Rolling | AP Physics C: Mech Unit 6 Study Guide

Getting Started

Consider a sphere rolling down an incline. Its motion is more complex than a simple sliding block; it is simultaneously translating (its center of mass is moving) and rotating about that center of mass. The core question is how to mathematically describe this composite motion and analyze the energy transformations involved. This requires us to synthesize the principles of linear and rotational dynamics into a single, coherent model.

What You Should Be able to Do

After studying this section, you will be able to:

Formulate the total kinetic energy of a rigid body as the sum of its translational and rotational components.
Construct free-body diagrams for rolling objects and apply Newton's second law for both translation ( $\sum F = m a_{c m}$ ) and rotation ( $\sum τ = I α$ ).
Use the calculus-based "no-slip" conditions ( $v_{c m} = r ω$ , $a_{c m} = r α$ ) to create a solvable system of equations.
Solve for the linear acceleration of an object rolling down an incline, demonstrating its dependence on the object's rotational inertia.
Apply the principle of conservation of mechanical energy to analyze the final speed of an object that rolls from a height.

Key Concepts & Mechanisms

This section analyzes rolling motion through the lens of Dynamics as Cause, where forces and torques are the agents that produce translational and rotational acceleration.

System & Preconditions

The system is a rigid body (e.g., a sphere, cylinder, or hoop) of mass $M$ , radius $R$ , and rotational inertia $I_{c m}$ about an axis through its center of mass. This body interacts with a stationary, unyielding surface and a uniform gravitational field.

Key Idealizations & Preconditions:

Rigid Body: The object does not deform under any applied forces. The distance between any two points on the body remains constant.
Symmetry: The object is typically assumed to be symmetric, so its center of mass is at its geometric center.
Rolling Without Slipping: This is the crucial constraint. The point on the rolling object in contact with the surface is instantaneously at rest relative to the surface. This implies a direct, linear relationship between the motion of the center of mass and the rotational motion of the object.

Key Steps / Relations

To analyze the dynamics of a rolling object, we follow a sequence of steps that connects the forces and torques to the resulting motion.

Represent Forces and Torques: Begin by drawing a complete free-body diagram for the object. Identify all external forces, such as gravity ( $M g$ ), the normal force ( $F_{N}$ ), and the force of static friction ( $f_{s}$ ). The force of static friction is essential; without it, the object would slide, not roll.
Apply Newton's Second Law for Translation: The translational motion of the center of mass is governed by the net external force.
$\sum F_{e x t} = M a_{c m}$
For an object on an incline of angle $θ$ , this vector equation is typically resolved into components parallel and perpendicular to the surface. For motion down the incline, the parallel component is:
$M g sin θ - f_{s} = M a_{c m}$
Apply Newton's Second Law for Rotation: The rotational motion about the center of mass is governed by the net external torque about that point.
$\sum τ_{c m} = I_{c m} α$
Neither gravity (acting at the center of mass) nor the normal force (acting through the center of mass) produces a torque about the center of mass. Only static friction, applied at the radius $R$ , generates a torque:
$τ_{c m} = R f_{s}$
This gives the rotational dynamics equation:
$R f_{s} = I_{c m} α$
Impose the No-Slip Constraint: The translational and rotational equations are linked by the condition of rolling without slipping. This kinematic relationship arises because the arc length subtended by a rotation $Δ θ$ must equal the distance the center of mass travels, $Δ x_{c m}$ . Taking successive time derivatives gives the full set of constraints:
$Δ x_{c m} = R Δ θ ⟹ v_{c m} = \frac{d x _{c m}}{d t} = R \frac{d θ}{d t} = R ω ⟹ a_{c m} = \frac{d v _{c m}}{d t} = R \frac{d ω}{d t} = R α$
The relation $a_{c m} = R α$ is the bridge between the two dynamics equations.
Solve the System of Equations: By substituting $α = a_{c m} / R$ into the torque equation, we can solve for $f_{s}$ and then substitute that into the force equation to find the acceleration $a_{c m}$ .

Outputs & Effects

Solving the system yields the linear acceleration of the center of mass:

$a_{c m} = \frac{g sin θ}{1 + I _{c m} / ( M R ^{2} )}$

This result shows that the acceleration is constant for a given incline and depends on the object's mass distribution, captured by the term $I_{c m} / (M R^{2})$ . Objects with larger rotational inertia (like a hoop, where $I = M R^{2}$ ) accelerate more slowly than objects with smaller rotational inertia (like a sphere, where $I = \frac{2}{5} M R^{2}$ ).

The total kinetic energy of the rolling object is the sum of the energy of its center-of-mass motion and the energy of rotation about the center of mass:

$K_{t o t a l} = K_{t r an s} + K_{ro t} = \frac{1}{2} M v_{c m}^{2} + \frac{1}{2} I_{c m} ω^{2}$

Using the no-slip condition $v_{c m} = R ω$ , this can be rewritten entirely in terms of $v_{c m}$ :

$K_{t o t a l} = \frac{1}{2} M v_{c m}^{2} (1 + \frac{I _{c m}}{M R ^{2}})$

Regulation & Limits

The model of rolling without slipping is valid only as long as the force of static friction required to maintain the motion is less than or equal to the maximum available static friction, $f_{s} \leq μ_{s} F_{N}$ . If the incline is too steep or the surface has too low a coefficient of static friction $μ_{s}$ , the object will begin to slip, at which point the constraint $a_{c m} = R α$ is no longer valid.

A critical consequence of the no-slip condition is that the force of static friction does no work. The work done by a force is $W = \int F \cdot d l$ , where $d l$ is the displacement of the point of application of the force. Since the point of contact on the object is instantaneously at rest, its displacement is zero, and no energy is dissipated by friction. Therefore, for ideal rolling, mechanical energy is conserved.

Key Models & Diagrams

The process of analyzing rolling dynamics can be mapped as follows:

Representation	Governing Equations (Differential Form)	Constraint	Predicted Observables
Free-Body Diagram showing $F_{g}$ , $F_{N}$ , $f_{s}$ on an incline.	Translation: $\sum F_{e x t} = M \frac{d ^{2} x _{c m}}{d t ^{2}}$ Rotation: $\sum τ_{c m} = I_{c m} \frac{d ^{2} θ}{d t ^{2}}$	No-Slip Condition: $a_{c m} = R α$ or $\frac{d ^{2} x _{c m}}{d t ^{2}} = R \frac{d ^{2} θ}{d t ^{2}}$	Linear Acceleration: $a_{c m}$ Final Velocity: $v_{c m}$ (from kinematics or energy conservation) Required Friction: $f_{s}$

Key Components & Evidence

Translational Kinetic Energy ( $K_{t r an s}$ ): The energy of motion of the center of mass. $K_{t r an s} = \frac{1}{2} M v_{c m}^{2}$ . Units: Joules (J).
Rotational Kinetic Energy ( $K_{ro t}$ ): The energy of motion about the center of mass. $K_{ro t} = \frac{1}{2} I_{c m} ω^{2}$ . Units: Joules (J).
Total Kinetic Energy ( $K_{t o t}$ ): The sum of translational and rotational kinetic energies for a rolling object. $K_{t o t} = K_{t r an s} + K_{ro t}$ .
Center of Mass Velocity ( $v_{c m}$ ): The instantaneous velocity vector of the object's center of mass. Units: m/s.
Angular Velocity ( $ω$ ): The instantaneous rate of rotation about the axis of rotation. Units: rad/s.
Rotational Inertia ( $I_{c m}$ ): A scalar quantity that measures an object's resistance to angular acceleration about its center of mass. It depends on mass and how that mass is distributed relative to the axis. Units: kg·m².
Static Friction ( $f_{s}$ ): The force exerted by the surface on the object that prevents slipping. It is the agent that provides the necessary torque for rotation. Units: Newtons (N).
No-Slip Condition: The set of kinematic equations ( $v_{c m} = R ω$ , $a_{c m} = R α$ ) that connect the translational and rotational motion for an object that rolls without slipping.

Skill Snapshots

Causation

A net force on the center of mass → causes translational acceleration ( $a_{c m}$ ).
A net torque about the center of mass (provided by static friction) → causes angular acceleration ( $α$ ).
An object's mass distribution, quantified by its rotational inertia $I_{c m}$ → determines the portion of its energy that becomes rotational, thereby affecting its final translational acceleration.

Comparison

Rolling vs. Sliding: A rolling object converts potential energy into both translational and rotational kinetic energy, while a purely sliding object converts it only into translational kinetic energy. Thus, for the same drop in height, the sliding object will have a greater center-of-mass speed.
Sphere vs. Hoop: A solid sphere ( $I = \frac{2}{5} M R^{2}$ ) will roll down an incline faster than a hollow hoop ( $I = M R^{2}$ ) of the same mass and radius because the hoop has a larger rotational inertia and requires more energy to be allocated to rotation for a given speed.
Static vs. Kinetic Friction: In ideal rolling, static friction provides the torque to change angular velocity but does no work. If the object slips, kinetic friction acts, which does negative work and dissipates mechanical energy as heat.

Change Over Time (CCOT)

Baseline: An object is at rest at the top of an incline, possessing only gravitational potential energy ( $U_{g} = M g h$ ).
Change 1: As it rolls down, $U_{g}$ is converted into $K_{t o t a l} = K_{t r an s} + K_{ro t}$ . The ratio of $K_{ro t}$ to $K_{t r an s}$ is constant and depends on the object's shape.
Change 2: The linear velocity $v_{c m}$ and angular velocity $ω$ both increase at a constant rate, governed by the constant accelerations $a_{c m}$ and $α$ .
Continuity: The total mechanical energy of the system ( $U_{g} + K_{t o t a l}$ ) remains constant throughout the motion, as static friction does no work.

Common Misconceptions & Clarifications

Misconception: The force of friction always opposes the direction of motion.
Clarification: For an object rolling down an incline, static friction points up the incline. It opposes the slipping that would occur (the bottom surface sliding down), not the overall motion of the center of mass. This upward force provides the torque that causes the object to spin.
Misconception: Friction always does negative work and removes energy from a system.
Clarification:Static friction in ideal rolling does zero work. The point of application of the force has zero velocity relative to the surface, so the power delivered by the force ( $P = F \cdot v$ ) is zero. Mechanical energy is conserved.
Misconception: The speed of every point on a rolling wheel is the same.
Clarification: The speed of points on the wheel varies with position. The point at the very top moves at $v_{t o p} = v_{c m} + R ω = 2 v_{c m}$ . The point at the bottom is instantaneously at rest ( $v_{b o tt o m} = v_{c m} - R ω = 0$ ).
Misconception: All round objects accelerate at the same rate when rolling down an incline.
Clarification: Acceleration depends on the rotational inertia. Objects that concentrate more mass away from their center (like a hoop) are harder to spin and thus accelerate more slowly than objects with mass concentrated at the center (like a sphere).

One-Paragraph Summary

Rolling motion is a composite of translation of the center of mass and rotation about the center of mass. Consequently, a rolling object's total kinetic energy is the sum of its translational ( $\frac{1}{2} M v_{c m}^{2}$ ) and rotational ( $\frac{1}{2} I_{c m} ω^{2}$ ) components. The analysis of its dynamics requires applying Newton's second law for both translation and rotation. These two descriptions are linked by the crucial no-slip condition, $v_{c m} = R ω$ and $a_{c m} = R α$ , which holds when static friction is sufficient to prevent slipping. This static friction force provides the necessary torque for rotation but, in an ideal case, does no work, allowing for the conservation of mechanical energy. The resulting acceleration depends not just on mass and external forces, but fundamentally on the object's mass distribution, as quantified by its rotational inertia.

Rolling - AP Physics C: Mechanics Study Guide