(MCQ II — one or more options correct) Which of th – Class 11 Physics

NCERT Exemplar · Q18

Q. (MCQ II — one or more options correct) Which of the following are not a unit of time?

(a) Second

(b) Parsec

(d) Light year

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Concept understanding — Unit Conversion

Unit Conversion: Why 1 Metre and 100 Centimetres Are the Same Thing

Imagine you're measuring the length of your desk. You pull out a ruler marked in centimetres and find it's 120 cm long. Your friend, using a metre stick, says it's 1.2 m. You're both right — you've just used different units to describe the same physical length.

That's the core idea: unit conversion is the process of changing how you express a quantity without changing the quantity itself.

The Intuition: Same Quantity, Different Labels

Think of a pizza. Whether you call it "one pizza" or "8 slices," the amount of pizza hasn't changed. You've just used a different unit (pizza vs. slice) to describe it.

Similarly, 1 metre and 100 centimetres are the same length — just like 1 pizza and 8 slices are the same amount. The number changes (1 becomes 100, or 1 becomes 8), but the actual thing being measured stays identical.

Note

This is the most important idea to hold onto: conversion changes the number, not the quantity. If you ever feel like the quantity has changed, you've made a mistake.

The Precise Statement

Unit conversion is the multiplication of a quantity by a conversion factor — a fraction equal to 1 — that cancels the old unit and introduces the new one.

A conversion factor looks like this:

\frac{\text{new unit}}{\text{old unit}} = 1

For length: $\frac{1 \text{ m}}{100 \text{ cm}} = 1$ and $\frac{100 \text{ cm}}{1 \text{ m}} = 1$ .

Why are these fractions equal to 1? Because 1 metre is 100 centimetres. The numerator and denominator describe the same physical length, so their ratio is exactly 1.

How to Convert: The Only Rule You Need

Multiply by a conversion factor that cancels the unit you have and leaves the unit you want.

Let's convert 120 cm to metres:

Start with what you have: $120 \text{ cm}$
Choose the conversion factor that has "cm" in the denominator (to cancel it) and "m" in the numerator: $\frac{1 \text{ m}}{100 \text{ cm}}$
Multiply:

120 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = \frac{120}{100} \text{ m} = 1.2 \text{ m}

The "cm" units cancel just like numbers do: $\frac{\text{cm}}{\text{cm}} = 1$ .

Tip

Always write the units explicitly. If the units don't cancel correctly, you've used the wrong conversion factor. This catches 90% of conversion mistakes.

The Reverse: Metres to Centimetres

Now convert 1.2 m to cm. This time, you want "cm" to remain and "m" to cancel. Use $\frac{100 \text{ cm}}{1 \text{ m}}$ :

1.2 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 1.2 \times 100 \text{ cm} = 120 \text{ cm}

Notice: when going from a larger unit (m) to a smaller unit (cm), the number gets larger (1.2 → 120). When going from smaller to larger, the number gets smaller (120 → 1.2). This is a useful sanity check.

Common Conversion Factors You'll Use

Quantity	Relationship	Conversion Factors
Length	1 m = 100 cm	$\frac{1 \text{ m}}{100 \text{ cm}}$ , $\frac{100 \text{ cm}}{1 \text{ m}}$
Mass	1 kg = 1000 g	$\frac{1 \text{ kg}}{1000 \text{ g}}$ , $\frac{1000 \text{ g}}{1 \text{ kg}}$
Time	1 h = 60 min	$\frac{1 \text{ h}}{60 \text{ min}}$ , $\frac{60 \text{ min}}{1 \text{ h}}$
Speed	1 km/h = $\frac{1000}{3600}$ m/s	$\frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}}$

Why this formula?

Dimensional Analysis: Why the Key Principles Hold

Dimensional Analysis is a powerful tool in physics and engineering that lets us check the consistency of equations, derive relationships, and convert units. But why does it work? Let's build the reasoning from the ground up.

1. The Core Idea: Physical Quantities Have Dimensions

Every physical quantity (like length, time, mass) can be expressed in terms of fundamental dimensions. The most common set in mechanics is:

L = Length
M = Mass
T = Time

For example:

Speed has dimensions $[L T^{-1}]$
Force has dimensions $[M L T^{-2}]$
Energy has dimensions $[M L^2 T^{-2}]$

Why this matters: Two quantities can only be meaningfully compared or equated if they have the same dimensions. You cannot add apples to oranges — and you cannot add length to time.

2. The Principle of Dimensional Homogeneity

The key formula that underpins everything is:

Every valid physical equation must be dimensionally homogeneous.

This means: the dimensions on the left-hand side must equal the dimensions on the right-hand side.

Why must this hold?

Consider an equation like:

v = u + at

Left side: $[v] = L T^{-1}$
Right side: $[u] = L T^{-1}$ , $[at] = (L T^{-2})(T) = L T^{-1}$

Both sides have dimensions $L T^{-1}$ . If they didn't match, the equation would be physically meaningless — you'd be comparing quantities that cannot be equal in any real experiment.

Reasoning: Physical laws describe relationships between measurable quantities. If the dimensions don't match, the equation cannot represent a real physical relationship, because the numerical value would depend on the arbitrary choice of units.

3. The Buckingham Pi Theorem: Why We Can Derive Relationships

This is the deeper mathematical reason. The Buckingham Pi Theorem states:

If a physical problem involves $n$ variables and $k$ fundamental dimensions, then it can be reduced to $n-k$ independent dimensionless groups (called $\pi$ groups).

Why does this work?

Imagine you have a relationship:

f(Q_1, Q_2, \dots, Q_n) = 0

where each $Q_i$ has dimensions. Because the equation must be dimensionally homogeneous, we can rearrange it into a function of dimensionless products only:

F(\pi_1, \pi_2, \dots, \pi_{n-k}) = 0

The reasoning: Dimensions act as constraints. Each fundamental dimension (M, L, T) gives one constraint. So if you have $n$ variables and $k$ constraints, you only have $n-k$ independent dimensionless combinations.

Example: For a simple pendulum, the period $T$ depends on length $L$ , mass $m$ , and gravity $g$ . That's 4 variables with 3 dimensions (M, L, T). So $4-3 = 1$ dimensionless group: $\pi = \frac{T^2 g}{L}$ . This tells us $T \propto \sqrt{L/g}$ without solving any differential equation.

4. Why We Can Convert Units Using Dimensional Analysis

The conversion factor formula:

\text{Value in new unit} = \text{Value in old unit} \times \left(\frac{\text{old unit}}{\text{new unit}}\right)^{\text{dimension exponent}}

Why this works: …

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This question tests your understanding of fundamental units. While "second" and "year" are units of time, "parsec" and "light year" are both units of distance, making them the correct answers.

Understanding units is fundamental in physics. Every physical quantity has a specific unit that defines its nature. For instance, length is measured in metres, mass in kilograms, and time in seconds. The key to solving this problem is to recall the definitions of each given option and determine whether it quantifies a duration (time) or a spatial extent (distance).

Let's break down each option:

Option (A) Second:

The second is the base unit of time in the International System of Units (SI). It is formally defined based on the radiation frequency of caesium-133 atoms. There is no ambiguity here; a second is unequivocally a unit of time.
Option (B) Parsec:

A parsec (pc) is a unit of length used to measure large distances to astronomical objects outside the Solar System. It is defined as the distance at which one astronomical unit (AU) subtends an angle of one arcsecond.
$1 \text{ parsec} \approx 3.086 \times 10^{16} \text{ metres} \approx 3.26 \text{ light years}$
Option (C) Year:

A year is a unit of time, commonly understood as the time it takes for the Earth to complete one orbit around the Sun. There are different definitions of a year (e.g., sidereal year, tropical year), but all represent a duration.
$1 \text{ year} \approx 365.25 \text{ days} \approx 3.156 \times 10^7 \text{ seconds}$
Option (D) Light year: …

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Here is the breakdown of the common mistakes students make on this question, along with the concept-first reasoning to avoid them.

The Core Concept: Distinguishing Time from Distance

The question tests a fundamental skill in physics: classifying physical quantities by their dimensions. Time has the dimension $[T]$ . Distance (or length) has the dimension $[L]$ .

Second (s) and Year (yr) are standard units of time.
Parsec (pc) and Light year (ly) are units of astronomical distance, not time.

A light year is the distance light travels in one year. A parsec is approximately 3.26 light years. Both measure length.

Common Mistake #1: Confusing "Year" with "Light Year"

The Mistake:

Students think that because "year" is in the name "light year," it must be a unit of time. They select Light year as a unit of time, or they fail to select Year as a unit of time.

Why it happens:

The word "year" triggers an association with time. The student doesn't stop to analyze the full phrase "light year" as a compound unit.

How to Avoid:

Always break down compound units. Ask: "What does this unit actually measure?"
- Year = time for Earth to orbit the Sun → Time.
- Light year = distance light travels in one year → Distance.
Use the definition as a filter. If the definition involves a speed (like the speed of light) multiplied by a time, the result is a distance.
Memorize the three common astronomical distance units: Light year, Parsec, and Astronomical Unit (AU). None of them are time units.

Common Mistake #2: Not Knowing What a "Parsec" Is

The Mistake:

Students have never heard of a parsec, or they confuse it with a unit of time or angle. They might leave it unselected (thinking it's a time unit) or select it incorrectly.

Why it happens:

"Parsec" sounds unfamiliar and technical. Without a clear memory of its definition, students guess.

How to Avoid:

Learn the definition: A parsec is a unit of length used in astronomy. It stands for "parallax of one arcsecond." It is approximately 3.26 light years.
Create a memory anchor: "Parsec = Parallax Second → Distance." The "second" here refers to an angle (arcsecond), not time. …

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Q.(MCQ II — one or more options correct) Which of the following are not a unit of time?

Concept understanding — Unit Conversion

Unit Conversion: Why 1 Metre and 100 Centimetres Are the Same Thing

The Intuition: Same Quantity, Different Labels

The Precise Statement

How to Convert: The Only Rule You Need

The Reverse: Metres to Centimetres

Common Conversion Factors You'll Use

Why this formula?

Dimensional Analysis: Why the Key Principles Hold

1. The Core Idea: Physical Quantities Have Dimensions

2. The Principle of Dimensional Homogeneity

Why must this hold?

3. The Buckingham Pi Theorem: Why We Can Derive Relationships

Why does this work?

4. Why We Can Convert Units Using Dimensional Analysis

Why this works: …

Concept: Units of Time vs. Units of Distance

Method: Unit Classification by Definition

Steps:

Applying the steps:

The Core Concept: Distinguishing Time from Distance

Common Mistake #1: Confusing "Year" with "Light Year"

Common Mistake #2: Not Knowing What a "Parsec" Is

Q.(MCQ II — one or more options correct) Which of the following are not a unit of time?

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