Force and Translational Dynamics

What this unit covers

The topics below follow the published Physics C: Mech course framework for Unit 2. This unit is worth 20–25% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Newton’s Laws & Free-Body Diagrams13 min · 3 objectivesState Newton’s three laws and apply the second law in the form ΣF = ma = m dv/dt · Draw free-body diagrams and resolve forces into perpendicular components · Solve for acceleration, normal force, and friction on a single object
• Drag Forces & Terminal Velocity15 min · 3 objectivesWrite Newton’s second law as a differential equation when a resistive force depends on speed · Determine terminal velocity from the condition that net force and dv/dt vanish · Describe qualitatively how velocity approaches its terminal value over time
• Systems & Connected Objects14 min · 3 objectivesApply Newton’s second law to systems of objects linked by strings and pulleys · Solve Atwood machines and blocks connected over an ideal pulley · Recognize that objects joined by an inextensible string share the same magnitude of acceleration
• Newton’s Law of Gravitation13 min · 3 objectivesState the law of universal gravitation and compute the force between two masses · Relate a planet’s surface gravity to its mass and radius through g = GM/R² · Predict how gravitational force changes with separation using the inverse-square law

Formulas in Unit 2

Every term in Unit 2

All 21 terms we publish for Force and Translational Dynamics, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Newton's second law in general form

ΣF = dp/dt. Reduces to ma only when mass is constant, which is why rocket problems need the general form.

Velocity-dependent drag

With F_drag = −bv, the equation of motion is a separable differential equation whose solution approaches terminal velocity exponentially.

Terminal velocity from a drag law

Set net force to zero: mg = bv gives v_t = mg/b; for quadratic drag mg = cv² gives v_t = √(mg/c).

Solving a separable equation of motion

Write m dv/dt = F(v), separate to dv/F(v) = dt/m, and integrate both sides with initial conditions as limits.

Static and kinetic friction

Static friction adjusts up to μsN; kinetic friction is μkN and constant during sliding. Static friction is generally the larger maximum.

Inclined plane analysis

Resolve weight into mg sin θ along and mg cos θ perpendicular. The angle at which sliding begins gives tan θ = μs.

Circular motion dynamics

The net inward force equals mv²/r = mω²r. Identify which real force supplies it before writing the equation.

Newton's law of gravitation

F = Gm₁m₂/r², measured center to center. Inside a uniform sphere, only the mass at smaller radius contributes.

Gravitational field inside a sphere

g rises linearly with r inside a uniform sphere and falls as 1/r² outside, peaking at the surface.

Orbits and Kepler's third law

Setting gravity equal to the centripetal requirement gives T² ∝ r³. Orbital speed v = √(GM/r) is independent of the orbiting mass.

Free-body diagram discipline

One diagram per object, showing only real interactions. Never draw ma or centripetal force as separate arrows — they are results, not causes.

Systems with multiple bodies

Treat the whole system to find acceleration (internal forces cancel), then isolate one body to find an internal force.

Pulley constraints

An inextensible string ties the accelerations of connected masses together in magnitude, which supplies the extra equation needed.

Non-inertial frames

In an accelerating frame, Newton's laws require a fictitious force. Physics C solves in an inertial frame instead to avoid it.

Solving the terminal velocity differential equation

From m dv/dt = mg − bv, separating and integrating gives v(t) = v_t(1 − e^(−bt/m)), approaching v_t asymptotically and never reaching it.

Impulse from a variable force

J = ∫F dt, the area under the force-time curve. This is how collision problems are handled when the force is not constant.

Vertical circular motion condition

At the top, the minimum speed for contact is where the normal force reaches zero, so mg alone supplies mv²/r and v_min = √(gr).

Banked curve with friction

Friction acts down the bank above the design speed and up the bank below it, which is why a range of safe speeds exists rather than one.

Gravitational potential energy sign

U = −GMm/r is negative because zero is taken at infinity and gravity is attractive. A more negative U means a more tightly bound orbit.

Total energy of a circular orbit

E = −GMm/2r, exactly half the potential energy. Negative total energy is what makes an orbit bound.

Kepler's second law

A line from the sun to a planet sweeps equal areas in equal times — a direct consequence of angular momentum conservation.

What examiners penalize here

• On the AP exam, draw the free-body diagram first and tilt your axes to match the motion. Write ΣF = ma along each axis separately. Getting the components of weight right — mg sin θ along a slope, mg cos θ into it — is where most points are won or lost.
• Keep the two formulas straight: F = Gm₁m₂/r² is the force between two masses, while g = GM/R² is the field (acceleration) one mass produces at distance R. Both are inverse-square — whenever a distance changes, square the ratio before scaling.

Practice Physics C: Mech

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 7 units of AP Physics C: Mechanics

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations