Straight-line motion with constant acceleration

$$v = v_0 + at$$ $$x = x_0 + v_0t + \frac{1}{2}at^2$$ $$v^2 = v_0^2 + 2a(x-x_0)$$ $$x - x_0 = \frac{v_0 + v}{2}t$$

Straight-line motion with variable acceleration

$$v = v_0 + \int_{0}^{t} a_x dt$$ $$x = x_0 + \int_{0}^{t} v_x dt$$

Freely falling bodies

Free fall is a case of motion with constant acceleration. The magnitude of the acceleration due to gravity is a positive quantity, g. The acceleration of a body in free fall is always downward.

Projectile motion (2D)

Assuming we have no air resistance, we get: $$a_x = 0\\ a_y = -g$$ $$v_x = v_0\cos\alpha_0\\ v_y = - gt + v_0\sin\alpha_0$$ $$x = (v_0\cos\alpha_0)t + x_0\\ y = -\frac{1}{2}gt^2 + (v_0\sin\alpha_0)t + y_0,$$ where $\alpha_0$ is the angle at which the projectile is launched.

To find the maximum height of the movement, set $v_y = 0$ to find the time $t_1$ at which the projectile reaches its maximum height, and then find $y(t_1)$.

To find the range of the movement, set $y = 0$ to find the time $t_2$ at which the projectile reaches the ground, and then find $x(t_2)$.

Uniform and nonuniform circular motion

(Needs verification) The 'rad' subscript denotes that the acceleration always lies on top of the radius (meaning always pointing to the center)

(Needs verification) The 'tan' subscript denotes the tangent acceleration. If an object moving in a circle is speeding up/slowing down, we can decompose the acceleration into two perpendicular vectors; radius and tangent ($a_{rad}$, $a_{tan}$)

$$a_{rad} = \frac{v^2}{R} = \frac{4\pi^2R}{T^2}\\ a_{tan} = \frac{d|\vec{v}|}{dt},$$ where $R$ is the radius of the circular path, and $T$ its period. This gives us $$a = \sqrt{a_{rad}^2 + a_{tan}^2}.$$ In the case of a uniform motion, $a_{tan} = 0$ and thus $a = a_{rad}$. Hence, for a circular uniform motion $$x(t) = R\cos\theta (t)\\ y(t) = R\sin\theta (t)\\ v(t) = \frac{2\pi R}{T}.$$

Weight

$$w = mg$$

Newton's first law

Newton's first law can be stated as: "A body will keep doing what it is doing as long as the sum of the forces acting on it is equal to 0". Or mathematically: $$\sum_{}{}\vec{F} = 0$$

Newton's second law

Newton's second law tells us what happens when the sum of the forces acting on a body is not equal to 0. Mathematically: $$\sum_{}{}\vec{F} = m\vec{a}$$

Newton's third law

Newton's third law states that when a body A exerts a force on another body B, then B will exert a force of equal magnitude in the opposite direction. Or mathematically: $$\vec{F}_{\textrm{A on B}} = -\vec{F}_{\textrm{B on A}}$$

Normal force

The normal force is the force that acts on a body resting on a surface, perpandicular to the surface. The normal force is expressed by: $$\vec{F}_N = m\vec{g}\cos\theta,$$ where $m$ is the body's mass and $\theta$ is the angle at which the surface is. This of course means that for a horizontal surface, $\theta = 0$ and hence $$\vec{F}_N = m\vec{g}.$$ If there is an external force acting on the body, the normal force is given by $$\vec{F}_N = mg \pm F\sin{\theta},$$ where $F$ is the external force, the sign of which is positive when it acts downwards and negative when it acts upwards.

Friction

There are 2 types of friction: kinetic friction and static friction. Kinetic friction is the force that acts on a body that slides on a surface, and its magnitude is given by: $$f_k = \mu_kn,$$ where $n$ is the magnitude of the normal force acting on the body, and $\mu_k$ is the friction coefficiant.

Static friction on the other hand is the force that acts on a body that is not moving on a surface. Its magnitude is anywhere between 0 and its maximum, which is given by: $$(f_s)_\textrm{max} = \mu_sn$$ Usually, $$\mu_s \geq \mu_k$$

The direction of the friction's force vector is in the opposite direction of the movement of the body (essentially "slowing" the body).

Air resistance

A decent approximation of the effects of air resistance is given by $$F = kv^2,$$ where $k$ is a constant.

An object in free fall reaches terminal velocity when its air resistance is equal to the gravitational force acting on it, and hence its acceleration becomes $0$. To find the terminal velocity, simply set $a_y = 0$.

Pulleys

In a system with 1 pulley with 2 masses $m_1$ and $m_2$ on each side of the rope, the tension force of the rope suspending the whole system is given by $$S = 2\frac{m_1m_2}{m_1 + m_2}g,$$ if we disregard the mass of the pulley itself.

Centripetal force

The centripetal force is the force necessary to keep an object in circular motion. Its value is $$F = m\frac{v^2}{R}.$$

Spring force

The force needed to compress or extend a spring by a distance x is given by $$F = -kx,$$ where $k$ is the spring constant. This is referred to as Hooke's law.

Work done by a constant force

A force does work if it produces a displacement. Note that work is a scalar quantity and not a vector. It is defined by: $$ W = \vec{F}\cdot \vec{s} = Fs\cos{\theta},$$ where $\vec{F}$ is the force acting on the object, $\vec{s}$ is the displacement and $\theta$ is the angle between $\vec{F}$ and $\vec{s}$.

Work done by a varying force

When the applied force varies and can be expressed by a function $F_x$, work is given by: $$\int_{x_1}^{x_2} F_x dx$$

Kinetic energy

The kinetic energy of a particle is the amount of work needed to accelerate the particle from rest to a speed $v$. Like work, it is a scalar, and is given by: $$K = \frac{1}{2}mv^2$$

The total amount of work done on a particule is equal to the change of kinetic energy it experiences when going from a speed $v_1$ to a speed $v_2$: $$W_\textrm{tot} = K_2 - K_1$$

Power

Power is the time rate of doing energy. It is a scalar like work and kinetic energy. The average power is given by: $$P = \frac{\Delta W}{\Delta t}$$ The instantaneous power is given by: $$P = \frac{dW}{dt}$$ The rate at which a force $\vec{F}$ does work to make a particle move at velocity $\vec{v}$ is given by: $$P = \vec{F}\cdot\vec{v}$$

Gravitational potential energy

The gravitational potential energy is the energy stored in an object due to its height. It is given by $$E_{pgrav} = mgh,$$ where $m$ is the mass of the object and $h$ its height.

Elastic potential energy

The elastic potential energy is the energy stored in an elastic object due to its stretching or compressing. It is given (in the case of an ideal spring following Hooke's law) by $$E_{pelastic} = \frac{1}{2}kx^2.$$

Mechanical energy

The total mechanical energy is given by $$E_{mec}=E_{pgrav} + E_{pelastic} + K = E_p + K,$$ where $K$ is the object's kinetic energy.

The law of conservation of energy

A force is either conservative or nonconservative. It is nonconservative if its work depends on other factors than kinetic and potential energy, for instance if the path taken affects its work: an example of this is the friction force. However, the following law holds for all forces $$\Delta K + \Delta E_p + \Delta U_{int} = 0,$$ where $U_{int}$ is the internal energy of the body.

Momentum

The momentum of an object refers to the quantity of motion that it has, and is given by $$\vec{p} = m\vec{v}.$$

Impulse

Impulse describes the change in momentum, given by $$J = \Delta p = F \cdot \Delta t,$$ where $F$ is the constant net force acting on the object. If the net force varies with time, we have $$J = \int_{t_1}^{t_2} \sum_{} F dt.$$

Conservation of momentum:

As long as the only forces acting on a system are the forces internal to the system, the total momentum of the system (the sum of all momenta in the system) is constant.

Collision

Center of mass

The center of mass of a body may be found by computing the weighted position of the various masses and dividing by the sum of the masses. Mathematically, the position of the center of mass in centimeters $$r_{cm} = \frac{1}{M_{tot}}\sum_{i=1} r_i m_i,$$ where $r_i$ is the position of the mass, and $m_i$ its mass. Note that the first position is $i=1$. One can do this in several dimensions by computing separately the $r_{cm}$ for every axis.

Rotational kinematics

Given a rigid body rotating about a stationary axis z, then the body's position is described by the angular coordinate $\theta$ and its angular velocity and acceleration are given by $$ \omega_z = \frac{d\theta}{dt}\\ \alpha_z = \frac{d\omega_z}{dt}.$$ If $\alpha_z$ is constant, then the same equations for motion in a straight line apply (although with $\theta$ instead of $x$).

Relating linear and angular kinematics

Given an object travelling at a distance $r$ from the rotation axis, we can derive its speed $v$ and acceleration $a_{tan}, a_{rad}$ from its angular velocity $\omega$ and acceleration $\alpha$ with $$v = r\omega\\ a_{tan} = r\alpha\\ a_{rad}= \omega^2r.$$

Moment of inertia and rotational kinetic energy

The moment of inertia of a body about an axis is a measure of its rotational inertia and is given by $$I = \sum_{i}m_ir_i^2,$$ with $i$ being the particles composing the body. The rotational kinetic energy is given by $$K = \frac{1}{2}I\omega^2.$$

Torque

Torque is a measure of the force that causes an object to acquire angular acceleration. It is given by $$\tau = F \cdot r\sin{\theta},$$ with $r$ being the distance between the axis of rotation $O$ and the point where the force is applied, and $\theta$ the angle between the force vector and $\vec{r}$.

Rotational dynamics

The rotational equivalent of Newton's second law states that $$\sum_{}\tau_z = I\alpha_z,$$ with $I$ the moment of inertia and $\alpha_z$ the angular acceleration.

Combined translation and rotation

In the case that a body is moving as well as rotating, we may express its kinetic energy as $$K = \frac{1}{2}Mv^{2} + \frac{1}{2}I\omega^{2},$$ with $M$ the body's mass, $v$ the velocity of its center of mass, $I$ the moment of inertia about the axis through the center of mass, and $\omega$ the angular velocity of the body. We may also state that the net external force on a body is defined by $$\sum_{}\vec{F}_{ext} = M\vec{a}.$$ The rotational motion about the center of mass is given by $$\sum_{}\tau_z = I\alpha_z.$$ If we are in a case where the body is tolling without slipping, we have $$v = R\omega,$$ where $R$ is the radius of the body.

If a body is made to rotate about an axis that is parallel to the old rotation axis, and at a distance $d$ of the latter, then: $$I = I_0 + md^2.$$

Work done by a torque

The work resultant of a torque that acts on a rotating body is given by $$W = \int_{\theta_1}^{\theta_2} \tau_z d\theta,$$ with $\theta_1$ being the initial angular position, and $\theta_2$ the final. Deriving from this, we get that the total done on a rotating rigid body is the difference between the final and initial rotational kinetic energy, or $$W_{tot} = \frac{1}{2}I\omega_2^{2} - \frac{1}{2}I\omega_2^{1}.$$ Finally, we get the power due to a torque acting on a rigid body, $$P = \tau_z\omega_z.$$

Angular momentum

The angular momentum of a particle is given by $$\vec{L} = \vec{r}\times m\vec{v},$$ where $\vec{r}$ is the position vector of the particle relative to the origin. The angular momentum of a rigid body rotating around a symmetry axis is given by $$\vec{L} = I\vec{\omega}.$$

Rotational dynamics and angular momentum

We may combine our previous discussions in order to conclude that the sum of external torques on the system is equal to the rate of change of the total angular momentum of the system, or $$\sum_{}\vec{\tau} = \frac{d\vec{L}}{dt}.$$

Pascal's Law

Pressure applied to an enclosed fluid is transmitte undiminished to every portion of the fluid and the walls of the containing vessel.

Oscillation

Period $T$: time for one cycle. Frequency $f = \frac{1}{T}.$ Angular frequency $\omega = 2\pi f.$

Simple harmonic motion

Simple harmonic motion (SHM) occurs when the oscilation does not change, for example when a spring obeys Hooke's law. We can define the restoring force exerted by an ideal spring in this case with $$F_x = -kx,$$ where $k$ is the force constant of the spring and $x$ the displacement. The acceleration of a body in SHM is given by $$a_x = -\frac{k}{m}x.$$ The angular frequency is given by $$\omega = \sqrt{\frac{k}{m}}.$$ Displacement can also be described with the angular frequency, amplitude $A$ and the phase angle $\phi$ by $$x = A\cos{(\omega t + \phi)}.$$ The total mechanical energy in SHM is $$E = \frac{1}{2}mv_x^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2 = \textrm{constant}.$$ In the case of a vertical spring, we may still have a SHM.

Simple pendulum

If there is a small amplitude, we have $$\omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{mgd}{I}}$$ $$T = 2\pi\sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{I}{mgd}},$$ where $L$ is the pendulum's length, $I$ the moment of inertia and $d$ the distance from rotation axis to center of gravity.

Damped oscillations

If there is a little damping, we get $$x = Ae^{-(b/2m)t}\cos{(\omega t + \phi)}$$ $$\omega = \sqrt{\frac{k}{m}-\frac{b^2}{4m^2}},$$ where $b$ is the damping constant.

Forced oscillation

The amplitude of a forced oscillator may be described by $$A = \frac{F_\textrm{max}}{\sqrt{(k - m\omega_d^2)^2+b^2\omega_d^2}},$$ where $F_\textrm{max}$ is the maximum value of the driving force and $b$ is the damping constant.

Thermal expansion

The length $L$ of a body changes proportionally to the change of temperature: $$\Delta L = \alpha L_0 \Delta T,$$ where $\alpha$ is the coefficient of linear expansion (in $K^{-1}$ or $°C^{-1}$) and $L_0$ is the original length. The volume also changes: $$\Delta V = \beta V_0 \Delta T,$$ where $\beta$ is the coefficiant of volume expansion.

Quantity of heat

The calorie is the amount of heat required to raise the temperature of 1 gram of water from 14.5°C to 15.5°C. $$1 cal = 4.186J$$ The heat required to change the temperature of a body of mass $m$ is defined by $$Q = mc\Delta T,$$ where $c$ is the specific heat of the material. The heat required to change the temperature of a certain number of moles is given by $$Q = nC\Delta T,$$ where $n$ is the number of moles of material and $C$ is the molar heat capacity of the material.

Calorimetry and phase changes

A phase is a specific state of matter. The heat stransfer in a phase change is given by $$Q = \pm mL,$$ where $L$ is the latent heat for this phase change. $Q > 0$ if heat enters the material, $Q < 0$ if heat leaves the material.

Mechanisms of heat transfer

Conduction is the transfer of heat within materials bulk motion of the materials. Its heat current is given by $$H = kA\frac{T_H - T_C}{L},$$ where $k$ is the thermal conductivity of the rod material, $A$ is the cross sectional area, $T_H$ and $T_C$ are the temperatures of the hot and cold ends of the rod respectively, and $L$ is the length of the rod.

Convection is complex.

Radiation is the transfer of heat by electromagnetic waves. Its heat current is given by $$H = Ae\sigma T^4,$$ where $A$ is the area of emitting surface, $e$ the emissivity of the surface, $\sigma$ the Stefan-Boltzmann constant, and $T$ the absolute temperature of the surface. The net rate of radiation of a body is given by $$H_\textrm{net} = Ae\sigma (T^4 - T_s^4),$$ where $T_s$ is the absolute temperature of the surroundings.

Equations of state

In order to "weight" a gas, we may use the following equation: $$m = nM,$$ where $n$ is the number of moles of the substance and $M$ its molar mass. An ideal gas is a gas for which the following equation holds for all pressures and temperatures: $$pV = nRT,$$ where $p$ is the gas' pressure, $V$ is its volume, $T$ its absolute temperature, $n$ the number of moles and $R$ the gas constant.

Molecular properties of matter

A mole is the amount of substance that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12.

The molar mass $M$ of a substance is the mass of 1 mole, and is given by $$M = N_Am,$$ where $m$ is the mass of a molecule of sustance and $N_A$ is Avogadro's number.

Work done during volume changes

When a gas changes volume, it produces work defined by $$W = \int_{V_1}^{V_2}pdV,$$ where $V_1, V_2$ are the initial and final volumes respectively of the gas, and $p$ its pressure.

Paths between thermodynamic states

There are several ways for a thermodynamic system to progress from an initial state to a final state, and during this progress the system passes through a series of intermediate states, which we refer to as a path.

Internal energy

The internal energy of any thermodynamic system depends only on its state. The change in internal energy in any process depends only on the initial and final states, not on the path. The internal energy of an isolated system is constant.

Kinds of thermodynamic processes

An adiabatic process is a process in which there is no heat transfer in and out of the system, ie. $Q = 0.$ In an adiabatic, temperature changes are due to work done by or on the system, not due to heat flow.

In an isochoric process, the volume is constant, ie. $W = 0.$

In an isobaric process, the pressure is constant, ie. $W = p(V_2 - V_1).$

In an isothermal process, the temperature is constant.

Ideal gases

The internal energy U of an ideal gas depends only on its temperature T, not on its pressure or volume.

We have the following definition for the molar heat capacity $C_p$ of an ideal gas at constant pressure: $$C_p = C_V + R,$$ where $C_V$ is the molar heat capacity at constant volume, and $R$ is the gas constant. The dimensionless ratio of heat capacities is $$\gamma = \frac{C_p}{C_V}.$$ If we are in an adiabatic process, the work done by an ideal gas is $$W = nC_V(T_1 - T_2),$$ where $n$ is the number of moles, $C_V$ the molar heat capacity at constant volume, and $T_1, T_2$ the initial and final temperatures respectively.

Engines

A heat engine converts heat $Q_H$ into work $W$ and discards waste heat $Q_C$. Its efficiency is given by $$ e = \frac{W}{Q_H} = 1 - |\frac{Q_C}{Q_H}|.$$

If an engine uses an Otto cycle, its thermal efficiency is $$e = 1 - \frac{1}{r^{\gamma - 1}},$$ where $r$ is the compression ratio.

The Carnot cycle is a hypothetical engine that yields the maximum possible efficiency without breaking the second law of thermodynamics. It consists of only reversible processes. Its efficiency is $$e = 1 - \frac{T_C}{T_H} = \frac{T_H - T_C}{T_H}.$$ Conversely, the Carnot refrigerator's coefficiant of performance is $$K = \frac{T_C}{T_H - T_C}.$$

Refrigerators

A refrigirator is essentially a reverse heat engine. Given $Q_C$ the heat removed from inside the refrigerator, $W$ the work input, and $_H$ the heat rejected into the ouside air, we have the coefficiant of performance of the refrigerator $$K = \frac{|{Q_C}|}{|{W}|} = \frac{|Q_C|}{|Q_H| - |Q_C|}.$$

Entropy

Entropy is a measurment of randomness of a system. In a reversible process, entropy change is defined by $$\Delta S = \int_{1}^{2}\frac{dQ}{T},$$ where $Q$ is the heat flow into the system, and $T$ the absolute temperature. All irreversable processes involve an incresae of entropy.