---
title: "AP Physics C: Mech Unit 2 Review: Force and Motion Dynamics"
description: "AP Physics C: Mechanics Unit 2 covers Spring Forces, Circular Motion, Resistive Forces, and Newton's Third Law. Study guides, practice questions, and key terms."
canonical: "https://fiveable.me/ap-physics-c-mechanics/unit-2"
type: "unit"
subject: "AP Physics C: Mechanics"
unit: "Unit 2 – Force and Motion Dynamics"
---

# AP Physics C: Mech Unit 2 Review: Force and Motion Dynamics

## Overview

Unit 2 develops the dynamics toolkit you will use in every subsequent unit. You start by defining systems and locating centers of mass, then build through Newton's three laws, gravitational and contact forces, Hooke's law, velocity-dependent drag, and circular motion including orbital mechanics.

## AP CED Alignment

This unit hub is organized around AP Course and Exam Description topics, skills, and exam task types when they are available in the source data.
- 2.1: Systems and Center of Mass
- 2.2: Forces and Free-Body Diagrams
- 2.3: Newton's Third Law
- 2.4: Newton's First Law
- 2.5: Newton's Second Law
- 2.6: Gravitational Force
- 2.7: Kinetic and Static Friction
- 2.8: Spring Forces
- 2.9: Resistive Forces
- 2.10: Circular Motion
- 2.3: Newton's Third Law and Tension
- 2.4-2.5: Newton's First and Second Laws
- 2.10: Circular Motion and Orbits
- Practice 3: Scientific Questioning and Argumentation
- FRQ 2 – Translation Between Representations
- FRQ 4 – Qualitative/Quantitative Translation
- FRQ 1 – Mathematical Routines

## Topics

- [2.1: Systems and Center of Mass](/ap-physics-c-mechanics/unit-2/1-properties-and-interactions-of-a-system/study-guide/Hw10Krhy0qtfeWAb): Define system boundaries, distinguish internal from external interactions, and calculate center of mass for discrete masses and continuous objects using summation and integration.
- [2.2: Forces and Free-Body Diagrams](/ap-physics-c-mechanics/unit-2/2-forces-and-free-body-diagrams/study-guide/2LH73zRqxtRXtAKH): Represent every external force on a single object as a vector arrow from the center-of-mass dot, choose a coordinate axis aligned with acceleration, and translate the diagram into Newton's law equations.
- [2.3: Newton's Third Law](/ap-physics-c-mechanics/unit-2/3-newtons-third-law/study-guide/SXl4nBHlUrotvxSj): Identify action-reaction pairs acting on different objects, explain why internal forces do not affect center-of-mass motion, and apply ideal string and pulley assumptions to tension problems.
- [2.4: Newton's First Law](/ap-physics-c-mechanics/unit-2/4-newtons-first-law/study-guide/t0eQsK3dx7BBjFSK): Apply translational equilibrium (sum of F = 0) to systems at rest or constant velocity, distinguish inertial from noninertial reference frames, and recognize when forces balance in one direction but not another.
- [2.5: Newton's Second Law](/ap-physics-c-mechanics/unit-2/5-newtons-second-law/study-guide/c4OMxeY505zPKE78): Use a_sys = F_net / m_sys component by component to find acceleration when net external force is nonzero; connect free-body diagrams directly to the algebraic equations of motion.
- [2.6: Gravitational Force](/ap-physics-c-mechanics/unit-2/6-gravitational-force/study-guide/CzrVgTyZ4BKEJNfh): Apply F_g = G m1 m2 / r^2, calculate gravitational field strength, distinguish true weight from apparent weight, and use Newton's shell theorem for objects inside and outside uniform spherical distributions.
- [2.7: Kinetic and Static Friction](/ap-physics-c-mechanics/unit-2/7-kinetic-and-static-friction/study-guide/D7dia71mCcEsurUu): Calculate kinetic friction with F_fk = mu_k F_N and apply the static friction inequality F_fs is less than or equal to mu_s F_N to determine whether an object slides or remains stationary.
- [2.8: Spring Forces](/ap-physics-c-mechanics/unit-2/8-spring-forces/study-guide/jtwF1NQEUJXZEYva): Apply Hooke's law F_s = -k delta x, identify the restoring force direction toward equilibrium, and calculate equivalent spring constants for series and parallel spring combinations.
- [2.9: Resistive Forces](/ap-physics-c-mechanics/unit-2/9-resistive-forces/study-guide/pXbIz3a4RtJYP8Gq): Model linear drag as F_r = -k v, set up and solve the resulting separable differential equation, and interpret the exponential velocity function and terminal speed in terms of the time constant tau = m/k.
- [2.10: Circular Motion](/ap-physics-c-mechanics/unit-2/10-circular-motion/study-guide/mSTvL7QY6udY9crx): Identify the real forces providing centripetal acceleration a_c = v^2 / r, analyze vertical loops and conical pendulums, and apply Kepler's third law T^2 = (4 pi^2 / GM) R^3 to circular satellite orbits.

## Hardest Topics And Analytics

Snapshot: practice snapshot
This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.
- **63% average MCQ accuracy** (Across 5.1k multiple-choice practice attempts for this unit.)
- **5.1k MCQ attempts** (Practice activity included in this snapshot.)
- **59% average FRQ score** (Across 3 scored free-response attempts for this unit.)
- **2.6: Gravitational Force**: 52% MCQ miss rate across 750 attempts. Review Gravitational Force with attention to how the concept appears in AP-style source and evidence questions.
- **2.1: Systems and Center of Mass**: 42% MCQ miss rate across 767 attempts. Review Systems and Center of Mass with attention to how the concept appears in AP-style source and evidence questions.
- **2.5: Newton's Second Law**: 37% MCQ miss rate across 328 attempts. Review Newton's Second Law with attention to how the concept appears in AP-style source and evidence questions.
- **2.8: Spring Forces**: 32% MCQ miss rate across 406 attempts. Review Spring Forces with attention to how the concept appears in AP-style source and evidence questions.

## Review Notes

### 2.1: Systems and Center of Mass

A system is a collection of objects whose interactions you choose to analyze together. If internal structure does not matter, you can treat the entire system as a point mass located at its center of mass. For discrete masses, x_cm = (sum of m_i x_i) / (sum of m_i). For a continuous object with variable density, r_cm = integral of r dm / integral of dm, where dm = lambda d-ell for a rod with linear mass density lambda = dm / d-ell. Symmetry lines always pass through the center of mass of a uniform object.

- **System boundary**: Defines which objects are internal; only external forces change the center-of-mass motion.
- **Center of mass (discrete)**: x_cm = (sum m_i x_i) / (sum m_i); mass-weighted average position of all constituent objects.
- **Center of mass (continuous)**: r_cm = integral of r dm / integral of dm; requires expressing dm in terms of a density function and a position variable.
- **Linear mass density**: lambda = dm / d-ell; used to set up the dm integral for rods or wires with nonuniform density.
- **Symmetry argument**: For any uniform object with a line of symmetry, the center of mass lies on that line, often eliminating the need for integration.

**Checkpoint:** A nonuniform rod of length L has linear mass density lambda(x) = 2x kg/m. Set up the integral to find x_cm and identify the limits of integration.

Mass distribution type | dm expression | Integration variable
--- | --- | ---
Uniform rod | lambda dx (constant lambda) | x from 0 to L
Nonuniform rod | lambda(x) dx | x from 0 to L
Uniform disk (radial) | sigma 2 pi r dr | r from 0 to R

### 2.2: Forces and Free-Body Diagrams

A force is a vector interaction between two objects; an object cannot exert a net force on itself. A free-body diagram shows every external force on a single object as an arrow originating from a dot representing the center of mass. Draw one arrow per force, label each with its type (weight, normal, tension, friction), and choose a coordinate axis aligned with the direction of acceleration to simplify the algebra. Do not include force components as separate arrows on the diagram.

- **Contact force**: Arises when two objects touch; macroscopic result of interatomic electric forces. Examples: normal force, friction, tension.
- **Free-body diagram rule**: Each force arrow starts at the center-of-mass dot and points in the direction the force acts; components are written in equations, not drawn as extra arrows.
- **Axis choice**: Rotating the coordinate system so one axis is parallel to the acceleration reduces the number of simultaneous equations needed.
- **Net force**: Vector sum of all external forces; equals m times a by Newton's second law.

**Checkpoint:** Draw a free-body diagram for a block sliding down a rough incline. Identify every force and explain your axis choice.

### 2.3: Newton's Third Law and Tension

Newton's third law states F_A on B = -F_B on A. The paired forces are equal in magnitude, opposite in direction, and act on different objects, so they never cancel in a single free-body diagram. Internal forces between objects within a system do not affect the system's center-of-mass acceleration. Tension in an ideal (massless, inextensible) string is the same at every point; in a string with nonnegligible mass, tension varies along the string. An ideal pulley has negligible mass and frictionless axle, so it only redirects tension without changing its magnitude.

- **Action-reaction pair**: Two forces that are equal in magnitude and opposite in direction, each acting on a different object in the interaction.
- **Internal forces**: Forces between objects within the chosen system; they cancel in pairs and do not change the system's center-of-mass motion.
- **Tension (ideal string)**: Uniform throughout a massless, inextensible string; the string transmits force without storing it.
- **Ideal pulley**: Massless and frictionless; redirects tension so its magnitude is the same on both sides of the pulley.

**Checkpoint:** Two blocks connected by an ideal string are pulled across a frictionless surface. Identify one Newton's third law pair and explain why the string tension is the same throughout.

String type | Tension uniform? | Effect on analysis
--- | --- | ---
Ideal (massless, inextensible) | Yes | Single tension value T throughout
Real (nonnegligible mass) | No | Tension varies; must treat string as extended object

### 2.4-2.5: Newton's First and Second Laws

Newton's first law: if the net force on a system is zero, its velocity is constant (translational equilibrium, sum of F_i = 0). Forces can be balanced along one axis and unbalanced along another; only the unbalanced direction produces acceleration. An inertial reference frame is one in which Newton's first law holds. Newton's second law: a_sys = F_net / m_sys. The acceleration of the center of mass points in the same direction as the net external force. Apply it component by component after drawing a free-body diagram.

- **Translational equilibrium**: Net force equals zero in every direction; the object moves at constant velocity or stays at rest.
- **Inertial reference frame**: A frame in which an object with zero net force moves at constant velocity; Newton's laws hold without fictitious forces.
- **Newton's second law**: a_sys = F_net / m_sys; acceleration is proportional to net external force and inversely proportional to mass.
- **Component form**: Apply sum of F_x = m a_x and sum of F_y = m a_y separately after choosing a coordinate system aligned with the acceleration.

**Checkpoint:** A 5 kg block on a frictionless surface is pulled by two horizontal forces: 20 N east and 8 N west. Find the acceleration magnitude and direction.

Condition | Net force | Motion result
--- | --- | ---
Translational equilibrium | Zero in all directions | Constant velocity (or rest)
Unbalanced in one direction | Nonzero along one axis | Acceleration along that axis only
Unbalanced in two directions | Nonzero along both axes | Acceleration with both components

### 2.6: Gravitational Force

Newton's law of universal gravitation: F_g = G m1 m2 / r^2, attractive, along the line connecting the two centers of mass. Near Earth's surface, g is approximately 10 N/kg and gravitational force is treated as constant (weight = mg). The gravitational field at a point is g = G M / r^2. Apparent weight equals the normal force; it differs from true weight when the system accelerates. The equivalence principle states that an observer in a noninertial frame cannot distinguish apparent weight from gravitational weight. Newton's shell theorem: a uniform spherical shell exerts zero net gravitational force on an object inside it, and acts as a point mass at its center for objects outside. An object inside a uniform solid sphere feels only the gravitational pull from the partial mass enclosed within its radius.

- **Universal gravitation**: F_g = G m1 m2 / r^2; force is attractive, acts along the line of centers, and follows an inverse-square law.
- **Gravitational field**: g = G M / r^2 at distance r from mass M; units are N/kg, numerically equal to free-fall acceleration.
- **Apparent weight**: Magnitude of the normal force on a system; equals mg only when acceleration is zero.
- **Newton's shell theorem**: Inside a uniform spherical shell: zero net force. Outside: shell acts as a point mass at its center.
- **Equivalence principle**: Inertial mass and gravitational mass are experimentally equal; an accelerating observer cannot distinguish their apparent weight from a gravitational field.

**Checkpoint:** An object is located at radius r inside a uniform solid sphere of mass M and radius R. Write an expression for the gravitational force on the object in terms of r, R, M, and G.

Object location | Net gravitational force from shell/sphere
--- | ---
Inside thin spherical shell | Zero
Outside thin spherical shell | G M_shell / r^2 (shell acts as point mass)
Inside uniform solid sphere at radius r | G M_partial / r^2, where M_partial = M(r/R)^3

### 2.7: Kinetic and Static Friction

Kinetic friction acts when two surfaces slide relative to each other: F_fk = mu_k F_N, directed opposite to the relative motion. Static friction acts when surfaces are not sliding; it adjusts in magnitude and direction to prevent motion up to a maximum of F_fs,max = mu_s F_N. Because mu_s is typically greater than mu_k, more force is needed to start sliding than to maintain it. Friction magnitude does not depend on contact area. Normal force is perpendicular to the surface, not always equal to mg (for example, on an incline or when an additional vertical force is applied).

- **Kinetic friction**: F_fk = mu_k F_N; constant magnitude once sliding begins, directed opposite to relative motion.
- **Static friction**: F_fs is less than or equal to mu_s F_N; self-adjusting force that prevents sliding until the applied force exceeds the maximum.
- **Normal force**: Perpendicular contact force from the surface; determines friction magnitude but is not always equal to mg.
- **mu_s greater than mu_k**: The coefficient of static friction exceeds kinetic friction for the same surfaces, so objects require more force to start moving than to keep moving.

**Checkpoint:** A 10 kg block sits on a surface with mu_s = 0.5 and mu_k = 0.3. What is the minimum horizontal force needed to start the block moving, and what is the friction force once it is sliding at constant velocity?

Friction type | Equation | When it applies
--- | --- | ---
Static | F_fs is less than or equal to mu_s F_N | Surfaces not moving relative to each other
Kinetic | F_fk = mu_k F_N | Surfaces sliding relative to each other

### 2.8: Spring Forces

An ideal spring has negligible mass and obeys Hooke's law: F_s = -k delta x, where delta x is displacement from the relaxed length and k is the spring constant in N/m. The negative sign means the force always points back toward equilibrium. For springs in series, 1/k_eq = sum of 1/k_i, giving an equivalent constant smaller than any individual spring. For springs in parallel, k_eq = sum of k_i, giving a stiffer combined spring. A nonideal spring either has nonnegligible mass or a force that is not proportional to displacement.

- **Hooke's law**: F_s = -k delta x; restoring force proportional to displacement from equilibrium, directed toward equilibrium.
- **Spring constant k**: Stiffness of the spring in N/m; larger k means more force per unit displacement.
- **Springs in series**: 1/k_eq = 1/k1 + 1/k2 + ...; equivalent constant is smaller than the smallest individual k.
- **Springs in parallel**: k_eq = k1 + k2 + ...; equivalent constant is the sum of all individual constants.

**Checkpoint:** Two springs with k1 = 200 N/m and k2 = 100 N/m are connected in series. Find k_eq and compare it to each individual spring constant.

Configuration | k_eq formula | k_eq relative to components
--- | --- | ---
Series | 1/k_eq = 1/k1 + 1/k2 + ... | Smaller than smallest k
Parallel | k_eq = k1 + k2 + ... | Larger than largest k

### 2.9: Resistive Forces

A linear resistive (drag) force is modeled as F_r = -k v, opposing the velocity. Applying Newton's second law gives m dv/dt = F_applied - k v, a first-order separable differential equation. Solving by separation of variables with initial condition v(0) = v0 yields v(t) = v_terminal + (v0 - v_terminal) e^(-kt/m), where v_terminal = F_applied / k and the time constant is tau = m/k. As t increases, velocity approaches terminal speed exponentially. Acceleration and position are also exponential functions of time. For a falling object, v_terminal = mg/k.

- **Linear drag force**: F_r = -k v; magnitude proportional to speed, direction always opposite to velocity.
- **Separation of variables**: Technique for solving m dv/dt = F - kv by rearranging to dv/(F - kv) = dt/m and integrating both sides.
- **Terminal speed**: v_terminal = F_applied / k; the constant speed at which drag equals the driving force and acceleration reaches zero.
- **Time constant tau**: tau = m/k; the time for the velocity difference from terminal speed to decrease by a factor of e.

**Checkpoint:** A 2 kg object falls from rest with drag coefficient k = 4 N/(m/s). Write the differential equation, identify v_terminal, and write v(t).

### 2.10: Circular Motion and Orbits

An object in circular motion has centripetal acceleration a_c = v^2 / r directed toward the center. The net inward force from all real forces (gravity, normal, tension, friction) must equal m v^2 / r. For a vertical loop, the minimum speed at the top occurs when gravity alone provides centripetal force: v_min = sqrt(g r). For a conical pendulum, the horizontal component of tension provides centripetal force and the vertical component balances gravity. For a satellite in circular orbit, gravity is the only centripetal force, giving Kepler's third law: T^2 = (4 pi^2 / GM) R^3.

- **Centripetal acceleration**: a_c = v^2 / r; always directed toward the center of the circular path, not a separate force but the result of the net inward force.
- **Net centripetal force**: Sum of inward force components from all real forces equals m v^2 / r; identify which forces point inward and which point outward.
- **Minimum speed at loop top**: v_min = sqrt(g r); at this speed, normal force is zero and gravity alone provides centripetal acceleration.
- **Kepler's third law**: T^2 = (4 pi^2 / GM) R^3 for a circular orbit; relates orbital period to orbital radius and the mass of the central body.
- **Tangential acceleration**: Rate of change of speed along the path; nonzero in nonuniform circular motion, perpendicular to centripetal acceleration.

**Checkpoint:** A car travels over a hill of radius r at speed v. Write the Newton's second law equation at the top of the hill and find the speed at which the car loses contact with the road.

Scenario | Force providing centripetal acceleration | Key equation
--- | --- | ---
Horizontal circle (string) | Tension component inward | T = m v^2 / r
Vertical loop (top) | Gravity + normal force inward | mg + N = m v^2 / r
Conical pendulum | Horizontal tension component | T sin(theta) = m v^2 / r
Satellite orbit | Gravity only | G M m / R^2 = m v^2 / R

## Study Guides

- [2.1 Properties and Interactions of a System](/ap-physics-c-mechanics/unit-2/1-properties-and-interactions-of-a-system/study-guide/Hw10Krhy0qtfeWAb)
- [2.2 Forces and Free-Body Diagrams](/ap-physics-c-mechanics/unit-2/2-forces-and-free-body-diagrams/study-guide/2LH73zRqxtRXtAKH)
- [2.3 Newton's Third Law](/ap-physics-c-mechanics/unit-2/3-newtons-third-law/study-guide/SXl4nBHlUrotvxSj)
- [2.4 Newton's First Law](/ap-physics-c-mechanics/unit-2/4-newtons-first-law/study-guide/t0eQsK3dx7BBjFSK)
- [2.5 Newton's Second Law](/ap-physics-c-mechanics/unit-2/5-newtons-second-law/study-guide/c4OMxeY505zPKE78)
- [2.6 Gravitational Force](/ap-physics-c-mechanics/unit-2/6-gravitational-force/study-guide/CzrVgTyZ4BKEJNfh)
- [2.7 Kinetic and Static Friction](/ap-physics-c-mechanics/unit-2/7-kinetic-and-static-friction/study-guide/D7dia71mCcEsurUu)
- [2.8 Spring Forces](/ap-physics-c-mechanics/unit-2/8-spring-forces/study-guide/jtwF1NQEUJXZEYva)
- [2.9 Resistive Forces](/ap-physics-c-mechanics/unit-2/9-resistive-forces/study-guide/pXbIz3a4RtJYP8Gq)
- [2.10 Circular Motion](/ap-physics-c-mechanics/unit-2/10-circular-motion/study-guide/mSTvL7QY6udY9crx)

## Practice Preview

### Multiple-choice practice

- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A student plots the square of the orbital period $$T^2$$ on the vertical axis versus the cube of the orbital radius $$r^3$$ on the horizontal axis for Jupiter's moons. The best-fit line has a slope $$S$$. Which expression correctly determines Jupiter's mass $$M_J$$ using this slope?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A crate of mass $$m$$ is pulled across a rough horizontal floor by a rope angled $$\theta$$ above the horizontal with tension $$T$$. Which claim correctly describes how the kinetic friction force $$f_k$$ depends on the angle $$\theta$$ (for $$0 < \theta < 90^\circ$$)?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A rectangular brick with dimensions $$L \times W \times H$$ ($$L > W > H$$) slides on a horizontal surface. First, it slides on its largest face ($$L \times W$$), and then it slides on its smallest face ($$W \times H$$). Assuming the coefficient of kinetic friction $$\mu_k$$ is constant, which claim correctly compares the kinetic friction forces $$f_1$$ and $$f_2$$ in these two cases?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A long slab B rests on a horizontal surface, and a smaller block A rests on top of B. The surface between A and B is rough. If slab B is pulled to the right with acceleration $$a$$ such that block A slides to the left relative to B, which claim correctly describes the kinetic friction force on block A?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A block of mass $$m$$ is held against a rough vertical wall by a horizontal force $$F$$. The block is released and slides down the wall with acceleration $$a$$. Which claim correctly represents the kinetic friction force $$f_k$$ acting on the block?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A block of mass $$m$$ slides on a rough horizontal surface with a coefficient of kinetic friction $$\mu_k$$. An external vertical force $$F(t) = ct$$ presses downward on the block, where $$c$$ is a positive constant and $$t$$ is time. Which claim correctly describes the magnitude of the kinetic friction force $$f_k$$ exerted on the block as it slides?

### FRQ practice

- **Circular motion, friction forces, rotating turntable**: FRQ 2 – Translation Between Representations | Circular motion, friction forces, rotating turntable
- **Block acceleration comparison on inclined planes**: FRQ 4 – Qualitative/Quantitative Translation | Block acceleration comparison on inclined planes
- **Spring compression, friction, energy dissipation**: FRQ 1 – Mathematical Routines | Spring compression, friction, energy dissipation

## Key Terms

- **translational equilibrium**: The condition in which the net force on a system is zero in every direction, expressed as sum of F_i = 0; the system moves at constant velocity or remains at rest.
- **contact force**: A force arising when two objects are in physical contact, such as normal force, friction, or tension; macroscopic result of interatomic electric forces.
- **tension**: The macroscopic net result of forces that segments of a string or cable exert on each other; uniform throughout an ideal massless, inextensible string.
- **internal forces**: Forces that objects within a system exert on each other; they cancel in Newton's third law pairs and do not change the system's center-of-mass acceleration.
- **differential mass element**: An infinitesimally small mass dm within a continuous distribution, used in the integral r_cm = integral of r dm / integral of dm to locate the center of mass.
- **gravitational field**: A field model giving the gravitational force per unit mass at a point in space; magnitude g = G M / r^2, with units N/kg equal numerically to free-fall acceleration.
- **apparent weight**: The magnitude of the normal force on a system; equals the gravitational force only when the system has zero acceleration.
- **Newton's shell theorem**: A uniform spherical shell exerts zero net gravitational force on an object inside it and acts as a point mass at its center for objects outside it.
- **terminal speed**: The constant speed reached when the drag force equals the net driving force, giving zero acceleration; for linear drag, v_terminal = F_applied / k.
- **springs in series**: An end-to-end spring arrangement where 1/k_eq = 1/k1 + 1/k2 + ...; the equivalent constant is smaller than any individual spring constant.
- **springs in parallel**: A side-by-side spring arrangement where k_eq = k1 + k2 + ...; the equivalent constant is the sum of all individual constants.
- **tangential acceleration**: The component of acceleration directed along the circular path, equal to the rate of change of speed; nonzero only in nonuniform circular motion.
- **orbital period**: The time T for a satellite to complete one circular orbit; related to orbital radius R and central body mass M by Kepler's third law: T^2 = (4 pi^2 / GM) R^3.
- **equivalence principle**: An observer in a noninertial reference frame cannot distinguish between apparent weight from acceleration and weight from a gravitational field; inertial and gravitational mass are experimentally equal.
- **Equal and opposite forces**: Newton's third law: when two objects interact, each exerts a force on the other that is equal in magnitude and opposite in direction, acting on different objects.

## Common Mistakes

- **Including internal forces in Newton's second law for a system**: Internal forces between objects within a system cancel by Newton's third law and do not appear in the net force equation for the system's center of mass. Only external forces change the center-of-mass acceleration.
- **Treating centripetal force as a separate, additional force**: Centripetal force is not a new force; it is the net inward component of real forces already on the free-body diagram. Adding a separate centripetal force arrow to a free-body diagram double-counts forces and produces incorrect equations.
- **Assuming normal force always equals mg**: Normal force equals mg only on a horizontal surface with no vertical acceleration and no additional vertical forces. On an incline, in an accelerating elevator, or in circular motion, the normal force must be found from Newton's second law, not assumed.
- **Applying the shell theorem result to nonuniform or non-spherical distributions**: Newton's shell theorem applies only to uniform spherical shells. For an object inside a solid sphere, only the mass enclosed within radius r contributes to the gravitational force; the outer shell contributes nothing.
- **Forgetting the negative sign in Hooke's law and drag force**: Both F_s = -k delta x and F_r = -k v include a negative sign indicating the force opposes the displacement or velocity. Dropping this sign reverses the direction of the force and leads to incorrect equations of motion.

## Exam Connections

- **Multi-object system analysis with free-body diagrams**: AP Physics C: Mechanics free-response problems frequently present two or more connected objects (blocks on pulleys, stacked masses, objects on inclines) and ask you to draw separate free-body diagrams, write Newton's second law equations for each object, and solve for acceleration and tension. Practicing the transition from diagram to equation in component form is the core skill tested.
- **Calculus-based force problems**: The exam tests calculus integration directly in this unit through center of mass calculations for nonuniform rods and through the resistive force differential equation. Expect to set up an integral with correct limits and a density function, or to separate variables and integrate to find v(t) and x(t) under drag. Showing the setup clearly earns partial credit even if arithmetic errors occur.
- **Circular motion and gravitation combined**: Problems combining universal gravitation with circular motion appear regularly, asking you to derive orbital speed, period, or the gravitational field at a given radius. The shell theorem also appears in problems where an object moves inside a planet of uniform density, requiring you to identify the partial mass and write the correct force expression as a function of radius.

## Final Review Checklist

- **Final Unit 2 review checklist: Center of mass calculations**: Set up and evaluate the center of mass integral for a nonuniform rod using lambda(x) = dm/dx. Confirm you can apply the discrete formula for multi-object systems and use symmetry to skip integration when appropriate.
- **Free-body diagrams and Newton's laws**: Draw a complete free-body diagram for at least three scenarios (inclined plane, Atwood machine, circular loop), then write the Newton's second law equations in component form directly from the diagram.
- **Gravitational force and shell theorem**: Calculate F_g using the universal law, find the gravitational field at a given distance, and apply the shell theorem to determine the force on an object inside a uniform sphere at radius r less than R.
- **Friction on inclines and flat surfaces**: Determine whether static or kinetic friction applies, calculate the normal force correctly (especially on inclines or with additional vertical forces), and find the threshold force needed to initiate sliding.
- **Spring combinations**: Calculate k_eq for springs in series and parallel, apply Hooke's law to find force or displacement, and confirm the direction of the restoring force relative to equilibrium.
- **Resistive force differential equation**: Write m dv/dt = F - kv, separate variables, integrate with initial conditions, and identify v_terminal and tau = m/k from the resulting exponential solution.
- **Circular motion force analysis**: For each circular motion scenario (horizontal circle, vertical loop, satellite orbit), write the net inward force equation equal to m v^2 / r and solve for the unknown quantity. Apply Kepler's third law for orbital problems.

## Study Plan

- **Step 1: Systems, center of mass, and free-body diagrams (Topics 2.1-2.2)**: Read the topic guides for 2.1 and 2.2. Practice setting up center of mass integrals for nonuniform rods and drawing complete free-body diagrams for at least five different physical setups. Check that every force arrow starts at the center-of-mass dot and that you have not included components as separate arrows.
- **Step 2: Newton's three laws (Topics 2.3-2.5)**: Work through the topic guides for 2.3, 2.4, and 2.5 in sequence. For each scenario, identify all Newton's third law pairs, decide whether the system is in translational equilibrium or accelerating, and write the component form of Newton's second law. Use the Atwood machine and inclined-plane problems as standard practice setups.
- **Step 3: Gravitational force and friction (Topics 2.6-2.7)**: Review the universal gravitation formula, gravitational field, apparent weight, and shell theorem from the 2.6 topic guide. Then work through 2.7 to practice distinguishing static from kinetic friction and calculating normal force correctly on inclines. Combine both topics in problems involving objects on inclined surfaces near a planet.
- **Step 4: Spring forces and resistive forces (Topics 2.8-2.9)**: Use the 2.8 topic guide to practice Hooke's law and series/parallel spring combinations. Then work through 2.9 to set up and solve the drag differential equation by separation of variables. Sketch v(t) and a(t) graphs and label v_terminal and tau on each graph.
- **Step 5: Circular motion and orbits, then full-unit FRQ practice (Topic 2.10)**: Study the 2.10 topic guide, focusing on identifying the inward force in each scenario and applying Kepler's third law. After completing all topic guides, use the 21 available FRQ practice problems to work through multi-part dynamics problems that combine force models from across the unit. Use the AP score calculator to estimate your exam performance.

## More Ways To Review

- [Topic study guides](/ap-physics-c-mechanics/unit-2#topics)
- [Practice questions](/ap-physics-c-mechanics/guided-practice?unitSlug=unit-2)
- [FRQ practice](/ap-physics-c-mechanics/frq-practice)
- [Key terms](/ap-physics-c-mechanics/key-terms)

## FAQs

### What topics are covered in AP Physics Mech Unit 2?

AP Physics C: Mechanics Unit 2 covers 10 topics: Systems and Center of Mass, Forces and Free-Body Diagrams, Newton's First, Second, and Third Laws, Gravitational Force, Kinetic and Static Friction, Spring Forces, Resistive Forces, and Circular Motion. Together these topics build the full framework for translational dynamics. See the full topic list and matched practice at [/ap-physics-c-mechanics/unit-2](/ap-physics-c-mechanics/unit-2).

### How much of the AP Physics Mech exam is Unit 2?

Unit 2 makes up 20-25% of the AP Physics C: Mechanics exam, making it one of the most heavily weighted units. It covers force and translational dynamics, including Newton's Laws, free-body diagrams, friction, spring forces, resistive forces, and circular motion. Expect multiple MCQ and FRQ questions drawn directly from these topics.

### What's on the AP Physics Mech Unit 2 progress check (MCQ and FRQ)?

The AP Physics C: Mechanics Unit 2 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 10 topics. MCQ questions test Newton's Laws, free-body diagrams, friction, spring forces, and circular motion. FRQ parts ask you to set up equations of motion, analyze forces on a system, and justify your reasoning in writing. For matched practice aligned to every progress check topic, visit [/ap-physics-c-mechanics/unit-2](/ap-physics-c-mechanics/unit-2).

### How do I practice AP Physics Mech Unit 2 FRQs?

AP Physics C: Mechanics Unit 2 FRQs most often come from Newton's Second Law, circular motion, friction, and spring forces. A typical question gives you a physical scenario, asks you to draw a free-body diagram, write net-force equations, and solve for an unknown. To practice, work through problems that require full algebraic solutions and written justifications, not just numerical answers. Find Unit 2 FRQ practice at [/ap-physics-c-mechanics/unit-2](/ap-physics-c-mechanics/unit-2).

### Where can I find AP Physics Mech Unit 2 practice questions?

The best place to find AP Physics C: Mechanics Unit 2 practice questions, including MCQ and practice test sets, is [/ap-physics-c-mechanics/unit-2](/ap-physics-c-mechanics/unit-2). You'll find questions covering all 10 topics, from Newton's Laws and free-body diagrams to circular motion and resistive forces. Mixing MCQ drills with full FRQ walkthroughs is the most effective way to prep for the 20-25% of the exam this unit represents.

### How should I study AP Physics Mech Unit 2?

Start with free-body diagrams, because every dynamics problem in this unit depends on drawing them correctly before writing a single equation. Then work through Newton's Laws in order, making sure you can apply the Second Law in both linear and circular contexts. After that, tackle friction, spring forces, and resistive forces as separate force models you plug into the same net-force framework. Concrete steps that work well: (1) sketch a free-body diagram for every practice problem, even simple ones. (2) Write out the sum-of-forces equation explicitly before solving. (3) Do at least one full FRQ per topic, checking that your written justification matches your math. (4) Return to circular motion last since it combines everything from earlier topics. All 10 topics and practice sets are at [/ap-physics-c-mechanics/unit-2](/ap-physics-c-mechanics/unit-2).

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