Newton's Second Law | AP Physics C: Mech Unit 5 Study Guide

Getting Started

Just as a net force causes an object's linear velocity to change, a net "twist" or torque causes its angular velocity to change. We will investigate the dynamics of a rotating rigid body, an object whose shape does not change as it moves. Our core question is: How can we formulate a law, analogous to Newton's Second Law for linear motion, that precisely describes the cause-and-effect relationship between the net torque on a rigid body and its resulting angular acceleration?

What You Should Be Able to Do

After working through this material, you will be able to:

Calculate the net vector torque on a rigid body by summing the cross products of position vectors and applied forces.
Formulate the equation of motion for a rigid body rotating about a fixed axis by relating the net torque to the body's rotational inertia and angular acceleration.
Set up and solve a system of coupled linear and rotational dynamics equations for systems where objects translate and rotate simultaneously (e.g., a massive pulley with a hanging block).
Analyze the relationship between the distribution of mass in a rigid body and its resulting angular acceleration under a given net torque.
Solve differential equations of the form $I \frac{d ^{2} θ}{d t ^{2}} = τ (θ, \dot{θ}, t)$ to find the angular position as a function of time.

Key Concepts & Mechanisms

System & Preconditions

The primary system we consider is the rigid body, an idealization of a solid object where the distance between any two internal points remains fixed. This assumption is crucial because it means the object's shape and mass distribution do not change during rotation, allowing its resistance to angular acceleration—its rotational inertia—to be treated as a constant. We will typically analyze rotation about a fixed axis, which simplifies the vector kinematics and dynamics, often allowing us to treat torque and angular acceleration as scalar quantities corresponding to the component along that axis.

Key Steps / Relations

The rotational form of Newton's Second Law is not a new fundamental law of nature but is derived directly from the linear form, $\sum F = m a$ . The derivation reveals the causal chain from force to angular acceleration.

Start with a Single Particle: Consider a single particle of mass $m$ at a position $r$ relative to an origin. A force $F$ acts on it, causing an acceleration $a$ . The torque on this particle is defined as the cross product $τ = r \times F$ .
Introduce Dynamics: Substitute Newton's Second Law, $F = m a$ , into the definition of torque:
$τ = r \times (m a) = m (r \times a)$ .
Connect Linear and Rotational Kinematics: For a particle in circular motion about a fixed axis, its tangential acceleration $a_{t}$ is related to its angular acceleration $α$ by $a_{t} = r α$ . The vector relationship is $a = α \times r + ω \times (ω \times r)$ . The torque is caused by the tangential component of acceleration, which is perpendicular to $r$ . The magnitude of the torque is thus $τ = r (m a_{t}) = r (m (r α)) = (m r^{2}) α$ .
Sum Over a Rigid Body: A rigid body is a collection of particles. To find the net external torque on the entire body, we sum the torques on all its constituent particles. The internal torques (from forces between particles) cancel out in pairs due to Newton's Third Law. Therefore, the net external torque is:
$\sum τ_{n e t, e x t} = \sum_{i} (m_{i} r_{i}^{2}) α$
Since the body is rigid, every particle has the same angular acceleration $α$ .
Define Rotational Inertia: We define the quantity in the parenthesis as the rotational inertia, $I$ , of the body about the axis of rotation. For a discrete collection of particles, $I = \sum_{i} m_{i} r_{i}^{2}$ . For a continuous body, this sum becomes an integral over the mass distribution: $I = \int r^{2} d m$ .
State the Governing Law: The summation leads to the central equation of rotational dynamics:
$\sum τ_{n e t} = I α$
This is Newton's Second Law in rotational form.

Outputs & Effects

The primary output of this law is the angular acceleration vector, $α$ , defined as the time derivative of the angular velocity vector, $α = d ω / d t$ . A non-zero net torque is the cause of a change in a system's rotational motion. The direction of $α$ is the same as the direction of the net torque $τ_{n e t}$ . If the net torque is constant, the angular velocity changes linearly with time. If the torque depends on position or velocity, this equation becomes a differential equation whose solution describes the object's rotational motion over time.

Regulation & Limits

The validity of $\sum τ = I α$ is contingent on the rigid body model. If an object deforms, some of the work done by the torques goes into changing the potential energy of deformation, and the relationship breaks down. The scalar form, $τ = I α$ , is valid for rotation about a fixed axis that is also an axis of symmetry (a principal axis). For rotation about an arbitrary axis, the rotational inertia becomes a tensor, and the angular acceleration vector $α$ may not be parallel to the net torque vector $τ_{n e t}$ .

Key Models & Diagrams

For systems involving both translation and rotation, a systematic approach is essential. The following flowchart maps the process from physical setup to quantitative prediction.

Step	Representation / Model	Governing Equations (Differential/Integral Form)	Predicted Observables
1. Isolate & Identify	Separate diagrams for each body. Use an Extended Free-Body Diagram for the rotating object, showing forces at their points of application.	N/A	Identification of all forces (Tension, Gravity, Normal Force, Friction).
2. Apply Newton's Laws	For each body, apply the appropriate form of Newton's Second Law.	Translational: $\sum F_{n e t} = m \frac{d ^{2} x}{d t ^{2}}$ Rotational: $\sum τ_{n e t} = I \frac{d ^{2} θ}{d t ^{2}}$	A set of equations relating forces, torques, and accelerations.
3. Link the Motions	Identify the Constraint Equation that connects the linear and rotational motion.	For a non-slipping rope on a pulley or rolling without slipping: $a = R α$ or, more formally, $∣ a ∣ = R ∣ α ∣$ .	An algebraic link between the variables $a$ and $α$ .
4. Solve the System	Combine the equations from steps 2 and 3 to form a solvable system of algebraic or differential equations.	System of linear equations.	Values for linear acceleration ( $a$ ), angular acceleration ( $α$ ), and internal forces (e.g., Tension).

Key Components & Evidence

Torque ( $τ$ ): The rotational analogue of force, representing the effectiveness of a force in causing rotation. It is a vector defined by the cross product $τ = r \times F$ , where $r$ is the vector from the axis of rotation to the point of force application. SI units: Newton-meters (N·m).
Rotational Inertia ( $I$ ): A scalar quantity that measures a body's resistance to being angularly accelerated about a given axis. It is the rotational analogue of mass and depends on the body's total mass and how that mass is distributed relative to the axis. SI units: kilogram-meters squared (kg·m²).
Angular Acceleration ( $α$ ): The vector representing the rate of change of angular velocity, $α = d ω / d t$ . SI units: radians per second squared (rad/s²).
Newton's Second Law (Rotational Form): The governing law $\sum τ_{n e t} = I α$ . It states that the net external torque on a rigid body is directly proportional to its angular acceleration.
Rigid Body Model: An idealization assuming an object's shape is immutable. This ensures that its rotational inertia, $I$ , is a constant during the motion.
Extended Free-Body Diagram: A crucial representation for rotational problems. Unlike a standard free-body diagram, it shows not just the forces but also where on the body they are applied, which is essential for calculating torques.
Constraint Equation: An equation that connects translational and rotational variables, such as $a = R α$ for a rope that does not slip over a pulley of radius $R$ . This is necessary to solve coupled systems.

Skill Snapshots

Causation

Driver → Change: A non-zero net external torque ( $\sum τ_{n e t} \neq = 0$ ) → causes the system's angular velocity to change over time ( $α = d ω / d t \neq = 0$ ).
Driver → Change: For a fixed net torque, an increase in the system's rotational inertia ( $I$ ) → causes a decrease in the magnitude of its angular acceleration ( $α = τ_{n e t} / I$ ).
Driver → Change: Applying a force $F$ at a point further from the axis of rotation (increasing the magnitude of the lever arm $r$ ) → causes a larger torque ( $τ = r F_{⊥}$ ) and thus a larger angular acceleration.

Comparison

Force vs. Torque: A net force causes the acceleration of a system's center of mass, whereas a net torque causes the system's angular acceleration about an axis.
Mass vs. Rotational Inertia: Mass is an intrinsic property measuring resistance to linear acceleration. Rotational inertia is an extrinsic property measuring resistance to angular acceleration, which depends on both mass and its spatial distribution relative to the axis of rotation.
Linear vs. Rotational Dynamics: The equation $\sum F = m a$ is structurally identical to $\sum τ = I α$ , with force corresponding to torque, mass to rotational inertia, and linear acceleration to angular acceleration.

Change and Continuity Over Time

Baseline: A rigid body rotating with constant angular velocity has zero angular acceleration, which implies the net torque on it must be zero.
Change: If a constant net torque is applied (e.g., by a motor), the body's angular velocity will change linearly with time according to $ω (t) = ω_{0} + α t$ , where $α = τ_{n e t} / I$ .
Change: If a restoring torque proportional to angular displacement is applied, $τ = - k θ$ , the system undergoes angular simple harmonic motion, described by the differential equation $I \frac{d ^{2} θ}{d t ^{2}} + k θ = 0$ .
Continuity: For a rigid body, its rotational inertia $I$ about a fixed axis remains constant throughout its motion, regardless of changes in its angular velocity.

Common Misconceptions & Clarifications

Misconception: Any force applied to an object will create a torque.
Clarification: A force creates a torque only if it has a component perpendicular to the lever arm (the position vector from the axis to the point of application). A force whose line of action passes through the axis of rotation produces zero torque.
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., a couple, which is a pair of equal and opposite forces applied at different points).
Misconception: Rotational inertia is a fixed property of an object, just like mass.
Clarification: Rotational inertia is not an intrinsic property; it depends on the chosen axis of rotation. A rod has a very small rotational inertia when rotated about its long axis but a much larger one when rotated about an axis through its center and perpendicular to its length.
Misconception: In problems with pulleys, the tension in the rope is the same on both sides.
Clarification: This is only true for an idealized, massless pulley. If the pulley has mass (and thus rotational inertia) and is accelerating angularly, there must be a net torque acting on it. This net torque can only be produced if the tensions on either side of the rope are different.

One-Paragraph Summary

Newton's Second Law in rotational form, $\sum τ_{n e t} = I α$ , provides the fundamental causal link between the net external torque on a system and its resulting angular acceleration. This principle is not new but is a direct consequence of applying the linear form, $\sum F = m a$ , to a collection of particles forming a rigid body. The rotational inertia, $I$ , acts as the proportionality constant, quantifying an object's resistance to changes in its rotational motion based on its mass and how that mass is distributed. For complex systems involving both translation and rotation, this law must be used in conjunction with its linear counterpart and any constraint equations that link the two types of motion. This powerful framework allows us to analyze and predict the motion of everything from a simple spinning disk to the complex dynamics of planetary orbits.

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