Newton's Second Law | AP Physics C: Mech Unit 2 Study Guide

Getting Started

We consider a system of particles, which could be anything from a single block to a complex satellite, existing within an environment that can exert forces upon it. The core question of dynamics is: how do interactions with the environment, quantified as forces, govern the change in the system's state of motion? This chapter introduces the fundamental law that provides a precise, quantitative, and causal link between the net force on a system and the resulting acceleration of its center of mass.

What You Should Be Able to Do

After studying this section, you should be able to:

Formulate a vector expression for the net external force acting on a system by identifying and summing all individual external forces.
Construct a second-order ordinary differential equation of motion for a system's center of mass using Newton's second law.
Solve for the instantaneous acceleration vector of a system's center of mass given the net force and system mass.
Determine the velocity and position vectors of a system's center of mass as functions of time by integrating the acceleration, applying appropriate initial conditions.

Key Concepts & Mechanisms

This section explores Newton's second law through the lens of Dynamics as Cause, where forces are the causal agents that produce changes in motion.

System & Preconditions

To apply the law, we first define our system: the collection of objects whose motion we wish to analyze. The boundary of the system separates it from its environment. Forces are categorized as internal (exerted by one part of the system on another) or external (exerted by an agent in the environment on the system).

The primary idealization is the treatment of the system's motion by tracking a single point: the center of mass. This is a position vector, $r_{c m}$ , representing the mass-weighted average position of all particles in the system. Newton's second law, in its most common form, precisely describes the motion of this point. The law is only valid when analyzed from the perspective of an inertial reference frame—a coordinate system that is not accelerating.

Key Steps / Relations

The application of Newton's second law follows a rigorous, causal sequence.

Identify and Represent Forces: Isolate the system from its environment and construct a Free-Body Diagram (FBD). This diagram represents the system as a point or simplified object and includes vector arrows for all external forces acting on it. Examples include gravity ( $F_{g}$ ), normal forces ( $N$ ), tension ( $T$ ), and friction ( $f$ ). Internal forces are omitted as they occur in equal and opposite pairs and their net effect on the system's overall motion is zero.
Vector Summation (Net Force): Establish a convenient coordinate system (e.g., Cartesian, polar). Resolve each force vector into its components within this system. The net force, denoted $\sum F$ , is the vector sum of all external forces:
$\sum F = F_{1} + F_{2} + \dots + F_{N}$
This sum is the ultimate driver of any change in the system's velocity.
Apply the Governing Law: Newton's second law states that the net external force on a system is directly proportional to the acceleration of its center of mass. The constant of proportionality is the total mass of the system, $m_{sys}$ . This is the fundamental equation of classical dynamics:
$\sum F = m_{sys} a_{c m}$
Since acceleration is the second time derivative of position, this is a second-order differential equation of motion:
$\sum F (t, r, v) = m_{sys} \frac{d ^{2} r _{c m}}{d t ^{2}}$
Solve for Motion: The vector equation is decomposed into a set of scalar equations for each coordinate axis (e.g., $\sum F_{x} = m a_{x}$ , $\sum F_{y} = m a_{y}$ ). These equations are then solved for the components of acceleration. If acceleration is constant, kinematic equations apply. If acceleration is a function of time, position, or velocity, calculus is required:
$v_{c m} (t) = v_{0} + \int_{0}^{t} a_{c m} (t^{'}) d t^{'}$
$r_{c m} (t) = r_{0} + \int_{0}^{t} v_{c m} (t^{'}) d t^{'}$

Outputs & Effects

The direct output of applying the law is the acceleration vector of the center of mass, $a_{c m}$ . This vector's magnitude and direction are identical to the magnitude and direction of the net force vector, scaled by the inverse of the mass. A non-zero net force is the only way to change the velocity of the system's center of mass. If $\sum F = 0$ , then $a_{c m} = 0$ , and the center of mass velocity remains constant (this is a restatement of Newton's first law).

Regulation & Limits

The validity of $\sum F = m_{sys} a_{c m}$ is constrained.

Inertial Frames: The law is only valid in non-accelerating reference frames. In an accelerating frame (e.g., inside a turning car), "fictitious" forces must be introduced to account for the frame's acceleration.
Constant Mass: This form of the law assumes the system's mass, $m_{sys}$ , is constant. For systems where mass changes (e.g., a rocket expelling fuel), the more general form relating force to the time derivative of momentum, $\sum F = d p / d t$ , must be used.
Center of Mass Motion: The law describes the translational motion of the center of mass. It does not, by itself, describe the system's rotational motion or the motion of its constituent parts relative to the center of mass.

Key Models & Diagrams

The procedural logic of applying Newton's second law can be mapped as a flowchart from physical situation to mathematical prediction.

Step	Representation	Governing Equation / Action	Predicted Observables
1. Setup	Physical scenario is abstracted into a Free-Body Diagram (FBD) with a chosen coordinate system.	Identify all external forces $F_{i}$ .	A complete inventory of interactions (gravity, contact forces, etc.).
2. Dynamics	Forces from the FBD are summed vectorially.	$\sum F = m_{sys} a_{c m}$	The net force vector $\sum F$ and the resulting acceleration vector $a_{c m}$ .
3. Kinematics	The acceleration is treated as the driver of changes in motion over time.	$v (t) = \int a (t) d t$ $r (t) = \int v (t) d t$	The system's velocity $v (t)$ and position $r (t)$ as functions of time.

Key Components & Evidence

Net Force ( $\sum F$ ): The vector sum of all external forces acting on a system. It is the direct cause of acceleration. Its SI unit is the Newton (N), where 1 N = 1 kg·m/s².
System Mass ( $m_{sys}$ ): A positive scalar quantity representing a system's inertia, or its resistance to acceleration. Its SI unit is the kilogram (kg).
Acceleration of Center of Mass ( $a_{c m}$ ): The time rate of change of the velocity of the system's center of mass, $a_{c m} = d v_{c m} / d t = d^{2} r_{c m} / d t^{2}$ . Its SI unit is meters per second squared (m/s²).
External Force ( $F_{e x t}$ ): An interaction between the system and an object in its environment. Only external forces contribute to the net force that can accelerate the center of mass.
Internal Force ( $F_{in t}$ ): A force that one part of a system exerts on another part. By Newton's third law, these always sum to zero and cannot accelerate the system's center of mass.
Inertial Reference Frame: A coordinate system in which an object with zero net force acting upon it moves with a constant velocity. Newton's laws are valid in these frames.
Free-Body Diagram (FBD): A crucial schematic representation that isolates a system and shows all external forces acting upon it as vectors originating from the center of mass.

Skill Snapshots

Causation

Driver: A constant net force, $\sum F = C$ .
Change: Produces a constant acceleration, $a = C / m$ , resulting in a velocity that changes linearly with time.
Driver: A net force that is a function of time, $\sum F (t)$ .
Change: Produces a time-varying acceleration, $a (t) = \sum F (t) / m$ , which must be integrated to find velocity.
Driver: A zero net force, $\sum F = 0$ .
Change: Produces zero acceleration, resulting in a constant velocity vector for the center of mass.

Comparison

A system with a single applied force will accelerate, whereas a system in static or dynamic equilibrium has a net force of zero and does not accelerate.
The motion of a system's center of mass is determined solely by net external forces, whereas the rotational motion of the system is determined by net external torques (a related but distinct concept).
Analysis in an inertial frame uses $\sum F = m a$ directly, whereas analysis in a non-inertial frame requires the introduction of fictitious forces to correctly predict motion relative to that frame.

Change and Continuity Over Time

Baseline: A system's center of mass moves with an initial constant velocity, $v_{0}$ , indicating the net external force is zero.
Change: A non-zero net external force is applied. The system's velocity vector begins to change at a rate given by $a = \sum F / m$ .
Change: The net external force is removed. The acceleration immediately becomes zero, and the system's velocity vector becomes constant at its new value.
Continuity: Throughout the process, the system's mass (inertia) is assumed to remain constant, providing a consistent relationship between force and acceleration.

Common Misconceptions & Clarifications

Misconception: Force causes velocity.
Clarification: A net force causes acceleration, which is the rate of change of velocity. An object can have a large velocity while experiencing zero net force (e.g., a hockey puck gliding on ice). Conversely, an object can have zero velocity at an instant while experiencing a large net force (e.g., a ball at the peak of its trajectory).
Misconception: The term $m a$ is a force (the "force of inertia").
Clarification: The expression $m a$ is the result of the net force; it is not a force itself. It belongs on one side of the cause = effect equation, while the sum of forces belongs on the other. The term $m a$ should never be drawn on a free-body diagram.
Misconception: If an object is at rest, no forces are acting on it.
Clarification: If an object is at rest, its acceleration is zero, which means the net force on it is zero. This is the condition of static equilibrium. There can be multiple, non-zero forces acting on the object (e.g., gravity pulling down and a table pushing up), but they must vectorially sum to zero.
Misconception: Internal forces can cause a system to accelerate.
Clarification: Only external forces can change the velocity of a system's center of mass. Internal forces, such as the forces between the atoms of a thrown baseball or the explosion of a firework, act in equal and opposite pairs according to Newton's third law. Their vector sum is always zero, so they cannot alter the motion of the system's center of mass.

One-Paragraph Summary

Newton's second law is the cornerstone of classical dynamics, providing the definitive causal link between force and changes in motion. It states that the acceleration of a system's center of mass is directly proportional to the net external force exerted on the system and inversely proportional to its total mass, expressed by the vector equation $\sum F_{e x t} = m_{sys} a_{c m}$ . This relationship, valid in inertial reference frames for systems of constant mass, is fundamentally a second-order differential equation of motion. By systematically identifying forces, constructing free-body diagrams, and applying this law in component form, we can solve for the acceleration and subsequently integrate to predict the complete trajectory of a system's center of mass. The law makes clear that forces produce acceleration, not velocity, and that only external forces can alter the motion of a system as a whole.

Newton's Second Law - AP Physics C: Mechanics Study Guide