Newton's First Law: Inertia and Equilibrium

Understanding Net Force and Equilibrium

To understand why objects move or stay still, we must look at the sum of all forces acting on them, known as the net force ($\sum F$). Newton's First Law can be mathematically summarized by stating that if the net force is zero, the acceleration is also zero.

Static Equilibrium

When an object is at rest and remains at rest, it is in a state of static equilibrium. In this state, all the forces acting on the object—such as gravity pulling it down and a table pushing it up—perfectly cancel each other out.

Dynamic Equilibrium

A common misconception is that "equilibrium" only applies to objects that are not moving. However, an object moving at a constant speed in a straight line is also in equilibrium. This is called dynamic equilibrium. In this scenario, the forces acting on the object (like engine thrust vs. air resistance) are balanced, resulting in a net force of zero.

The mathematical condition for equilibrium is:

$$\sum F = 0$$

When this condition is met, the acceleration ($a$) must be zero:

$$\begin{aligned} a &= 0 \end{aligned}$$

Worked Example: The Floating Book

Let's apply these concepts to a practical scenario to see how we calculate equilibrium.

Problem: A book with a mass of $2.0\text{ kg}$ is resting on a flat table. What is the net force acting on the book?

Step 1: Identify the forces. There are two main vertical forces acting on the book:

1. The force of gravity ($F_g$) pulling the book downward.
2. The normal force ($F_N$) from the table pushing the book upward.

Step 2: Calculate the force of gravity. Using the formula $F_g = m \cdot g$, where $g \approx 9.8\text{ m/s}^2$:

$$\begin{aligned} F_g &= 2.0\text{ kg} \cdot 9.8\text{ m/s}^2 \\ F_g &= 19.6\text{ N (downward)} \end{aligned}$$

Step 3: Determine the net force. Since the book is at rest (not accelerating), we know from Newton's First Law that the system must be in equilibrium. Therefore, the normal force must be equal and opposite to the gravitational force:

$$\begin{aligned} F_N &= 19.6\text{ N (upward)} \end{aligned}$$

The net force is:

$$\begin{aligned} \sum F &= F_N - F_g \\ \sum F &= 19.6\text{ N} - 19.6\text{ N} \\ \sum F &= 0\text{ N} \end{aligned}$$

Because the net force is zero, the book remains in its state of rest, perfectly demonstrating Newton's First Law.

Summary Table

By mastering the relationship between mass, force, and equilibrium, you lay the groundwork for Newton's Second Law, which explores exactly what happens when that net force is not zero.