AP Physics C Mechanics Formula Sheet: Calculus Study Guide

This AP Physics C mechanics formula sheet is a calculus based study guide to the mechanics course, built on Newton's second law in the form $\sum \vec{F} = \frac{d\vec{p}}{dt}$ . For constant mass it reduces to $\sum F = ma$ , but the derivative form also handles rockets, conveyor belts and anything else whose momentum changes in less tidy ways.

It is a summary for revision and not the exam board's own sheet, which comes with the exam along with a table of constants. The tables follow the usual order of the course: motion, forces, energy, momentum, rotation, oscillations and gravitation. For the second C course, see the E&M sheet; for how the two fit together, see the AP Physics C overview or the formula sheets hub.

Quick facts

What this is
A study guide, not the official equation table
Core law
$\sum \vec{F} = \frac{d\vec{p}}{dt}$
Energy link
$F_x = -\frac{dU}{dx}$
Rotation link
$\sum \tau = I\alpha = \frac{dL}{dt}$
Math used
Derivatives, definite integrals, simple differential equations
Value of g
9.8 m/s² unless told otherwise

Kinematics and forces

Motion and Newton's laws in calculus form
Relation	Use	SI unit	Explained on
$v = \frac{dx}{dt},\ a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$	velocity and acceleration from x(t)	m/s, m/s²	Instantaneous velocity
$x = x_0 + \int v\,dt$	position from a velocity function	m	Displacement
$\sum \vec{F} = m\vec{a}$	constant mass dynamics	N	Net force
$F_{drag} = -bv$	linear drag at low speed	N	Drag force
$v(t) = v_T\left(1 - e^{-t/\tau}\right),\ \tau = \frac{m}{b}$	falling with linear drag, from rest	m/s	Terminal velocity
$v_T = \frac{mg}{b}$	terminal speed with linear drag	m/s	Terminal velocity

The drag result comes from solving $m\frac{dv}{dt} = mg - bv$ , a first order differential equation. Expect to set it up, separate the variables and sketch the exponential approach to $v_T$ .

Work, energy, power and momentum

Energy and momentum relations
Relation	Use	SI unit	Explained on
$W = \int_{x_1}^{x_2} F_x\,dx$	work when F changes with x	J	Work
$\Delta K = W_{net}$	work energy theorem	J	Work energy theorem
$F_x = -\frac{dU}{dx}$	force from a potential energy curve	N	Potential energy
$U_s = \frac{1}{2}kx^2,\ U_G = -\frac{Gm_1m_2}{r}$	spring and gravitational potential energy	J	Spring energy
$P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}$	instantaneous power	W	Work and energy
$\vec{J} = \int \vec{F}\,dt = \Delta\vec{p}$	impulse of a time varying force	N s	Impulse
$x_{cm} = \frac{\int x\,dm}{\int dm}$	center of mass (centre of mass) of a continuous body	m	Center of mass

Example 1: work by a variable force

A force $F(x) = 6x^2$ newtons (x in metres) pushes a 1.5 kg block from rest at x = 0 to x = 1.5 m on a frictionless surface. Find the work done and the final speed.

$W = \int_0^{1.5} 6x^2\,dx = \left[2x^3\right]_0^{1.5} = 2(3.375) = 6.75\ \text{J}$ .

Work energy theorem: $\frac{1}{2}(1.5)v^2 = 6.75$ , so $v^2 = 9.0$ and $v = 3.0\ \text{m/s}$ .

Answer: 6.75 J (about 6.8 J) and 3.0 m/s.

Rotation

Rotational dynamics and angular momentum
Relation	Use	SI unit	Explained on
$\omega = \frac{d\theta}{dt},\ \alpha = \frac{d\omega}{dt}$	angular velocity and acceleration	rad/s, rad/s²	Angular velocity
$\vec{\tau} = \vec{r}\times\vec{F}$	torque as a cross product	N m	Torque
$I = \int r^2\,dm$	rotational inertia by integration	kg m²	Moment of inertia
$I = I_{cm} + Md^2$	inertia about a shifted, parallel axis	kg m²	Moment of inertia
$\sum\tau = I\alpha$	Newton's second law for rotation	N m	Angular acceleration
$\vec{L} = \vec{r}\times\vec{p},\ L = I\omega$	angular momentum of a particle and of a rigid body	kg m²/s	Angular momentum
$K_{rot} = \frac{1}{2}I\omega^2$	kinetic energy of spinning	J	Kinetic energy

Example 2: rotational inertia of a rod about one end

Find I for a uniform rod of mass 0.60 kg and length 1.2 m, pivoted at one end.

Mass per length $\lambda = M/L$ , so $I = \int_0^L \frac{M}{L}x^2\,dx = \frac{1}{3}ML^2$ .

$I = \frac{1}{3}(0.60)(1.2)^2 = \frac{0.864}{3} = 0.288\ \text{kg m}^2$ . Answer: about 0.29 kg m². Check with the parallel axis theorem: $\frac{1}{12}ML^2 + M\left(\frac{L}{2}\right)^2 = \frac{1}{3}ML^2$ .

Oscillations and gravitation

Simple harmonic motion and orbits
Relation	Use	SI unit	Explained on
$x(t) = A\cos(\omega t + \phi)$	position in simple harmonic motion	m	Amplitude
$\omega = \sqrt{\frac{k}{m}},\ T = 2\pi\sqrt{\frac{m}{k}}$	mass on a spring	rad/s, s	Period
$T = 2\pi\sqrt{\frac{\ell}{g}}$	simple pendulum, small angles	s	Period
$T = 2\pi\sqrt{\frac{I}{mgd}}$	physical pendulum, d from pivot to center of mass	s	Frequency
$F_G = \frac{Gm_1m_2}{r^2}$	universal gravitation	N	Gravitation
$v = \sqrt{\frac{GM}{r}}$	speed in a circular orbit	m/s	Orbital velocity

Any system whose equation of motion has the form $\frac{d^2x}{dt^2} = -\omega^2 x$ is simple harmonic, and ω can be read straight off. That is the quickest route to the period of an unfamiliar oscillator.

Exam habits for AP Physics C mechanics formula sheet questions

State the law, then the calculus. Write ΣF = ma or ΣF = dp/dt before you integrate.
Choose dm carefully. For a rod use λ dx, for a disk use σ 2πr dr.
Keep signs in F = −dU/dx. Force points downhill on a U against x graph.
Check the limits. A drag answer should give v = 0 at t = 0 and v → v_T at large t.

If a question says a quantity is proportional to time or position, it is telling you to integrate rather than use the constant acceleration equations.

Frequently asked questions

Is this the exam board's physics C mechanics formula sheet?

No. It is an independent study guide. The official equation tables are provided with the exam, so practise past questions with that version too.

What topics are on the AP Physics C mechanics exam?

In general terms: kinematics, Newton's laws, work, energy and power, linear momentum, rotation and angular momentum, oscillations, and gravitation. Check the current course outline for details.

Where do I get the AP Physics C: Mechanics formula sheet itself?

From the exam board, which publishes the current equation tables in its course materials and provides them in the exam. This page explains the same physics in plain words.

Do I need integrals for AP Physics C mechanics?

Yes. Typical uses are work from a varying force, impulse from a varying force, center of mass and rotational inertia of continuous bodies, and motion with drag.

How does AP Physics C mechanics compare with AP Physics 1?

They cover much of the same mechanics, but AP Physics C uses calculus throughout and goes into more depth on topics such as drag, variable forces and rotational inertia.