Newton's Second Law | AP Physics 1 Unit 2 Study Guide

Getting Started

Why does a kicked soccer ball speed up, or a car slow down when the brakes are applied? This chapter explores the fundamental connection between forces—the pushes and pulls that objects exert on one another—and the resulting changes in motion. We will investigate how interactions cause an object's velocity to change, providing the quantitative tools to predict the motion of systems ranging from a falling apple to a satellite in orbit.

What You Should Be Able to Do

After completing this section, you will be able to:

Describe the conditions under which an object's velocity will change.
Explain the relationship between net force, mass, and acceleration.
Calculate the acceleration of a system when the net force and mass are known.
Predict the direction of a system's acceleration based on the direction of the net force acting on it.
Differentiate between a state of balanced forces (no change in velocity) and unbalanced forces (a change in velocity).

Key Concepts & Mechanisms

This section examines motion through the lens of interactions and causation. The core idea is that forces are interactions that cause a change in a system's state of motion, a concept quantified by Newton's Second Law.

System & Preconditions

To analyze motion, we first define the system, which is the object or collection of objects we are interested in. Forces are categorized as either internal (acting between objects within the system) or external (acting on the system from an outside agent). Newton's Second Law is concerned with external forces, as these are the interactions that can change the motion of the system as a whole.

Our analysis makes several idealizations:

We often treat objects as point masses, meaning we consider all their mass to be concentrated at a single point (the center of mass), ignoring size, shape, and rotation.
We operate within an inertial frame of reference, a non-accelerating perspective from which we observe the motion. For most purposes on Earth, this is a valid assumption.

Key Steps / Relations

The process of linking forces to a change in motion follows a clear, logical sequence.

Identify All External Interactions (Forces): The first step is to identify every external push or pull acting on the system. These forces can be contact forces (like tension, normal force, or friction) or non-contact forces (like gravity). A Free-Body Diagram is the essential tool for visualizing and cataloging these forces.
Determine the Net Force: Individual forces rarely act alone. The crucial quantity is the net force (symbol: $Σ F$ or $F_{n e t}$ ), which is the vector sum of all external forces acting on the system. If forces are balanced, they cancel each other out, and the net force is zero. If they are unbalanced, there is a non-zero net force. The SI unit for force is the Newton (N).
Apply Newton's Second Law: This law provides the direct causal link between the net force and the system's response. It states that the acceleration of a system's center of mass is directly proportional to the net force and inversely proportional to the system's mass. The mathematical relationship is:
$Σ F = m a$
- $Σ F$ is the net force in Newtons (N).
- $m$ is the system's mass in kilograms (kg). Mass is a scalar measure of an object's inertia, its intrinsic resistance to a change in its state of motion.
- $a$ is the resulting acceleration in meters per second squared (m/s²). Acceleration is the rate at which the system's velocity changes.
Crucially, this is a vector equation. The direction of the acceleration vector ( $a$ ) is always the same as the direction of the net force vector ( $Σ F$ ).

Outputs & Effects

The application of a net force produces a clear and predictable effect on the system's motion.

If $Σ F \neq = 0$ (Unbalanced Forces): The system must accelerate ( $a \neq = 0$ ). This means its velocity must change. The change can be in its speed (speeding up or slowing down), its direction of motion, or both.
If $Σ F = 0$ (Balanced Forces): The system's acceleration is zero ( $a = 0$ ). This does not mean the system is at rest. It means its velocity is constant. This is the condition for equilibrium, encompassing both objects at rest (static equilibrium) and objects moving at a constant velocity (dynamic equilibrium). This is also a restatement of Newton's First Law.

Regulation & Limits

The law applies to the motion of the system's center of mass. It doesn't describe the rotation or deformation of the object.
The mass ( $m$ ) of the system is assumed to be constant. This is a valid assumption for nearly all introductory physics problems.
It is the net force that matters. A system can have many strong forces acting on it, but if they are perfectly balanced, the effect on its velocity is the same as if there were no forces at all.

Key Models & Diagrams

The process of applying Newton's Second Law can be summarized by connecting a physical situation to its representation, its mathematical model, and its predicted outcome.

Physical Situation & Representation	Mathematical Model (Equations)	Predicted Observable (Motion)
Situation: A 10 kg crate is pulled to the right with a 50 N force on a frictionless horizontal surface. Free-Body Diagram: Shows gravity down, normal force up, and the 50 N pull to the right.	Vertical (y-axis): $Σ F_{y} = F_{N} - F_{g} = 0$ . The vertical forces are balanced. Horizontal (x-axis): $Σ F_{x} = 50 N$ . The net force is 50 N to the right. Newton's 2nd Law: $Σ F_{x} = m a_{x} ⟹ 50 N = (10 kg) a_{x}$	Acceleration: $a_{x} = 5.0 m/s^{2}$ . Change in Velocity: The crate's velocity will increase by 5.0 m/s every second in the rightward direction.
Situation: A 1 kg book rests on a table. Free-Body Diagram: Shows gravity ( $F_{g} \approx 9.8$ N) down and the normal force ( $F_{N}$ ) from the table up.	Vertical (y-axis): The book is not accelerating, so $Σ F_{y} = 0$ . $Σ F_{y} = F_{N} - F_{g} = 0 ⟹ F_{N} = F_{g}$ . The forces are balanced.	Acceleration: $a = 0 m/s^{2}$ . Change in Velocity: The book's velocity remains constant (at zero). It stays at rest.

Key Components & Evidence

Force ( $F$ ): A push or a pull on an object resulting from an interaction with another object. Its role is to cause acceleration. The SI unit is the Newton (N).
Net Force ( $Σ F$ ): The vector sum of all forces acting on a system. It is the direct cause of a change in the system's velocity. The SI unit is the Newton (N).
Mass ( $m$ ): A scalar quantity representing an object's inertia, or its resistance to being accelerated. The SI unit is the kilogram (kg).
Acceleration ( $a$ ): The vector quantity representing the rate of change of velocity. It is the effect produced by a net force. The SI unit is meters per second squared (m/s²).
Velocity ( $v$ ): The vector quantity representing the rate of change of position. A change in velocity is evidence that a net force has acted on the system. The SI unit is meters per second (m/s).
Newton's Second Law ( $Σ F = m a$ ): The fundamental law of motion that quantitatively connects the cause (net force) with the effect (acceleration).
Balanced Forces: A configuration where the net force on a system is zero ( $Σ F = 0$ ). This results in zero acceleration and constant velocity.
Unbalanced Forces: A configuration where the net force on a system is not zero ( $Σ F \neq = 0$ ). This is the necessary condition for a system's velocity to change.
Free-Body Diagram: A required schematic that isolates a single object and shows all external forces acting on it as vectors originating from the center of the object.

Skill Snapshots

Causation

A nonzero net force exerted on a system causes the system's center of mass to accelerate.
For a constant net force, increasing the mass of the system causes the magnitude of its acceleration to decrease.
An unbalanced force applied opposite to the direction of an object's velocity causes the object to slow down.

Comparison

A system subject to balanced forces maintains a constant velocity, whereas a system subject to unbalanced forces experiences a change in velocity.
Mass is an intrinsic property measuring inertia, whereas weight is the gravitational force acting on that mass.
Velocity describes how an object's position is changing, whereas acceleration describes how its velocity is changing.

Change Over Time

Baseline: An object moving at a constant velocity of 2 m/s to the right has zero net force acting on it.
Change 1: If a constant net force is applied in the direction of motion, the object's velocity will steadily increase over time.
Change 2: If that net force is then reversed (applied opposite to the direction of motion), the object will slow down, momentarily stop, and then begin to speed up in the opposite direction.
Continuity: Throughout this entire process, the mass of the object remains constant.

Common Misconceptions & Clarifications

Misconception: Any force on an object causes it to move.
Clarification: Only an unbalanced or net force causes a change in motion (acceleration). A book resting on a table has both gravity and a normal force acting on it, but because they are balanced, its velocity does not change.
Misconception: Acceleration is always in the same direction as an object's velocity.
Clarification: Acceleration is always in the same direction as the net force. When a car brakes, its velocity is forward, but the net force (from friction and air resistance) is backward. Therefore, its acceleration is backward, causing the forward velocity to decrease.
Misconception: If an object's velocity is zero, the net force on it must be zero.
Clarification: This is only true if the object's velocity is zero and remains zero. If you throw a ball straight up, its velocity is momentarily zero at the very peak of its path, but the force of gravity is still acting on it. This net force causes its velocity to change from upward to downward.
Misconception: A constant force produces a constant velocity.
Clarification: A constant net force produces a constant acceleration, which means the velocity is constantly changing. To maintain a constant velocity, the net force must be zero.

One-Paragraph Summary

Newton's Second Law is the cornerstone of dynamics, providing the definitive link between cause (force) and effect (a change in motion). It states that the acceleration of a system is directly proportional to the net external force exerted on it and inversely proportional to its mass, expressed as $Σ F = m a$ . This relationship clarifies that only an unbalanced force—a non-zero net force—can alter a system's velocity. If the forces on a system are balanced, its velocity remains constant. By defining mass as the measure of inertia, this law allows us to quantitatively predict how an object's motion will change in response to its interactions with the surrounding world, forming the predictive basis for classical mechanics.

Newton's Second Law - AP Physics 1: Algebra-Based Study Guide