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## Measurement and Units

### Formulas

Estimated uncertainty $=\Delta x$

Percent uncertainty $=\frac{\Delta x}{x}\times 100\%$

### Formulas with Summary

#### Uncertainty

The estimated uncertainty is the best guess for the uncertainty of a measurement, which may be limited by the precision of the measuring apparatus. Suppose you measure a board with a ruler whose smallest marks are millimeters and find it to be 4.5 cm. A reasonable estimate of the uncertainty would be $\pm$0.1 cm so that the answer with uncertainty would be 4.5$\pm$ 0.1 cm.

The percent uncertainty is simply the ratio of uncertainty to the measured value multiplied by 100. For example, in the previous example, the percent uncertainty is

$\frac{0.1}{4.5}\times 100%\approx 2%$

here the $\approx$ symbols means "approximately equal to".

#### Significant Figures

The number of significant figures is the number of reliably known digits in a number. For example, the number 143.8 has four significant figures while the number 0.0081 has only two significant figures. When multiplying, dividing, adding or subtracting numbers with different numbers of significant figures, the final answer should contain no more significant figures than the number with the least number of significant figures. However, it is best to keep all significant figures throughout the calculation and round the answer at the end. For example, if we measure a rectangle to have base 18.9 cm and height 16.1 cm, the area should be quoted as 304 cm2, not 304.29 cm2. This is because the numbers we started with only had three significant figures so the final answer can have no more than three.

#### Scientific Notation

Numbers are commonly expressed in scientific notation. In this convention, there is only one digit to the left of the decimal place. For example, the number 18900 would be 1.89 x 104 and the number .00000037 would be 3.7 x 10-7

## Linear Motion

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: x=position, v=velocity, a=acceleration

$v=v_0+at\ \ \ _{:bc:}$

$x=x_0+v_0t+\frac{1}{2}at^2\ \ \ _{:bc:}$

$v^2=v_0^2+2a(x-x_0)\ \ \ _{:bc:}$

$\text{If}\ at^2+bt+c=0\ \text{then}\ t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ (quadratic formula)

### Formulas with Summary

#### Position and Displacement

In one dimension, the position of an object can be specified by one number, x. The displacement of an object is its change in position. If an object is at position x1 at time t1 and position x2 at time t2 then its displacement is given by

$\Delta x = x_2-x_1$

The displacement of an object may be different from the distance traveled, which is always positive. For example, if a dog walks all the way around a circle of circumference 100 meters its total distance traveled will be 100 meters, but its displacement will be zero since it returned to its starting position.

#### Velocity

The average speed refers to how fast an object moved, regardless of direction. It is defined by

$\text{average speed}=\frac{\text{distance traveled}}{\text{time elapsed}}$

because distance traveled and time elapsed are always positive numbers, the speed is always positive. In contrast, the average velocity depends on the direction of displacement as well as its magnitude, and can be positive or negative. It is defined by

$\text{average speed}=\frac{\text{displacement}}{\text{time elapsed}}$

In symbols this is given by

$\bar{v}=\frac{x_2-x_1}{t_2-t_1}$

The instantaneous velocity is the velocity an object has at a particular instant in time. This can be found by calculating the average velocity over an infinitesimal time interval, or equivalently, differentiating the position with respect to time:

$v=\text{lim}_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}$

#### Acceleration

The average acceleration is defined as the change in velocity divided by the change in time

$\text{average acceleration}=\frac{\text{change in velocity}}{\text{time elapsed}}$

In symbols this is given by

$\bar{a}=\frac{v_2-v_1}{t_2-t_1}$

The instantaneous acceleration is defined analogously to the instantaneous velocity as

$a=\text{lim}_{\Delta t\rightarrow 0}\frac{\Delta v}{\Delta t}=\frac{dv}{dt}=\frac{d^2x}{dt^2}$

#### Motion at Constant Acceleration

When an object moves with constant acceleration its position and velocity can be simply related to the acceleration and elapsed time. These relationships can be expressed as the familiar equations of one dimensional motion listed in the formulas section above. Suppose that an object is initially at position x0 and has initial velocity v0 and accelerates with acceleration a. After a time t the velocity is given by

$v=v_0+at$

Its position after a time t is given by

$x=x_0+v_0t+\frac{1}{2}at^2$

These formulas can be combined to eliminate the t variable in order to yield an expression which relates the velocity to the displacement and acceleration:

$v^2=v_0^2+2a(x-x_0)$

Together, these equations can be applied to solve a wide variety of one dimensional motion problems with constant acceleration.

#### Falling Motion

A particularly important class of 1-D motion problems study the motion of an object under the influence of gravity. In these problems, the effects of air resistance are often neglected and the magnitude of the acceleration a is none other than g=9.8 m/s2, the acceleration of gravity at the earth's surface. Although g is usually defined as a positive number, one must remember that if the coordinate system is oriented so the positive direction is up, then the acceleration of the object is actually negative, that is, a=-9.8 m/s2.

## Vectors and Projectile Motion

### Formulas

Key: vi=component of velocity in the i direction, ai=component of acceleration in the i direction, $\theta$= angle projectile makes with the horizontal.

$v_x=v\cos\theta$

$v_y=v\sin\theta$

$\tan\theta=v_y/v_x$

$v=\sqrt{v_x^2+v_y^2}$

$v_x=v_{x0}+a_xt$

$x=x_0+v_{x0}t+\frac{1}{2}a_xt^2$

$v_x^2=v_{x0}^2+2a_x(x-x_0)$

$v_y=v_{y0}+a_yt$

$y=y_0+v_{y0}t+\frac{1}{2}a_yt^2$

$v_y^2=v_{y0}^2+2a_y(y-y_0)$

### Formulas with Summary

#### Vectors and Scalars

In physics, many objects are vectors which means they have both a magnitude and direction. Examples of vectors are velocity, force and electric field. Objects which only have a magnitude are called scalars. Examples of scalars are mass, temperature and speed.

Vectors can be represented as arrows. Two vectors, V1 and V2, can be added graphically by moving the end of V2 to the tip of V1 and drawing a new arrow from the end of V1 to the tip of V2 (Fig. 3.1)

We can subtract V2 from V1 by adding (-V2). We find the negative of a vector simply by reversing the direction of the arrow (Fig 3.2)

In addition to representing and adding vectors graphically, we can also represent and add them by components. We represent a vector in component form as a list of numbers in parentheses that specify its (x,y,z) coordinates. In two dimensions we have V1=(v1x,v1y) and V2=(v2x,v2y). Then to add the vectors we simply add their components:

$\vec{V}_1+\vec{V}_2=(v_{1x}+v_{2x}, v_{1y}+v_{2y})$

#### Vector Decomposition

In many physics problems, however, a vector is not given to us in components, but rather, as a magnitude and direction. In this case, we have to use trigonometry to find the components before we can add them. This is particularly common in projectile motion where an object is usually specified to have an initial speed of, say, 20 m/s and an angle of 30° with respect to the horizontal. If | V | is the magnitude of the vector and θ is the angle it makes with the horizontal (Fig 3.3), then in two dimensions we have

$v_x=|V|\cos\theta$

$v_y=|V|\sin\theta$

We can also find the magnitude and angle of a vector from its components by using the relations

$V^2=v_x^2+v_y^2$

$\tan\theta=\frac{v_y}{v_x}$

#### Projectile Motion

In projectile motion, the linear motion equations are valid for each coordinate separately. In three dimensions, this gives three identical sets of equations:

$v_x=v_{x0}+a_xt$

$x=x+v_{x0}t+\frac{1}{2}a_xt^2$

$v_x^2=v_{x0}^2+2a_x(x-x_0)$

The equations for $y$ and $z$ are obtained by making the substitutions $x\rightarrow y$ or $x\rightarrow z$ respectively. Vectors can also be specified by a magnitude and angles. Let $\theta$ be the angle measured with respect to horizontal. In two dimensions we have:

$v_x=v\cos\theta$

$v_y=v\sin\theta$

$v=\sqrt{v_x^2+v_y^2}$

## Newton's Laws

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: F=force, FG=force of gravity, Ffric=force of friction, FN=normal force.

$\sum\mathbf{F}=\mathbf{F_{net}}=m\mathbf{a}\ \ \ _{:bc:}$

$\mathbf{F}_G=m\mathbf{g}$

$\mathbf{F}_{fric}\le\mu \mathbf{F}_N\ \ \ _{:bc:}$

$\mathbf{F}_N=mg\cos\theta$

$\mathbf{F}_{AB}=-\mathbf{F}_{BA}$

### Formulas with Summary

Let $\mathbf{F}_{fric}$= force of friction, $\mu$ = coefficient of kinetic friction and $F_N$ = normal force. When the object is sliding $\mu=\mu_k$ is the coefficient of kinetic friction and when the object is at rest $\mu=\mu_s$ is the coefficient of static friction. Note that we always have $\mu_k\le\mu_s$ which means it takes the same or more force to get an object sliding than it does to keep it sliding.

## Energy and Work

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: W=work, KE=kinetic energy, PE=potential energy, P=power, E=energy.

$W=F\Delta r\cos\theta\ \ \ _{:b:}$

$W=\int_C\mathbf{F}\cdot d\mathbf{r}\ \ \ _{:c:}$

$KE=\frac{1}{2}mv^2\ \ \ _{:bc:}$

$PE_{grav}=mgh\ \ \ _{:bc:}$

$PE_{spring}=\frac{1}{2}kx^2\ \ \ _{:b:}$

$P_{avg}=\frac{W}{\Delta t}\ \ \ _{:b:}$

$P=\frac{dW}{dt}\ \ \ _{:c:}$

$P=Fv\cos\theta\ \ \ _{:b:}$

$P=\mathbf{F}\cdot\mathbf{v}\ \ \ _{:c:}$

$E_{initial}=E_{final}$

### Formulas with Summary

The work done on an object that is displaced a distance $\Delta r$ by a force $F$ which makes an angle $\theta$ with the direction of motion is given by

$W=F\Delta r\cos\theta\ \ \ _{:b:}$

For an object moved along a curve $C$, this expression can be expressed as a line integral

$W=\int_C\mathbf{F}\cdot d\mathbf{r}\ \ \ _{:c:}$

## Momentum

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: p=momentum, J=impulse, xCM=position of center of mass.

$\mathbf{p}=m\mathbf{v}\ \ \ _{:bc:}$

$\mathbf{J}=\mathbf{F}\Delta t=\Delta \mathbf{p}\ \ \ _{:b:}$

$\mathbf{J}=\int\mathbf{F}dt=\Delta \mathbf{p}\ \ \ _{:c:}$

$\mathbf{F}=\frac{d\mathbf{p}}{dt}\ \ \ _{:c:}$

$x_{CM}=(m_ax_a+m_bx_b+...)/(m_a+m_b+...)$

$\sum E_i=\sum E_f$ and $\ \sum p_i=\sum p_f$ (elastic collision)

$\sum p_i=\sum p_f$ (inelastic collision)

## Rotational Motion

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: ac=centripetal acceleration, T=period, f=frequency, θ=angular position, ω=angular velocity, α=angular acceleration.

$a_c=\frac{v^2}{r}=r\omega^2\ \ \ _{:bc:}$

$T=\frac{1}{f}\ \ \ _{:b:}$

$v=r\omega\ \ \ _{:c:}$

$\omega = \omega_0 +\alpha t\ \ \ _{:c:}$

$\theta=\theta_0+\omega_0t+\frac{1}{2}\alpha t^2\ \ \ _{:c:}$

$\omega=\frac{\Delta \theta}{\Delta t}$

$\alpha =\frac{\Delta \omega}{\Delta t}$

$\omega = 2\pi f$

$\omega^2 = \omega_0^2+2\alpha\Delta\theta$

## Torque and Angular Momentum

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: τ=torque, I=moment of inertia, rCM=position of center of mass, L=angular momentum.

$\tau = rF\sin\theta\ \ \ _{:b:}$

$\mathbf{\tau}=\mathbf{r}\times\mathbf{F}\ \ \ _{:c:}$

$\sum\mathbf{\tau} =\mathbf{\tau}_{net}=I\mathbf{\alpha}\ \ \ _{:c:}$

$I=\int r^2dm=\sum mr^2\ \ \ _{:c:}$

$\mathbf{r}_{CM}=\sum m\mathbf{r}/\sum m\ \ \ _{:c:}$

$\mathbf{L}=\mathbf{r}\times\mathbf{p}=I\mathbf{\omega}\ \ \ _{:c:}$

## Gravitation

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: FG=force of gravity, UG=gravitational potential energy.

$F_G=-\frac{Gm_1m_2}{r^2}\ \ \ _{:b:}$

$U_G=-\frac{Gm_1m_2}{r}\ \ \ _{:b:}$

## Static Equilibrium and Elasticity

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: Fs=spring force, k=spring constant, x=displacement, ΔL=change in length, ΔV=change in volume, A=area, ΔP=change in pressure, E=elastic (Young's) modulus, B=bulk modulus.

Static equilibrium means $\sum\mathbf{F}=0$ and $\sum\mathbf{\tau}=0$

$F_s=-kx$

$\Delta L=\frac{1}{E}\frac{F}{A}L_0$

$\Delta V=-\frac{1}{B}V_0\Delta P$

## Fluids

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: ρ=density, P=pressure, g=acceleration of gravity, h=height, Fbuoy=buoyant force, A=area, v=velocity, y=vertical position.

$\rho=m/V\ \ \ _{:b:}$

$P=F/A\ \ \ _{:b:}$

$P=P_0+\rho g h\ \ \ _{:b:}$

$P_{out}=P_{in}\ \ \ _{:b:}$

$F_{buoy}=\rho_{liquid}V_{object}g\ \ \ _{:b:}$

$A_1v_1=A_2v_2\ \ \ _{:b:}$

$P+\rho g y+\frac{1}{2}\rho v^2=constant\ \ \ _{:b:}$

$v_1=\sqrt{2g(y_2-y_1)}$

## Vibrations and Waves

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: Fs=spring force, Ts/p=spring/pendulum period, T=period, f=frequency, v=velocity of wave, λ=wavelength, ESHO=energy of simple harmonic oscillator, Fpend=restoring force of pendulum.

$F_s=-kx\ \ \ _{:b:}$

$T_s=2\pi\sqrt{m/k}\ \ \ _{:b:}$

$T_p=2\pi\sqrt{l/g}\ \ \ _{:b:}$

$T=1/f\ \ \ _{:b:}$

$v=f\lambda\ \ \ _{:b:}$

$E_{SHO}=\frac{1}{2}mv^2+\frac{1}{2}kx^2$

$x(t)=A\cos(\omega t+\theta_0)$

$v(t)=-\omega A\sin(\omega t+\theta_0)$

$a(t)=-\omega^2A\cos(\omega t+\theta_0)$

$F_{pend}=-mg\sin\theta\approx -mg\theta$

## Sound

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: v=velocity of sound, β=sound level, λ=wavelength, f'=doppler shifted frequency.

$v=f\lambda\ \ \ _{:b:}$

$\beta \text{(in dB)} =10\log \frac{I}{I_0}$

$f_{beats}=f_1-f_2$

$\lambda_1=2L$ (open tube)

$f_n=nf_1$ (open tube)

$\lambda_1=4L$ (closed at one end)

$f_n=(2n+1)f_1$ (closed at one end)

$f'=f/(1-\frac{v_{source}}{v_{sound}})$ (source moving toward observer)

$f'=f/(1+\frac{v_{source}}{v_{sound}})$ (source moving away from observer)

## Temperature and Thermal Expansion

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: T=temperature, ΔL=change in length, α=coefficient of linear expansion, ΔV=change in volume, β=coefficient of volume expansion, n=number of mols, R=universal gas constant, kB=Boltzmann's constant, KEavg=average kinetic energy of gas molecules at temperature T, vrms=root mean square velocity.

$T(K)=T(C^{\circ})+273.15$

$\Delta L=\alpha L_0\Delta T\ \ \ _{:b:}$

$\Delta V=\beta V_0\Delta T$

$PV=nRT=Nk_BT\ \ \ _{:b:}$

$KE_{avg}=\frac{3}{2}k_BT$ (ideal gas) $\ \ _{:b:}$

$v_{rms}=\sqrt{3RT/M}=\sqrt{3k_BT/\mu}\ \ \ _{:b:}$

$W=-P\Delta V$ (work done on a system) $\ \ _{:b:}$

## Heat

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: U=internal energy, Q=heat, m=mass, c=specific heat, Lf=heat of fusion, LV=heat of vaporization, H=rate of heat transfer, k thermal conductivity, A=area, L=length, I=intensity of radiation emitted by blackbody, e=emissivity, σ=Stefan-Boltzmann constant.

$U=\frac{3}{2}nRT$

$Q=mc\Delta T$

$Q=mL_f$ (melting/freezing)

$Q=mL_V$ (vaporizing/condensing)

$H=(kA\Delta T)/L$ (Heat transfer) $\ \ \ _{:b:}$

$I=e\sigma AT^4$

## Thermodynamics

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: ΔU=change in internal energy, e=efficiency, TH=temperature of hot reservoir, TC=temperature of cold reservoir.

$\Delta U=Q+W\ \ \ _{:b:}$

$W=-P\Delta V\ \ \ _{:b:}$ (work done on a system)

$e=|W|/|Q_H|\ \ \ _{:b:}$

$e=(T_H-T_C)/T_H\ \ \ _{:b:}$

$\Delta Q=T\Delta S$ (T is constant)

## Electric Charge Forces and Fields

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: q=charge, r =separation, k=Coulomb's law constant, ε0=vacuum permittivity, Q=total charge enclosed.

$F=\frac{kq_1q_2}{r^2}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\ \ \ _{:bc:}$

$\mathbf{E}=\frac{\mathbf{F}}{q}\ \ \ _{:bc:}$

$\oint \mathbf{E}\cdot d\mathbf{A}=Q/\epsilon_0\ \ \ _{:c:}$

## Electric Potential and Capacitance

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: UE=electric potential energy, V=voltage, E=electric field, C=Capacitance, Q=charge on capacitor, A=area of capacitor plate, d=separation of plates, UC=energy stored in capacitor.

$U_E=qV=\frac{kq_1q_2}{r}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\ \ \ _{:bc:}$

$E_{avg}=-V/d\ \ \ _{:b:}$

$E=-\frac{dV}{dr}\ \ \ _{:c:}$

$V=k(\frac{q_1}{r_1}+\frac{q_2}{r_2}+\frac{q_3}{r_3}+...)\ \ \ _{:b:}$

$V=\frac{1}{4\pi\epsilon_0}\sum\frac{q_i}{r_i}\ \ \ _{:c:}$

$C=Q/V\ \ \ _{:bc:}$

$C=\epsilon_0A/d\ \ \ _{:bc:}$

$U_C=\frac{1}{2}QV=\frac{1}{2}CV^2\ \ \ _{:bc:}$

## Currents and Resistance

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: I=current, R=resistance, P=power, ρ=resistivity, L=length, A=area, N=number of charge carriers per unit volume, vd=drift velocity, J=current density.

$I_{avg}=\frac{\Delta Q}{\Delta t}\ \ \ _{:b:}$

$I=\frac{dQ}{dt}\ \ \ _{:c:}$

$V=IR\ \ \ _{:bc:}$

$P=IV\ \ \ _{:bc:}$

$R=\rho L/A\ \ \ _{:bc:}$

$I=Nev_dA\ \ \ _{:c:}$

$\mathbf{E}=\rho\mathbf{J}\ \ \ _{:c:}$

## Circuits

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: R=resistance, C=capacitance, Vc=voltage on capacitor, Qc=charge on capacitor, τ=time constant.

$R_{series}=R_1+R_2+R_3+...\ \ \ _{:b:}$

$R_{series}=\sum_iR_i\ \ \ _{:c:}$

$1/R_{parallel}=1/R_1+1/R_2+1/R_3+...\ \ \ _{:b:}$

$1/R_{parallel}=\sum_i 1/R_i\ \ \ _{:c:}$

$C_{parallel}=C_1+C_2+C_3+...\ \ \ _{:b:}$

$C_{parallel}=\sum_iC_i\ \ \ _{:c:}$

$1/C_{series}=1/C_1+1/C_2+1/C_3+...\ \ \ _{:b:}$

$1/C_{series}=\sum_i 1/C_i\ \ \ _{:c:}$

$V_C=V(1-e^{-t/RC})$ (charging)

$Q_C=Q_0(1-e^{-t/RC})$ (charging)

$V_C=V_0e^{-t/RC}$ (discharging)

$Q_C=Q_0e^{-t/RC}$ (discharging)

$\tau=RC$ (time constant)

## Magnetism

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: q=charge, FB=magnetic force, v=velocity, B=magnetic field, θ=angle, L=length, μ0=vacuum permeability, r =separation, n=number of loops of wire per unit length, l=length of one loop of wire, I=current, τ=torque on current loop, N=number of loops, A=area.

$F_B=qvB\sin\theta\ \ \ _{:b:}$

$\mathbf{F}_B=q\mathbf{v}\times\mathbf{B}\ \ \ _{:c:}$

$F_B=BIL\sin\theta\ \ \ _{:b:}$

$\mathbf{F}=\int Id\mathbf{l}\times\mathbf{B}\ \ \ _{:c:}$

$\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I\ \ \ _{:c:}$

$B=\frac{\mu_0I}{2\pi r}\ \ \ _{:b:}$

$d\mathbf{B}=\frac{\mu_0}{4\pi}\frac{Id\mathbf{l}\times\mathbf{r}}{r^3}\ \ \ _{:c:}$

$B_s=\mu_0nI/l\ \ \ _{:c:}$

$\tau=NIAB\sin\theta$

## Electromagnetic Induction

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: φm=magnetic flux, ε=EMF, v=velocity, VP/S=voltage on primary/secondary coil, NP/S=turns of wire in primary/secondary coil, UL=energy stored in an inductor, L=inductance.

$\phi_m=BA\cos\theta\ \ \ _{:b:}$

$\phi_m=\int\mathbf{B}\cdot d\mathbf{A}\ \ \ _{:c:}$

$\mathcal{E}_{avg}=-\frac{\Delta\phi_m}{\Delta t}\ \ \ _{:b:}$

$\mathcal{E}=\oint\mathbf{E}\cdot d\mathbf{l}=-\frac{d\phi_m}{dt}\ \ \ _{:c:}$

$\mathcal{E}=Blv\ \ \ _{:b:}$

$\frac{V_S}{V_P}=\frac{N_S}{N_P}$ (Transformer equation)

$\mathcal{E}=-L\frac{dI}{dt}\ \ \ _{:c:}$

$U_L=\frac{1}{2}LI^2\ \ \ _{:c:}$

## Electromagnetic Waves

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: B=magnetic field, J=current density, E=electric field, UEM=energy stored in electromagnetic field, $\bar{I}$=average intensity, c=speed of light, P=radiation pressure.

$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$

$v=f\lambda\ \ \ _{:b:}$

$U_{EM}=\frac{1}{2}\epsilon_0E^2+\frac{1}{2\mu_0}B^2=\epsilon_0E^2$

$\bar{I}=\frac{1}{2}\epsilon_0cE_0^2$

$P=\frac{\bar{I}}{c}$ (total absorption)

$P=\frac{2\bar{I}}{c}$ (total reflection)

## Reflection and Refraction

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: θ=angle, f=focal length, R=radius of curvature, di/o=distance to image/object, hi/o=height of image/object, m=magnification, n=index of refraction, θC=critical angle.

$\theta_{incident}=\theta_{reflected}$

$f=\frac{R}{2}\ \ \ _{:b:}$

$\frac{1}{d_i}+\frac{1}{d_o}=\frac{1}{f}\ \ \ _{:b:}$

$m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}\ \ \ _{:b:}$

$n=\frac{c}{v}\ \ \ _{:b:}$

$n_1\sin\theta_1=n_2\sin\theta_2\ \ \ _{:b:}$

$\sin\theta_C=\frac{n_2}{n_1}\ \ \ _{:b:}$

## Interference and Diffraction

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: d=distance between slits, λ=wavelength, m=an integer, D=width of single slit, I=intensity of light, θp=polarizing angle.

$d\sin\theta = m\lambda$ (constructive) $\ \ \ _{:b:}$

$d\sin\theta=(m+\frac{1}{2})\lambda$ (destructive)

$D\sin\theta=m\lambda,\ m\ne0$ (single slit minima)

$d \sin\theta=m\lambda$ (diffraction grating maxima)

$I_{out}=I_{in}\cos^2\theta$ (polarization)

$\tan\theta_p=n_2/n_1$

## Optical Instruments

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: M=magnifying power, f=focal length, N=eye's near point (usually use N=25 cm), fo/e focal length of objective/eyepiece, θm=minimum resolvable angular separation, D=diameter of telescope.

$M=\frac{N}{f}$ (magnifying glass; eye focused at infinity)

$M=\frac{N}{f}+1$ (magnifying glass; eye focuses at near point N)

$M=-\frac{f_o}{f_e}$ (telescope)

$\theta_m=1.22\lambda/D$

## Special Relativity

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: γ=relativistic factor, Δt0=separation of time in rest frame, L0=length in rest frame, p=momentum, m0=rest mass, c=speed of light, E=energy, v3= the sum of velocities v1 and v2 when added relativistically.

$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$

$\Delta t=\gamma\Delta t_0$

$L=L_0/\gamma$

$p=\gamma m_0 v$

$E_0=m_0c^2$

$E=\gamma m_0c^2$

$E^2=p^2c^2+m_0^2c^4$

$v_3=\frac{v_1+v_2}{1+v_1v_2/c^2}$

## Quantum Theory and the Atom

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: E=energy, h=Planck's constant, f=frequency, KE=kinetic energy of ejected electron, φ=work function, p=momentum, θ=Compton scattering angle, R=Rydberg constant, n=an integer, Z=atomic number, Δx=uncertainty in position.

$E=hf\ \ \ _{:b:}$

$KE=hf-\phi$ (photoelectric effect) $\ \ \ _{:b:}$

$\lambda = h/p\ \ \ _{:b:}$

$\lambda'=\lambda+\frac{h}{m_0c}(1-\cos\theta)$

$\frac{1}{\lambda}=R(\frac{1}{2^2}-\frac{1}{n^2}),\ n\ge3$ (Balmer series of H)

$\frac{1}{\lambda}=R(\frac{1}{1^2}-\frac{1}{n^2}),\ n\ge2$ (Lyman series of H)

$\frac{1}{\lambda}=R(\frac{1}{3^2}-\frac{1}{n^2}),\ n\ge4$ (Paschen series of H)

$E_n=-(13.6 eV)\frac{Z^2}{n^2}$

$h/(2\pi)\le\Delta x\Delta p$

$h/(2\pi)\le\Delta E\Delta t$

### Formulas

Note: The $_{:b:}\ ,\ _{:c:}$ and $_{:bc:}$ symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets.

Key: E=energy, m=mass, N=number of radioactive nuclei, λ=decay constant, T1/2=half-life.

$E= mc^2\ \ \ _{:b:}$

$\alpha$ particle = 2 neutrons + 2 protons

$\beta$-decay: neutron $\rightarrow$ proton+electron+neutrino

$N=N_0e^{-\lambda t}$

$T_{1/2}=(\ln 2)/\lambda$

# References

## Fundamental Constants

Note: This material is included in the AP physics B and C equation sheets.

Quantity Symbol Approximate Value
Proton mass mp 1.67(26) x 10-27 kg
Neutron mass mn 1.67(49) x 10-27 kg
Electron mass me 9.11 x 10-31 kg
Avogadro's number N0 6.02 x 1023 mol-1
Universal gas constant R 8.31 J/(mol $\cdot$ K)
Boltzmann's constant kB 1.38 x 10-23 J/K
Electron charge e 1.6 x 10-19 C
Electron volt eV 1.6 x 10-19 J
Speed of light in vacuum c 3.00 x 108 m/s
Gravitational constant G 6.67 x 10-11 m3/(kg$\cdot$ s2)
ag at Earth's surface g 9.8 m/s2
Atomic mass unit u 1.66 x 10-27 kg = 931 MeV/c2
Planck's constant h 6.63 x 10-34 J $\cdot$ s= 4.14 x 10-15 eV$\cdot$s
Planck's constant times c hc 1.99 x 10-25 J$\cdot$m= 1.24 x 103 eV$\cdot$nm
Vacuum permittivity ε0 8.85 x 10-12 C2/(N $\cdot$ m2)
Coulomb's law constant k=1/(4πε0) 9.0 x 109 N$\cdot$m2/C2
Vacuum permeability μ0 4π x (T$\cdot$m)/A
Magnetic constant k'=μ0/4π 1 x (T$\cdot$m)/A
Atmospheric pressure atm 1 x 105 Pa
Stefan-Boltzmann constant σ 5.67 x 10-8 W/(m2$\cdot$K4)
Rydberg constant R 1.0974 x 107 m-1

## Unit Symbols and Definitions

Some of this information is on the AP physics B and C equation sheets.

Quantity Unit Abbreviation Base Units
Length meter m m
Mass kilogram kg kg
Time second s s
Force newton N kg$\cdot$ m/s2
Energy and Work joule J kg$\cdot$ m2/s2
Power watt W kg$\cdot$ m2/s3
Pressure pascal Pa kg/( m$\cdot$ s2)
Frequency hertz Hz s-1
Temperature kelvin K K
Current ampere A A
Electrical charge coulomb C A$\cdot$ s
Electric potential volt V kg$\cdot$ m2/(A$\cdot$ s3)
Electric resistance ohm $\Omega$ kg$\cdot$ m2/(A2$\cdot$ s3)
Capacitance farad F A2$\cdot$ s4/(kg$\cdot$ m2)
Magnetic field tesla T kg/(A$\cdot$ s2)
Magnetic flux weber Wb kg$\cdot$ m2/(A$\cdot$ s2)
Inductance henry H kg$\cdot$ m2/(A2$\cdot$ s2)
Energy electron-volt eV kg$\cdot$ m2/s2

## Geometry and Trigonometry

This material can be found on both the AP physics B and C equation sheets.

Key: $A$=area, $C$=circumference, $V$=volume, $S$=surface area, $b$=base, $h$=height, $l$=length, $w$=width, $r$=radius.

### Rectangle

$A=bh$

### Triangle

$A=\frac{1}{2}bh$

### Circle

$A=\pi r^2$

$C=2\pi r$

### Rectangular Solid

$V=lwh$

### Cylinder

$V=\pi r^2l$

$S=2\pi rl+2\pi r^2$

### Sphere

$V=\frac{4}{3}\pi r^3$

$S=4\pi r^2$

### Right Triangle

$a^2+b^2=c^2$

$\sin\theta=a/c$

$\cos\theta = b/c$

$\tan\theta = a/b$

## Calculus

This material is found on the AP physics C equation sheet only.

$\frac{df}{dx}=\frac{df}{du}\frac{du}{dx}$

$\frac{d}{dx}(x^n)=nx^{n-1}$

$\frac{d}{dx}(e^{a\cdot x})=ae^{a\cdot x}$

$\frac{d}{dx}(\ln x)=\frac{1}{x}$

$\frac{d}{dx}(\sin x)=\cos x$

$\frac{d}{dx}(\cos x)=-\sin x$

$\int x^n\ dx=\frac{1}{n+1} x^{n+1},\ n\ne -1$

$\int e^{a\cdot x}\ dx=\frac{1}{a}e^x$

$\int\frac{dx}{x}=ln|x|$

$\int\cos x\ dx=\sin x$

$\int\sin x\ dx=-\cos x$

## Prefixes

Note: This material is included in the AP physics B and C equation sheets.

Factor Prefix Symbol
109 giga G
106 mega M
103 kilo k
10-2 centi c
10-3 milli m
10-6 micro μ
10-9 nano n
10-12 pico p

## Unit Conversions

### Length

1 in = 2.54 cm

1 cm = 0.3937 in

1 ft = 30.48 cm

1 m = 39.37 in = 3.281 ft

1 mi = 5280 ft = 1.609 km

1 fermi = 1 femtometer(fm)=10-15 m

1 Angstrom = 10-10 m

1 light-year (ly)=9.461 x 1015 m

1 parsec = 3.26 ly=3.06 x 1016 m

### Volume

1 liter(L)=1000 mL =1000 cm3=1.0 x 10-3 m3

1 gal(U.S) = 4qt (U.S) = 3.785 L

1 quart (U.S.)= 2 pints (U.S.)=586 mL

1 pint (British) = 1.20 pints (U.S.) = 586 mL

1 m3=35.31 ft3

### Speed

1 mi/h = 1.467 ft/s = 1.609 km/h = .447 m/s

1 km/h = .278 m/s = .621 mi/h

1 ft/s = 0.305 m/s = 0.682 mi/h

1 m/s = 3.281 ft/s = 3.600 km/h = 2.237 mi/h

1 knot = 1.151 mi/h = 0.5144 m/s

### Time

1 day = 8.64 x 104 s

1 year =3.156 x 107 s

### Mass

1 atomic mass unit (u) =1.6605 x 10-27 kg

1 kg = 0.0685 slug

[1 kg has a weight of 2.20 lb where g = 9.80 m/s2 ]

### Force

1 lb = 4.45 N

1 N = 105 dyne = 0.225 lb

### Energy and Work

1 J = 107 ergs = 0.738 ft$\cdot$lb

1 kcal =4.186 x 103 J = 3.97 BTU

1 eV =1.602 x 10-19 J

1 kWh =3.60 x 106 J

### Power

1 W = 1J/s=0.738 ft$\cdot$lb/s = 3.42 BTU

1 hp =550 ft$\cdot$lb/s=746 W

### Pressure

1 atm = 1.013 bar = 1.013 x 105 N/m2 =760 torr

1 Pa = 1 N/m2

For more help with physics check out the app Physics X on our website Knowvio

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