What is the SI unit of mass? How is it defined?

Exercises · 1.14

Q. What is the SI unit of mass? How is it defined?

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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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The SI unit of mass is the kilogram ( $\text{kg}$ ), which is now defined by fixing the numerical value of the Planck constant ( $h$ ).

The International System of Units (SI) provides a globally accepted framework for measurement. It's crucial for scientific and engineering consistency, ensuring that a measurement taken in one part of the world is understood and reproducible elsewhere. At its core, the SI system defines seven base units, from which all other units are derived. Mass is one of these fundamental quantities, and its base unit, the kilogram, has a particularly interesting history and a very modern definition.

Understanding why the definition of the kilogram changed helps appreciate the precision and universality sought in modern metrology. Historically, many units were defined by physical artifacts, but these have inherent limitations.

Here's how the SI unit of mass is defined:

Identifying the SI Unit of Mass: The SI unit of mass is the kilogram, denoted by the symbol $\text{kg}$ . It is one of the seven base units of the SI system.
Historical Definition (and its limitations): For a long time, the kilogram was unique among the SI base units because its definition relied on a physical artifact. From 1889 until 2019, one kilogram was defined as the mass of the International Prototype of the Kilogram (IPK), a cylinder made of a platinum-iridium alloy, stored in a vault at the International Bureau of Weights and Measures (BIPM) in Sèvres, France.
Watch out
Relying on a physical artifact like the IPK had significant drawbacks:
- Instability: The mass of the IPK could subtly change over time due to surface contamination or cleaning, making the definition inherently unstable.
- Accessibility: It was difficult to make precise comparisons with the IPK, and its physical nature meant it couldn't be universally accessed or perfectly replicated.
- Lack of Universality: Its definition was tied to a specific object, not a fundamental constant of nature.
Modern Definition (since 2019): To overcome these limitations, the kilogram was redefined on May 20, 2019, based on a fundamental physical constant. The kilogram is now defined by fixing the numerical value of the Planck constant ( $h$ ).

Important

The Planck constant, $h$ , is defined to be exactly $6.62607015 \times 10^{-34} \text{ J}\cdot\text{s}$ (joule-seconds).

Since a joule ( $\text{J}$ ) is defined as $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}$ , the Planck constant can also be expressed as $6.62607015 \times 10^{-34} \text{ kg}\cdot\text{m}^2\cdot\text{s}^{-1}$ .

This means that the kilogram is now defined in terms of $h$ , along with the definitions of the metre ( $\text{m}$ ) and the second ( $\text{s}$ ), which are themselves defined by other fundamental constants:
- The speed of light in vacuum ( $c$ ) defines the metre. …

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SI Unit of Mass: The Kilogram

1. Concept First — The Idea Being Tested

This question tests your understanding of SI base units — the seven fundamental units from which all other units are derived. Mass is one of the most intuitive physical quantities: it measures how much matter an object contains. But defining it precisely for science and trade is surprisingly tricky.

Why does this matter?

Without a standard unit, measurements from different labs or countries wouldn't match.
Mass affects everything from grocery scales to satellite orbits.
The definition has evolved as measurement technology improved — from a physical object to a fundamental constant.

The key idea: The SI unit of mass is the kilogram (kg), and its modern definition is based on a fixed value of Planck's constant, not a physical object.

2. Step-by-Step — Understanding the Definition

Step 1: What is the SI unit of mass?

The SI unit of mass is the kilogram, symbol kg.

It is one of the seven SI base units.
Note: The kilogram is the only base unit with a prefix (kilo-) built into its name. This is a historical quirk — the gram was originally the base, but the kilogram became the practical standard.

Step 2: How was it defined historically?

From 1889 to 2019, the kilogram was defined by a physical artifact:

The International Prototype of the Kilogram (IPK), a cylinder of platinum-iridium alloy kept in a vault near Paris.
All mass measurements worldwide were traceable to this single object.

Problem: The IPK's mass changed slightly over time (due to surface contamination, wear, etc.), causing inconsistencies. This motivated a shift to a definition based on an invariant constant of nature.

Step 3: The modern definition (since 20 May 2019)

The kilogram is now defined by fixing the numerical value of Planck's constant $h$ :

The kilogram is defined by taking the fixed numerical value of Planck's constant $h$ to be $6.62607015 \times 10^{-34}$ when expressed in the unit $\text{J·s}$ , which is equal to $\text{kg·m}^2\text{·s}^{-1}$ .

What does this mean in plain language?

Planck's constant $h$ is a fundamental constant from quantum mechanics.
By fixing $h$ exactly, we define the kilogram in terms of the second (time) and the metre (length), which are themselves defined by atomic clocks and the speed of light.
This makes the kilogram universal and unchanging — no physical object to degrade.

Step 4: How is this definition realized in practice?

The definition is realized using a Kibble balance (formerly called a watt balance):

A mass is placed on one side of a balance.
An electric current through a coil in a magnetic field produces a force to balance the weight.
By measuring the current, voltage, and velocity of the coil, the mass can be calculated from $h$ , the metre, and the second. …

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Common Mistakes: SI Unit of Mass — and How to Avoid Them

The Correct Answer First

SI unit of mass: kilogram (kg)
Definition: The kilogram is defined by taking the fixed numerical value of the Planck constant $h$ to be $6.62607015 \times 10^{-34}$ when expressed in the unit $\text{J·s}$ , which is equal to $\text{kg·m}^2\text{·s}^{-1}$ , where the metre and the second are defined in terms of $c$ and $\Delta \nu_{\text{Cs}}$ .

(This is the current definition, revised in 2019. The old definition — the mass of the international prototype kilogram — is no longer valid.)

Mistake #1: Writing "gram" instead of "kilogram"

What students do wrong:

They write "gram (g)" as the SI unit of mass.

Why it's wrong:

The SI base unit of mass is the kilogram, not the gram. The gram is a submultiple (1 kg = 1000 g), but it is not the base unit.

How to avoid:

Memorise: kg is the only SI base unit with a prefix (kilo-) built in.
Remember the mnemonic: "The kilogram is the king of mass units."

Mistake #2: Giving the old definition (the "Paris prototype")

What students do wrong:

They say: "One kilogram is the mass of the international prototype kilogram kept at Sèvres, France."

Why it's wrong:

This definition was replaced in 2019. The prototype is no longer the standard — it was retired because its mass changed slightly over time.

How to avoid:

Know the 2019 revision — the kilogram is now defined using the Planck constant $h$ .
Use the phrase: "Defined in terms of fundamental constants, not a physical object."

Mistake #3: Confusing mass with weight

What students do wrong:

They say: "The SI unit of mass is the newton (N)" or "Mass is measured in newtons."

Why it's wrong:

The newton is the SI unit of force (weight), not mass. Mass is a scalar quantity; weight is a force that depends on gravity.

How to avoid:

Remember: kg for mass, N for weight.
Think: "Mass is what you are; weight is what the scale shows under gravity."

Mistake #4: Omitting the Planck constant in the definition

What students do wrong:

They give a vague definition like: "It is defined using the Planck constant." — without the exact value or the equation.

Why it's wrong:

Exams expect precision. The definition must include:

The fixed value: $h = 6.62607015 \times 10^{-34} \ \text{J·s}$
The relationship: $\text{kg} = \frac{h}{6.62607015 \times 10^{-34} \ \text{m}^2\text{s}^{-1}}$

How to avoid:

Write the definition in two parts:
1. "The kilogram is defined by fixing $h = 6.62607015 \times 10^{-34} \ \text{J·s}$ ."
2. "This implies $1 \ \text{kg} = \frac{h}{6.62607015 \times 10^{-34}} \ \text{m}^{-2}\text{s}$ ."
Practise writing it exactly as in the official SI brochure.

Mistake #5: Forgetting the role of the metre and second

What students do wrong:

They define the kilogram in isolation, without mentioning that the definition depends on the definitions of the metre and second.

Why it's wrong: …

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Q.What is the SI unit of mass? How is it defined?

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