Chapter 1 · Introduction CHAPTER 2 · VECTORS →

1.2 Units

In statics, we work primarily with two unit systems: the International System of Units (SI) and the U.S. Customary (English) system. Understanding how fundamental quantities such as length, mass, and time relate to derived quantities like force is essential for setting up problems correctly.

This section reviews the common units used in statics, clarifies the difference between mass and weight, and connects Newton’s law of gravitation to the familiar acceleration due to gravity \(g\).

1.2.1 Unit Systems

In statics, we primarily use two unit systems: the International System of Units (SI) and the U.S. Customary (or English) system of units. The table below lists common quantities and their units in each system.

The units of length, time, and mass are considered fundamental units and serve as the building blocks for other units of measurement. Quantities such as force are derived units, created by combining fundamental units.

Quantity	SI Units	English Units
Length	meter (m)	foot (ft)
Mass	kilogram (kg)	slug
Time	second (s)	second (s)
Force	newton (N)	pound (lb)

CAUTION! One of the most common mistakes is treating both kg and lb as units of mass. In reality, kg is a unit of mass, while lb is a unit of force (weight). Mass represents the amount of matter in an object and remains constant regardless of location. Weight is the gravitational force acting on that mass and depends on the local value of gravity.

The unit of force is a derived unit obtained from Newton’s second law, \( F = m a \). In SI units, mass is measured in kilograms and acceleration in m/s\(^2\), so:

\[ [F] = [m][a] = \text{kg} \cdot \frac{\text{m}}{\text{s}^2} \equiv \text{N} \]

Similarly, in English units, the pound can be expressed as

\[ \text{lb} = \text{slug} \cdot \frac{\text{ft}}{\text{s}^2}. \]

1.2.2 Gravity Values

Gravity is not a measured quantity in statics problems, but the acceleration due to gravity appears frequently when converting between mass and weight. Typical values used in coursework are shown below.

SI Units	English Units
9.81 \( \dfrac{\text{m}}{\text{s}^2} \)	32.2 \( \dfrac{\text{ft}}{\text{s}^2} \)

ℹ

Gravity varies slightly: The value of gravitational acceleration depends on latitude and altitude. For engineering mechanics, the standard values \( g = 9.81~\text{m/s}^2 \) and \( g = 32.2~\text{ft/s}^2 \) are sufficiently accurate for all problems you will encounter.

1.2.3 Newton's Law of Gravitation

Example Gravity & weight

Calculating the Weight of a Person on Earth

What is the weight (force in newtons) of a person with a mass of \( 75~\text{kg} \) on Earth?

Why this matters: This example connects Newton’s law of gravitation to the familiar expression \( W = mg \) and clarifies why weight depends on location, while mass does not.

Person of mass m standing on the surface of Earth — Earth of mass \(M_\text{E}\) and radius \(r\); person of mass \(m\).

Compute the gravitational acceleration

Newton’s law of universal gravitation gives the force between two masses:

\[ F = \frac{G M_\text{E} m}{r^2} \]

For a person at Earth’s surface, use:

\( G = 6.67 \times 10^{-11}\, \dfrac{\text{m}^3}{\text{kg}\cdot\text{s}^2} \)
\( M_\text{E} \approx 5.97 \times 10^{24}~\text{kg} \)
\( r \approx 6{,}371{,}000~\text{m} \)

Grouping terms independent of the person’s mass \(m\) gives the surface acceleration:

\[ g = \frac{G M_\text{E}}{r^2} \approx 9.81~\frac{\text{m}}{\text{s}^2}. \]

In statics we usually take \( g = 9.81~\text{m/s}^2 \) as a standard value and do not recompute it.

Compute the weight as a force

The person’s weight \( W \) is the gravitational force acting on their mass:

\[ W = m g = 75~\text{kg} \cdot 9.81~\frac{\text{m}}{\text{s}^2} \approx 735.75~\text{N}. \]

To three significant figures, this is \( W \approx 736~\text{N} \).

Common mistake: Treating \(75~\text{kg}\) as a force. Kilograms are units of mass; the weight must come from \( W = mg \) with units of newtons.

Key ideas:

Mass \(m = 75~\text{kg}\) is an intrinsic property and does not change with location.
Weight \(W \approx 736~\text{N}\) is the gravitational force and depends on the local value of \(g\).
Near Earth’s surface we model weight as \( W = mg \).

▶ Video solution: coming soon.

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