Review AP Physics C: Mechanics Unit 2 to build the core framework for analyzing forces and motion. This unit covers Newton's three laws, free-body diagrams, gravity, friction, spring forces, resistive forces, and circular motion, all of which appear throughout the rest of the course.
Use the topic guides, key terms, and FRQ practice available for this unit to work through every major force model before exam day.
Force and Translational Dynamics is the largest conceptual foundation in AP Physics C: Mechanics. Every later unit on energy, momentum, rotation, and oscillations depends on the force models and problem-solving strategies introduced here.
Topics 2.1-2.5 establish how to define a system, locate its center of mass, draw a correct free-body diagram, and apply Newton's first and second laws. The center of mass equation for discrete masses is x_cm = (sum of m_i x_i) / (sum of m_i), and for continuous objects you integrate r_cm = integral of r dm / integral of dm. Newton's second law gives a_sys = F_net / m_sys.
Topics 2.6-2.9 introduce four force laws you must apply quantitatively. Universal gravitation: F_g = G m1 m2 / r^2. Friction: kinetic F_fk = mu_k F_N; static F_fs is less than or equal to mu_s F_N. Hooke's law: F_s = -k delta x. Linear drag: F_r = -k v, which produces exponential velocity approaching terminal speed.
Topic 2.10 requires identifying which real forces provide the net inward (centripetal) force. Centripetal acceleration is a_c = v^2 / r, always directed toward the center. For satellites in circular orbit, gravity supplies all centripetal force, and Kepler's third law gives T^2 = (4 pi^2 / GM) R^3.
Every force in this unit is an interaction between two objects. Newton's third law guarantees a paired force on the other object, internal forces never change a system's center-of-mass motion, and the net external force alone determines acceleration. This interaction-based view is the thread connecting free-body diagrams, gravity, friction, springs, drag, and circular motion throughout the unit.
Define system boundaries, distinguish internal from external interactions, and calculate center of mass for discrete masses and continuous objects using summation and integration.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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Review Gravitational Force with attention to how the concept appears in AP-style source and evidence questions.
Review Systems and Center of Mass with attention to how the concept appears in AP-style source and evidence questions.
Review Newton's Second Law with attention to how the concept appears in AP-style source and evidence questions.
Review Spring Forces with attention to how the concept appears in AP-style source and evidence questions.
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.
| 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 |
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.
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.
| 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 |
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.
| 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 |
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.
| 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 |
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).
| 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 |
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.
| 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 |
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.
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.
| 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 |
Try AP-style multiple-choice questions and written prompts after you review the notes.
2. A small block of mass m = 0.50 kg is placed on a horizontal turntable at a distance r = 0.40 m from the center of the turntable, as shown in Figure 1. The turntable rotates about a vertical axis through its center. The coefficient of static friction between the block and the turntable is μₛ = 0.60, and the coefficient of kinetic friction is μₖ = 0.40.
Figure 1: Top view of turntable with block at radius r = 0.40 m (counterclockwise rotation)
Figure 2: Dot represents the block in Scenario 1 (viewed from above)
On Figure 2, draw and label a free-body diagram showing the forces (not components) exerted on the block in Scenario 1 as viewed from above the turntable. Draw the relative lengths of all vectors to reflect the relative magnitudes of all the forces. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. In Scenario 1, the turntable rotates at constant angular speed ω₁ = 3.0 rad/s.
Derive an expression for ω₂, the angular speed at which the block begins to slip. Express your answer in terms of m, r, μₛ, μₖ, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. In Scenario 2, the angular speed is gradually increased until the block just begins to slip at angular speed ω₂.
Figure 3: Axes for graphing friction force magnitude f versus angular speed ω
On the axes in Figure 3, sketch a graph of the magnitude of the friction force f as a function of the angular speed ω. The graph should include the entire range from ω = 0 rad/s to ω = 6.0 rad/s. A student conducts an experiment in which the angular speed ω of the turntable is varied from 0 rad/s to 6.0 rad/s. For each angular speed, the student measures the magnitude of the friction force f exerted on the block.
Calculate the magnitude of the acceleration of the block relative to the ground at this instant. In Scenario 3, the block is slipping on the turntable while the turntable rotates at constant angular speed ω₃ = 5.0 rad/s. At one instant, the block is at distance r = 0.40 m from the center and is moving tangentially with speed v = 1.5 m/s relative to the ground.
4. Two identical wooden blocks, each of mass m = 2.0 kg, are placed on two different inclined planes, as shown in Figure 1 and Figure 2. Both planes make an angle θ = 30° with the horizontal. The coefficient of kinetic friction between each block and its respective surface is μk = 0.20.
Figure 1. Scenario 1
Figure 2. Scenario 2
While Block 1 is sliding up the incline in Scenario 1, the magnitude of its acceleration is a₁. While Block 2 is sliding down the incline in Scenario 2, the magnitude of its acceleration is a₂.
Indicate whether a₁ is greater than, less than, or equal to a₂ by writing one of the following.
Justify your answer using qualitative reasoning beyond referencing equations.
Derive an expression for the distance d₁ that Block 1 travels up the incline before momentarily coming to rest. Express your answer in terms of m, θ, μk, v₀, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider Scenario 1, where Block 1 slides up the incline with initial speed v₀ = 4.0 m/s.
In Scenario 3, Block 1 is replaced with a new block of mass M = 4.0 kg. The new block is given the same initial speed v₀ = 4.0 m/s up the incline. The coefficient of kinetic friction between the new block and the surface in Scenario 3 is the same as in Scenario 1 (μk = 0.20).
Indicate whether the distance d₃ that the new block travels up the incline before coming to rest is greater than, less than, or equal to the distance d₁ from Scenario 1.
Briefly justify your answer.
1. A block of mass M is placed on a horizontal surface with coefficient of kinetic friction μ_k. One end of an ideal spring with spring constant k is attached to the block, and the other end is attached to a fixed wall. The block is initially at position x = -x_0, where the spring is compressed by a distance x_0 from its relaxed length. The relaxed position of the spring corresponds to x = 0. The block is released from rest at x = -x_0 and slides along the surface in the +x-direction, as shown in Figure 1. The block reaches position x = d before coming to rest.
Figure 1. Block–spring system on a rough horizontal surface. The block is released from rest at x = −x₀ (spring compressed) and later comes to rest at x = d.
Figure 2. Draw and label the forces acting on the block while it slides on the horizontal surface between x = −x₀ and x = 0.
On the dot in Figure 2, draw and label arrows to represent all the forces exerted on the block at an instant while it is sliding between x = -x_0 and x = 0. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.
The block slides from x = -x_0 to x = d, where it comes to rest. During this motion, the force exerted on the block by the spring varies with position according to F_s(x) = -kx, where k is the spring constant.
Derive an expression for the distance d in terms of x_0, μ_k, M, g, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Calculate the numerical value of the distance d at which the block comes to rest. Consider a new scenario where the block has mass M = 0.80 kg, the spring has spring constant k = 120 N/m, the coefficient of kinetic friction is μ_k = 0.25, and the initial compression is x_0 = 0.30 m. The block is released from rest at x = -x_0.
| Term | Definition |
|---|---|
| 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. |
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.
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.
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.
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.
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.
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.
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.
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.
Open the individual guides for Unit 2 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP 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.
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.
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 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.
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. 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.
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.