option 4 — kg m s⁻¹
📖 Given
• Mass of the body, m, measured in kilogram (kg)
• Velocity of the body, v, measured in metre per second (m s⁻¹)
✍️ Formula
Momentum, p = m × v
🔢 Solution
• Unit of p = (unit of mass) × (unit of velocity)
• Unit of p = kg × m s⁻¹
• Unit of p = kg m s⁻¹ (also written as kg m/s)
Momentum measures the quantity of motion in a body. A heavy truck moving slowly and a light bullet moving fast can both have a large momentum, which is why both are hard to stop.
❌ Why the other options are incorrect
✗ kg m — the second is missing altogether; mass × distance is not momentum.
✗ kg m⁻¹ s⁻¹ — here metre carries a negative power; momentum has metre in the numerator, since velocity is m/s.
✗ kg m s — this multiplies by time instead of dividing by it, which is the exact opposite of m/s.
📌 SSC & Railway Exam Facts
• p = mv; momentum is a vector quantity and points along the velocity.
• Newton's second law: F = ma = dp/dt — force is the rate of change of momentum.
• Impulse = F × t = change in momentum, measured in N s, and 1 N s = 1 kg m s⁻¹.
• Law of conservation of momentum: in the absence of an external force, total momentum stays constant.
• Recoil of a gun, rocket propulsion and collisions are all explained by conservation of momentum.

Correct Answer: Option 4 — Newton
The SI unit of force is the newton (N), named after Sir Isaac Newton. It is defined from Newton's second law, F = ma: one newton is the force that gives a body of mass 1 kilogram an acceleration of 1 metre per second squared. In base units, 1 N = 1 kg m s⁻².
Force is a vector quantity — it has both magnitude and direction — and its effect is to change a body's state of rest or of uniform motion, or to change its shape. The weight of a 1 kg body on the earth is W = mg = 1 × 9.8 = 9.8 N, which is why a newton is roughly the weight of a 100 g object.
❌ Why the other options are incorrect
✗ rad/s — the unit of angular velocity (ω), telling how fast an object turns, not how hard it is pushed.
✗ Dyne — this is a unit of force, but it belongs to the CGS system, not SI: 1 dyne = 1 g cm s⁻² = 10⁻⁵ N.
✗ m/s — the unit of speed or velocity; it measures how fast a body moves, not the force acting on it.
📌 SSC & Railway Exam Facts
• F = ma; 1 N = 1 kg m s⁻² = 10⁵ dyne.
• 1 kgf (kilogram-force) = 9.8 N; weight W = mg.
• Other derived SI units: work/energy → joule (J), power → watt (W), pressure → pascal (Pa).
• Momentum p = mv in kg m/s; impulse = F × t in N s.
• The seven SI base units are metre, kilogram, second, ampere, kelvin, mole and candela.

Correct Answer: Option 2 — Pascal
Pressure is defined as the force acting perpendicularly per unit area:
P = F/A
Its SI unit is the pascal (Pa), named after the French scientist Blaise Pascal. By definition 1 pascal = 1 newton per square metre (1 N/m²), i.e. the pressure produced when a force of 1 N acts on an area of 1 m².
Because the pascal is a very small unit, larger units such as the bar (1 bar = 10⁵ Pa) and the atmosphere (1 atm = 1.01 × 10⁵ Pa) are used in practice. Gas pressure inside a cylinder, atmospheric pressure and blood pressure are all pressures, though blood pressure is still quoted in mm of Hg.
❌ Why the other options are incorrect
✗ Kelvin — the SI unit of temperature; K = °C + 273, with 0 K as absolute zero.
✗ Cubic metre — the SI unit of volume (m³); 1 m³ = 1000 litres.
✗ Newton — the SI unit of force, not pressure; pressure is force divided by area, so its unit is N/m².
📌 SSC & Railway Exam Facts
• 1 pascal = 1 N/m²; the unit is named after Blaise Pascal.
• 1 atm = 1.01 × 10⁵ Pa = 760 mm of Hg = 1.013 bar.
• SI units: length → metre, mass → kilogram, time → second, temperature → kelvin, current → ampere, amount → mole, luminous intensity → candela.
• Atmospheric pressure is measured by a barometer; gas pressure by a manometer.
• Pascal's law: pressure applied to an enclosed fluid is transmitted equally in all directions — the basis of the hydraulic lift and brakes.

Correct Answer: Option 3 — Pascal
Pressure is the force acting normally per unit area, P = F/A. Substituting the SI units of force (newton, N) and of area (square metre, m²) gives the SI unit of pressure as N/m², which is given the special name pascal (Pa) after Blaise Pascal.
So 1 pascal = 1 N/m² — the pressure exerted when a force of 1 newton is spread over an area of 1 square metre. Since the pascal is very small, the bar (10⁵ Pa) and the atmosphere (1.01 × 10⁵ Pa) are used for larger pressures.
This is also why a sharp knife cuts better (small area → large pressure) and why a camel's broad feet do not sink in sand (large area → small pressure).
❌ Why the other options are incorrect
✗ Newton — the unit of force, not pressure; pressure needs force divided by area.
✗ Newton/meter — this is N/m, the unit of surface tension or of a spring constant; pressure requires N/m².
✗ Angstrom — a unit of length, 1 Å = 10⁻¹⁰ m, used for atomic and molecular sizes.
📌 SSC & Railway Exam Facts
• Pressure = Force / Area; SI unit pascal (Pa) = N/m².
• 1 bar = 10⁵ Pa; 1 atm = 1.01 × 10⁵ Pa = 760 mm of Hg.
• 1 Å = 10⁻¹⁰ m; 1 nm = 10⁻⁹ m.
• N/m is the unit of surface tension; N·m (joule) is the unit of work/energy.
• Pascal's law explains the hydraulic press, hydraulic lift and hydraulic brakes.