Homework Help: Introductory Physics Formulary

[Doc Al]

Homework Statement

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Relevant Equations

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Introductory Physics Formulary

This sticky is meant to be a repository for some basic equations commonly used in elementary physics that might prove handy to homework posters and homework helpers. The equations will be in Latex format so you can quickly copy them. (For more about Latex, go here: Learn LaTeX for Math Equations.)

This is not meant to be an exhaustive reference list of all physics formulae, just a handy guide and time saver.

The posts are not intended as tutorials: You'd better know what an equation means before you use it. (Although I will include a few notes.)

If you have suggestions or corrections, just send me a PM. I'll be adding posts on various topics as time permits.

- Doc Al

Quick Links:
Basic Equations of 1-D Kinematics
Basic Equations of Special Relativity
Newton's laws / Particle Dynamics
Linear Momentum and Collisions
Basic Equations of Electromagnetism
Work, Mechanical Energy, and Power

 

[Doc Al]

Basic Equations of 1-D Kinematics

Basic Equations of 1-D Kinematics

General:

average velocity:
$$v_{ave} = \Delta x / \Delta t$$

average acceleration:
$$a_{ave} = \Delta v / \Delta t$$

Uniform Acceleration:

$$v_{ave} = (v_i + v_f)/2$$

Three key variables (displacement, time, velocity) lead to three key relationships relating each pair:

velocity & time:
$$v = v_0 + a t$$

displacement & time:
$$x = x_0 + v_0 t + (1/2) a t^2$$

velocity & displacement:
$$v^2 = v_0^2 + 2 a \Delta x$$

 

[Doc Al]

Basic Equations of Special Relativity

Basic Equations of Special Relativity

Lorentz transformations:

$$\Delta x = \gamma(\Delta x' + v\Delta t')$$
$$\Delta t = \gamma(\Delta t' + v\Delta x'/c^2)$$

$$\Delta x' = \gamma(\Delta x - v\Delta t)$$
$$\Delta t' = \gamma(\Delta t - v\Delta x/c^2)$$

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

Viewed from the unprimed frame, the primed frame is moving with speed v in the +x direction.


The invariant spacetime interval:

$$(\Delta s)^2 = c^2 (\Delta t)^2 - (\Delta x)^2 = c^2 (\Delta t')^2 - (\Delta x')^2$$


Behavior of moving rods and clocks:

Length contraction: Moving rods shrink along their direction of motion by a factor of $\gamma$:
$$L = L_0/\gamma$$

Time dilation: Moving clocks run slow by a factor of $\gamma$:
$$\Delta T = \gamma \Delta T_0$$

Clock desynchronization: Moving clocks, synchronized in their rest frame but separated by a distance D along their direction of motion, are not synchronized in the stationary frame; The front clock lags the rear clock by an amount:
$$T = Dv/c^2$$



Addition of (parallel) velocities:

Low-speed (Galilean) addition of velocities:
$$V_{a/c} = V_{a/b} + V_{b/c}$$

Relativistic addition of velocities:
$$V_{a/c} = \frac{V_{a/b} + V_{b/c}}{1 + (V_{a/b} V_{b/c})/c^2}$$

 

[Doc Al]

Newton's laws / Particle Dynamics

Newton's laws / Particle Dynamics

Newton's 2nd Law:

$$\vec{F}_{net} = \Sigma \vec{F} = m \vec{a}$$

Newton's 3rd Law:

Whenever body A exerts a force on body B, body B exerts an equal and opposite force on body A:

$$\vec{F}_{A/B} = - \vec{F}_{B/A}$$

Note that these forces (called "3rd Law pairs") act on different bodies and thus never "cancel" or produce equilibrium.

Equilibrium condition:

$$\vec{a} = 0$$
$$\vec{F}_{net} = \Sigma \vec{F} = 0$$

or, in component form:
$$(\vec{F}_{net})_x = \Sigma F_x = 0$$
$$(\vec{F}_{net})_y = \Sigma F_y = 0$$

Some typical forces:​

Friction, which always opposes slipping between the surfaces

Static friction:
$$f_s \leq \mu_s N$$
(Note that static friction is often less than the maximum value.)

Kinetic friction:
$$f_k = \mu_k N$$

Weight (near the Earth's surface):

$$w = m g$$

(Do not confuse real weight, which is the Earth's gravitational force on a mass, with apparent weight, which is the magnitude of the contact force supporting an object. "Weightlessness" is when the apparent weight equals zero.)

 

[Doc Al]

Linear Momentum and Collisions

Linear Momentum and Collisions

Linear Momentum:

$$\vec{p} \equiv m\vec{v}$$

Impulse:

$$\mbox{impulse} \equiv \int \vec{F(t)} dt = \vec{F}_{ave}\Delta t = \vec{F} \Delta t$$

Impulse-Momentum Theorem:

$$\vec{F} \Delta t = \Delta \vec{p} = m(\vec{v}_f - \vec{v}_i)$$

Conservation of Momentum Principle:

If the net external force on a system is zero, the total momentum of the system remains constant.

Simple Collisions:

Momentum is conserved in any collision:

$$m_1\vec{v}_1 + m_2\vec{v}_2 = m_1\vec{v}_1{ '} + m_2\vec{v}_2{ '}$$

Elastic Collisions:

In an elastic collision, mechanical energy (as well as momentum) is conserved:

$$\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$$

Inelastic Collisions:

An inelastic collision is one in which mechanical energy is not conserved (but momentum is conserved);

In a totally inelastic collision, the colliding objects stick together and move with a common velocity:

$$m_1\vec{v}_1 + m_2\vec{v}_2 = (m_1 + m_2)\vec{v}_{f}$$

Special Case: Elastic Collisions in one dimension:

For a perfectly elastic straight-line collision, the relative velocity is reversed during the collision:

$$v_1 - v_2 = v_2' - v_1'$$

Collisions in two dimensions:

Note that Conservation of Momentum yields two equations, one for each coordinate.

Explosions and throwing of objects:

An explosion or throwing of an object can be considered the opposite of a collision; momentum is conserved.

 

[Redbelly98]

Basic Equations of Electromagnetism

Basic Equations of Electromagnetism

Values of constants

$$\begin{equation*} \begin{split} \epsilon_0 & = & & 8.854 \times 10^{-12} \ \frac{C^2}{N \cdot m^2} \\ \\ \frac{1}{4 \pi \epsilon_0} = k & = & & 8.988 \times 10^9 \ N \cdot m^2 / C^2 \\ \\ \mu_0 & = & & 4 \pi \times 10^{-7} \ T \cdot m / A \\ \\ \frac{\mu_0}{2 \pi} & = & & 2 \times 10^{-7} \ T \cdot m / A \\ \\ e & = & & 1.602 \times 10^{-19} \ C \\ \end{split} \end{equation*}$$

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Electrostatics

Coulomb's Law:

$$F = k \ \frac{q_1 q_2}{r^2} \ \mbox{ or } \ \frac{1}{4 \pi \epsilon_0} \ \frac{q_1 q_2}{r^2}$$​

Electric field and potential for a point charge:

$$\begin{equation*} \begin{split} E & = & & k \ \frac{q}{r^2} \ \mbox{ or } \ \frac{1}{4 \pi \epsilon_0} \ \frac{q}{r^2}\\ \\ V & = & & k \ \frac{q}{r} \ \mbox{ or } \ \frac{1}{4 \pi \epsilon_0} \ \frac{q}{r}\\ \\ \end{split} \end{equation*}$$​

Gauss's Law:

$$\sum E_{\perp} \cdot \Delta A = \frac{Q_{enclosed}}{\epsilon_0} \ \text{, } \ \ \ \text{ or } \ \ \ \oint _S \vec{E} \cdot \vec{dA} = \frac{Q_{enclosed}}{\epsilon_0}$$​

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Magnetism

Magnetic force on a moving charge:

$$F = q \ v \ B \ \sin(\theta) \ \text{, } \ \ \ \text{ or } \ \ \ \vec{F} = q \ \vec{v} \times \vec{B}$$​

Magnetic force on a wire of length L:

$$F = L \ I \ B \ \sin(\theta) \ \text{, } \ \ \ \text{ or } \ \ \ \vec{F} = L \ \vec{I} \times \vec{B}$$​

Magnetic field of a long straight wire:

$$B = \frac{\mu_0}{2 \pi} \ \frac{I}{r}$$​

Force-per-length between two parallel wires:

$$\frac{F}{L} = \frac{\mu_0}{2 \pi} \ \frac{I_1 I_2}{d}$$​

Magnetic field inside a solenoid (n = turns-per-length):

$$B = \mu_0 \ n \ I$$​

Torque on a current loop:

$$T = N \ I \ A \ B \ \sin(\theta) = M \ B \ \sin(\theta)\ \text{, } \ \ \ \text{ or } \ \ \ \vec{T} = N \ I \ \vec{A} \times \vec{B} = \vec{M} \times \vec{B}$$​

Ampere's Law:

$$\sum B_{\parallel} \cdot \Delta l = \mu_0 \ I_{enclosed} \ \text{, } \ \ \ \text{ or } \ \ \ \oint \vec{B} \cdot \vec{dr} = \mu_0 \ I_{enclosed}$$​

Faraday's Law of Induction:

$$EMF = -N \ \frac{\Delta \Phi_B}{\Delta t} \ \ \mbox{or} \ \ -N \ \frac{d\Phi_B }{dt}$$​

Biot-Savart Law:

$$\vec{B} = \frac{\mu_0}{4 \pi} \ \int \frac{ I \ \vec{dl}\times \hat{r}}{r^2} \ \text{ or } \ \frac{\mu_0}{4 \pi} \ \int \frac{ I \ \vec{dl}\times \vec{r}}{r^3}$$​

 

[Redbelly98]

Work, Mechanical Energy, and Power

Work, Mechanical Energy, and Power

Work done on an object:
$$W = \vec{F} \cdot \vec{d} \ \text{ or } \ F \ d \cos\theta$$

Kinetic energy:
$$K = \frac{1}{2}m \ v^2 = \frac{p^2}{2m}$$

Work-energy theorem:
$$W_{net} = \Delta K$$

Potential energy:
$$\begin{align} U &= m \ g \ h && \small \text{ gravitational potential near the surface of Earth} \normalsize \\ \\ U &= -\frac{G \ m_1 \ m_2}{r} && \small \text{ gravitational potential for two masses } \normalsize \\ \\ U &= \frac{1}{2} k \ x^2 && \small \text{ elastic potential for a spring} \normalsize \end{align}$$

Conservation of total mechanical energy:
$$K_1 + U_1 = K_2 + U_2$$

Work done by non-conservative forces:
$$W_{nc} = \Delta E = (K_2 + U_2) - (K_1 + U_1)$$

Power:
$$P = \frac{\Delta W}{\Delta t} = \vec F \cdot \vec v \ \text{ or } \ F \ v \cos\theta$$