Newton’s Second Law | AP Physics 1 Unit 5 Study Guide

Getting Started

Imagine trying to spin a playground merry-go-round. Pushing it near the center is difficult, but pushing on the outer edge makes it spin much more easily. This chapter explores the relationship between the forces applied to an object, where they are applied, and how quickly that object's rotation changes, providing a rotational version of Newton's Second Law.

What You Should Be Able to Do

After completing this chapter, you will be able to:

Identify the conditions that cause an object's rate of rotation to change.
Relate the net torque on a rigid object to its resulting angular acceleration and its rotational inertia.
Calculate the angular acceleration of an object for a given net torque and rotational inertia.
Analyze complex systems where both linear and rotational motion occur simultaneously, such as a massive pulley with a hanging block.

Key Concepts & Mechanisms

The most effective way to understand the cause of rotational acceleration is to compare it directly to its linear counterpart, Newton's Second Law. The underlying physical principle—that an unbalanced interaction causes an acceleration—is identical. The mathematical structure and physical quantities are direct analogs.

Feature	Linear Motion Model ( $\sum F = ma$ )	Rotational Motion Model ( $\sum τ = I α$ )	Why It Matters
Cause of Acceleration	Net Force ( $\sum F$ ). The vector sum of all forces acting on an object. A non-zero net force causes the object's center of mass to accelerate. Units: Newtons (N).	Net Torque ( $\sum τ$ ). The rotational equivalent of force, which depends on the applied force, the distance from the pivot, and the angle. A non-zero net torque causes the object's angular velocity to change. Units: Newton-meters (N·m).	Understanding that torque, not just force, is the cause of angular acceleration is crucial. A large force can produce zero torque if applied at the axis of rotation.
Inertia (Resistance to Acceleration)	Mass ( $m$ ). An intrinsic property of an object that measures its resistance to a change in its linear velocity. Units: kilograms (kg).	Rotational Inertia ( $I$ ). A property that measures an object's resistance to a change in its angular velocity. It depends not only on the object's mass but also on how that mass is distributed relative to the axis of rotation. Units: kilogram-meter squared (kg·m²).	This distinction explains why a hollow hoop is harder to spin than a solid disk of the same mass and radius. The distribution of mass is a key factor in rotational dynamics.
The Law	$\sum F = ma$ . The net force on an object is equal to its mass times its linear acceleration.	$\sum τ = I α$ . The net torque on a rigid object is equal to its rotational inertia times its angular acceleration.	This powerful analogy allows you to solve rotational problems using the same logical steps you learned for linear problems: identify interactions (torques), account for inertia ( $I$ ), and find the resulting acceleration ( $α$ ).
Motion Variables	Linear Velocity ( $v$ ) and Linear Acceleration ( $a$ ). Describe the motion of the object's center of mass. Units: m/s and m/s².	Angular Velocity ( $ω$ ) and Angular Acceleration ( $α$ ). Describe how the orientation of the object changes over time. Units: rad/s and rad/s².	For an object that is both moving and rotating (like a rolling ball), you must analyze both its linear and rotational motion. These two types of motion are often linked by a constraint, such as the no-slip condition $a = r α$ .

Key Models & Diagrams

To solve problems involving rotational dynamics, we use a systematic approach that combines visual representations with the laws of motion. This process often requires analyzing both the linear forces and the rotational torques independently and then connecting them.

Physical System	Representation	Governing Equations	Predicted Observables
A solid disk pulley with mass $M$ and radius $R$ has a rope wrapped around it, attached to a hanging block of mass $m$ .	Two Free-Body Diagrams: 1. Block (m): Shows gravity ( $m g$ ) down and tension ( $F_{T}$ ) up. 2. Pulley (M): An extended FBD showing the pivot point, the pulley's weight, the support force from the axle, and the tension force ( $F_{T}$ ) acting tangentially at radius $R$ .	For the Block (Linear): $\sum F_{y} = m g - F_{T} = ma$ For the Pulley (Rotational): $\sum τ = F_{T} \cdot R = I α$ Constraint Equation: $a = R α$	The linear acceleration ( $a$ ) of the block and the angular acceleration ( $α$ ) of the pulley. By substituting and solving the system of equations, both can be determined.

Key Components & Evidence

Net Torque ( $\sum τ$ ): The rotational cause of angular acceleration. It is the sum of all individual torques acting on a rigid body, taking into account their direction (clockwise or counter-clockwise). Its SI unit is the Newton-meter (N·m).
Rotational Inertia ( $I$ ): A scalar quantity that measures an object's resistance to being angularly accelerated. It depends on the mass and its distribution around the axis of rotation. Its SI unit is kg·m².
Angular Acceleration ( $α$ ): The rate at which an object's angular velocity changes. It is a vector quantity, with its direction indicating the axis and sense of the rotational change. Its SI unit is radians per second squared (rad/s²).
Newton's Second Law for Rotation: The fundamental relationship $\sum τ = I α$ . This law states that angular acceleration is directly proportional to net torque and inversely proportional to rotational inertia.
Extended Free-Body Diagram: A diagram that shows not only the magnitude and direction of all forces acting on an object but also their points of application. This is essential for calculating torques.
Axis of Rotation: The line about which a rigid body rotates. The choice of axis is critical for calculating both torques and rotational inertia.
Lever Arm: The perpendicular distance from the axis of rotation to the line of action of an applied force. Torque is the product of the force and the lever arm.
Constraint Equation: A mathematical relationship that connects the linear motion of one part of a system to the rotational motion of another. For a rope unwinding from a pulley without slipping, this is $a = r α$ .

Skill Snapshots

Causation

A non-zero net torque exerted on a bicycle wheel causes its angular velocity to change.
The magnitude of the angular acceleration is caused by the ratio of the net torque to the object's rotational inertia.
Applying two equal and opposite forces that are not collinear (a couple) causes a pure rotational acceleration without any linear acceleration of the center of mass.

Comparison

Force vs. Torque: A force is a push or pull, while a torque is the rotational equivalent of a force that depends on where the force is applied.
Mass vs. Rotational Inertia: Mass resists changes in linear motion, while rotational inertia resists changes in rotational motion. An object with a large mass can have a small rotational inertia if its mass is concentrated near the axis of rotation.
Linear vs. Rotational Second Law: The equation $\sum F = ma$ is structurally identical to $\sum τ = I α$ , with torque replacing force, rotational inertia replacing mass, and angular acceleration replacing linear acceleration.

Change Over Time

Baseline: A spinning top is rotating with a constant angular velocity. The net torque on it is zero (ignoring friction).
Change 1: A small frictional torque from the air and the point of contact acts on the top. This net torque causes a negative angular acceleration, and the top's angular velocity gradually decreases.
Change 2: A child gently pushes the side of the top tangentially. This applied force creates a torque in the direction of motion, causing a positive angular acceleration and a temporary increase in the top's angular velocity.
Continuity: Throughout the process, the top's rotational inertia ( $I$ ) remains constant, as its mass and shape do not change.

Common Misconceptions & Clarifications

Misconception: Any force applied to an object will make it rotate.
Clarification: A force only produces a torque if it has a component perpendicular to the line connecting the axis of rotation and the point of application. A force directed through the axis of rotation (e.g., pushing directly on a hinge) produces zero torque and no angular acceleration.
Misconception: Objects with more mass are always harder to spin.
Clarification: Rotational inertia ( $I$ ), not mass, determines the resistance to rotation. The distribution of mass is key. A 1 kg hollow sphere is harder to spin than a 1 kg solid sphere of the same radius because more of its mass is located far from the center.
Misconception: In a pulley system, the tension is the same everywhere in the rope.
Clarification: This is only true for an idealized, massless pulley. If a pulley has mass (and thus rotational inertia) and is angularly accelerating, there must be a net torque acting on it. This requires the tension on one side of the rope to be greater than the tension on the other side.
Misconception: An object with zero net force cannot be accelerating.
Clarification: An object with zero net force has zero linear acceleration of its center of mass. However, it can still have an angular acceleration if it is subjected to a non-zero net torque (e.g., from two equal and opposite forces applied at different points, known as a couple).

One-Paragraph Summary

Newton's Second Law for Rotation, expressed as $\sum τ = I α$ , is the foundational principle governing changes in rotational motion. It establishes that a non-zero net torque is the exclusive cause of angular acceleration. In this relationship, rotational inertia ( $I$ ) acts as the measure of an object's resistance to changes in its rotational state, analogous to how mass resists changes in linear motion. This law is not merely a theoretical parallel to its linear counterpart ( $\sum F = ma$ ); it is an essential tool for analyzing real-world systems where objects both translate and rotate, such as rolling wheels or massive pulleys. By applying both linear and rotational analyses, we can fully predict the dynamics of complex mechanical systems.

Newton’s Second Law in Rotational Form - AP Physics 1: Algebra-Based Study Guide