When no acceleration occurs in a system due to forces cancelling out, equilibrium is obtained
\(\sum_{k=1}^{n} \textbf{F}_k = 0\)
For angular systems such that the torque is calculated with the radius to some chosen fulcrum, when no rotation occurs equilibrium is obtained
\(\sum_{k=1}^{n} \tau_k = 0\)
Fixed point on some lever
Partition some lever into two lengths \(a,b : \ell = a+b\), where \(a\) is the length of the left partiton and \(b\) the length of the right. Let \(F_{L},F_{R}\) be two forces at the leftmost and rightmost sides of the lever respectfully, orthogonal to the lever pushing up and let \(F_{D}\) be a force pushing orthogonally down at the point on the lever where \(a\) and \(b\) meet, then \(F_{D}a = F_{R}(a+b) \land F_{D}b=F_{L}(a+b)\)
When falling in a fluid, equilibrium is obtained by accounting for drag. The velocity at chich this occurs is calld terminal velocity and denoted \(v_{\infty}\)
\(v_{\infty} : F_{g} - F_{D}(v_{\infty})= 0 \)
\( v_{\infty} = \sqrt{\frac{2F_{g}}{C\rho A}} \)
Noting a falling object's acceleration as \(m\dot{v} = F_{g}-F_{D}\), dividing both sides by \(m\) and writing the expression in terms of \(v_{\infty}\) produces the following DE:
\( \dot{v} = g( 1- \frac{v^2}{v^{2}_{\infty}}) \)
\( v(t) = v_{\infty} \tanh (\frac{gt}{v_{\infty}}) \)
Scalar quantity representing amount of space traversed
\(\displaystyle s(t) = \int^{t}_{0} |\textbf{r}'(t')| dt'\)
Vector quantity representing point in space relative to some defined origin
\(\displaystyle \textbf{r}(t)=\begin{pmatrix} x(t) \\ y(t) \\ z(t) \end{pmatrix}\)
Vector quantity representing difference from original position and current position. Displacement is 'relative position' to the stating position
\(\textbf{x}(t)= \textbf{r}(t) - \textbf{r}(0)\)
Angle quantity representing radians between original and current position vectors
\(\theta\) is an angular displacement function
\(\theta (t) = \frac{s(t)}{r} \)
Vector quantity representing speed and its direction, the norm of velocity is called speed
\(\textbf{v}=\dot{\textbf{x}}\)
Angle quantity representing the radians the angular displacement increases per second
\(\omega =\dot{\theta} \) is an angular frequency function
\( \|\textbf{v}(t)\| = \omega(t) r\)
Pseudovector quantity normal to the circle of rotation with magnitude of the angular frequency
\(\| \boldsymbol{\omega} \| = \omega =\dot{\theta} \)
Vector quantity representing the speed at which the velocity increases and its direction
\( \textbf{a}=\dot{\textbf{v}}=\ddot{\textbf{r}} = \textbf{v} \cdot \frac{d\textbf{v}}{ds}\)
Pseudovector quantity normal to the circle of rotation with magnitude representing the radians the angular velocity increases per second
\(\alpha=\dot{\omega}=\ddot{\theta}\)
\( \textbf{x}(t) = r \hat{\textbf{r}}\)
\( \textbf{v}(t) = \omega r \hat{\boldsymbol{\theta}}\)
\( \textbf{a}(t) = -r \omega^2 \hat{\textbf{r}} + r \alpha \hat{\boldsymbol{\theta}} \)
\( \textbf{x}(t) = r \hat{\textbf{r}}\)
\( \textbf{v}(t) = r \hat{\textbf{r}} + r\omega \hat{\boldsymbol{\theta}}\)
\( \textbf{a}(t) = [ \ddot{r}-r \omega^2 ] \hat{\textbf{r}} + [ r\alpha + 2\dot{r}\omega ] \hat{\boldsymbol{\theta}}\)
Fictious acceleration representing the effect of inertia in a non-inertail reference frame when centripetal acceleration occurs on the inertail reference frame
Measure of resistance to change of speed
Position such that the weighted relative position of all points within an object sum to zero, the balance point of an object.
\(\displaystyle \textbf{R} = \frac{1}{M} \sum^{n}_{i=1} m_i \textbf{r}_i \)
\(\displaystyle \textbf{R} = \frac{1}{M}\iiint_{Q}\rho(\textbf{r})\textbf{r} dV \)
\(\textbf{p} = m\textbf{v}\)
Closed, isolated physical systems conserve momentum
\( \textbf{p}\text{ is the system's momentum} \implies \Delta \textbf{p} = \textbf{0}\)
Empirical observations formulated as definitions for Newtonian mechanics that hold within inertial reference frames (non accelerating reference frames)
Velocity of body is only changed when external forces act upon it; absence of force implies constant velocity
\(\sum \textbf{F}_n =\textbf{0} \iff \Delta \textbf{v} = \textbf{0}\)
\(S, S' \text{ are inertial reference frames } \iff \sum \textbf{F}_{S',n} =\textbf{0} \implies \sum \textbf{F}_{S,n} =\textbf{0}\)
Force is a vector \(\textbf{F}\) representing Influence that accelerates an object
\(\textbf{F}=m\textbf{a}= \frac{d\textbf{p}}{dt}\)
For each body applying force to another, there exists a pair of forces of equal magnitude and opposite direction, one force acting on each of the bodies involved
\( \textbf{F}_{\alpha \beta}=-\textbf{F}_{\beta \alpha} \)
Theorem relating rotational inertia of axis' of rotation parallel to the axix of rotation passing through the center of mass.
\(I = I_{com} + Mh^2\)
Change in momentum of an object, calculated by the forces acting upon the object (and hence spurring the momentum)
\(\displaystyle \textbf{J} = \Delta \textbf{p} = \int^{t}_{0} \textbf{F} dt\)
Event of two bodies exerting forces on eachother for a relatively short time
Funamental interaction creating attraction between bodies with mass
\(\textbf{F}_{g}= \begin{pmatrix} 0 \\ -mg \end{pmatrix}\)
Proportionality constant of the magnitude of a gravitational force
\(G\)
Two bodies have a gravitational force directed at one another with the following magnitude:
\(|\textbf{F}| = G \frac{m_1 m_2}{r^2}\)
\(\textbf{F} = G \frac{m_1 m_2}{\| \boldsymbol{\rho}\|^2} \hat{\rho}\)
Force acting on falling object that is connected to some fixed point. By N3L
Force acting on the surface against motion that occurs. Since gravitational force creates the contact with a surface, friction is therefore proportional to the normal of this graviational force
\(|F_{F}| \propto |F_{N}|\)
\(|F_{F}| = \mu_{s} |F_{N}|\)
\(|F_{F}| = \mu_{k} |F_{N}|\)
\(\mu : [0,1]\)
\(|F_{f}|=\mu |F_{n}|\)
\( \dot{\textbf{x}}(t) = \begin{pmatrix} v_0 \cos (\theta) \\ -gt +v_0 \sin (\theta) \end{pmatrix} \)
\( y(x) = \frac{-gx^2}{2v_{0}^{2}} (1+ \tan^2 (\theta)) + x \tan (\theta) \)
Solving the aforementioned quadratic gives the total horizontal displacement
\( x = \frac{v_0^2}{g} \cos (2\theta )\)
Influence to rotate some object, analogous to linear force but for rotational systems
\( \boldsymbol{\tau} = \textbf{r} \times \textbf{F}\)
\( \| \boldsymbol{\tau} \| = I \alpha\)
As for linear forces a body's inertia is exactly its mass, rotational forces have inertia based on the geometry of the body with its radius to its fulcrum and its mass
\(\displaystyle I = \iiint_Q \rho ( \textbf{r}' )\|\textbf{r}\|^2 dV \)
Origin, orientation and scale defined in a physical space
Due to relative speeds within frames of references, if a frame in motion slows down,
Inertial force
Coriolis force
Reference frame that can be represented as a non accelerating transfrom of the original frame
Transform between two inertial reference frames
\(\textbf{r}_{S'} = \textbf{r}_{S} - \textbf{G}\)
\(\textbf{G}\) is the vector from the origin of \(S\) to the origin of \(S'\), since it is nonaccelerating it is an affine function
\(\textbf{r}_{S}, \textbf{r}_{S'}\) are vectors representing the same point in the reference frames \(S,S'\) respectively
Energy associated with motion, for constant acceleration is calculated as:
\(K = \frac{m\| \textbf{v}\|^2}{2}\)
The energy transfered to an body when a force is applied.
\(W = \Delta K = \textbf{F} \cdot \textbf{x} \)
For variable forces, a line integral allows all the infinitesimal work forces to be captured
\(\displaystyle W = \int_{\mathcal{C}} \textbf{F} \cdot d\textbf{x} = \int^{t}_{0} \textbf{F} \cdot \textbf{v} dt' \)
\(\displaystyle W = \int^{t}_{0} \boldsymbol{\tau} \cdot \boldsymbol{\omega} dt' \)
Theorem stating that work is the catalyst for change in kinetic energy. It provides a means of calculating kinetic energy through the definition of work
\(\Delta K = W \)
Treating gravity as the hypotenuse and the plane as one side of a right angled triangle, geometry provides the following gravitational force on the axis of the plane. This is an example of work of gravity
\(F_{\hat{u}}=\textbf{F}_{g} \cdot \hat{u} = mg \cos (\theta)\)
Solution to the ODE from Hooke's law
\(\ddot{x} = \frac{k}{m}x\)
\( x(t) = x_0 \sin(\sqrt{\frac{k}{m}}x) + \frac{v_0}{\sqrt{\frac{k}{m}}} \cos (\sqrt{\frac{k}{m}}t)\)
\( U(x) = \frac{k x^2}{2}\)
\( A \sin [\frac{2\pi}{T}(t+P)]\)
Empirical law stating that the force applied to a string is proportional to the displacement it is stretched to
\(\textbf{F} = -k\textbf{x}\)
The rate of work that can be done per unit of time
\(P = \frac{dW}{dt} = \textbf{F} \cdot \textbf{v}\)
Energy due to the stress caused by a body's relative position to another body, that is, energy with the potential to be actualized into work. Potential energy may be converted to and from kinetic energy.
\(\Delta U = -W\)
\(\Delta U = U_1 - U_0\)
Forces such that its work is independent of path of the force
In other words, one can partition the kinetic work such that \(W_{0}=-W_{1}\)
Sum of poential and kinetic energy
\(E_{\text{mec}} = K + U\)
isolated systems conserve mechanical energy.
\(\Delta E_{\text{mec}} = \Delta K + \Delta U = 0 \)
This can be derived by noting by noting that by the definition of potential energy and the kinetic energy-work theorem implies that all kinetic energy lost is just work that is transferred to potential energy.
Non conservative forces change this equation to the following:
\(\Delta E_{\text{mec}} = \Delta K + \Delta U = W_{\text{NC}} \)
Note that although mechanical energy is not conserved, the energy is still transferred to some other type of energy that mechanical energy does not account for. No exception to energy conservation has been documented
\(\Delta v = v_e\ln (\frac{m_0}{m_1})\)
Motion arising from physical systems that apply a restoring force proportional to displacement, or more precisely obeying the following differential equation
\( \ddot{x} = -n^2(x-c) \)
\(x(t) = r\sin (nt + c) \)
Wave perpendicular to source
Wave parallel to source
Morphing substance with freely moving molecules that take the shape of its container
Fluid where the viscous stress is proportional to the strain rate
\(\tau = \mu \frac{du}{dy}\)
Denoted \(p\), refers to uniform amount of perpendicularly applied force on a unit of surface area
\(p=\frac{ \|\textbf{F}\|}{A}\)
Principle that a change in pressure to any point in an enclosed incompressible fluid changes the pressure in the whole fluid by this exact amount
\( \Delta p_{\textbf{r}_1} = \Delta p_{\textbf{r}_2} \)
Due to gravity acting on the fluid above, pressure is greater
\(p(h)= \rho g h\)
Law that the buoyancy force equals the weight of displaced fluid
\(F_{b} = \rho_{f}V g\)
This law implies that an object floats in a fluid iff the mass of the object is less than the mass of the fluid it displaces
Fluid that is
For ideal flow, equation governing the constant volume of a fluid parcel passing through any surface per unit of time;
\( \frac{\Delta V}{ \Delta t} = Au\)
\( A_1 u_1 = A_2 u_2\)
Corollary of the continuity equation and conservation of energy which governs ideal flow of steady velocity in uniform density fluid. It encapsulated the idea that increasing fluid velocity decreases pressure normal to the velocity
\( \frac{\rho u^2}{2} + p + \rho g h = k \)
The scenario of gravitational force as the only force in some physical system
Denoted \(\rho\), the measure of mass per unit of volume
\(\rho = \frac{m}{V}\)
Drag is resistance of a force in a fluid, and its force is given as follows
Empirical law for calculating drag force in a fluid
\( \| \textbf{F}_{D} \| = \frac{\rho v^2 C A}{2} \)
Law determining the drag force of a sphere flowing in a viscous fluid with low Reynold's number
\( \| \textbf{F}_{d} \| = 6 \pi \mu Rv \)
Denoted \(\mu\), the measure of deformation of a fluid with respect to time
\( \nu - \frac{\mu}{\rho}\)
Dimensionless number representing ratio of flow velocity to the speed of sound
\(M= \frac{u}{c}\)
Dimensionless number representing
\(\text{Re}= \frac{\rho u \ell}{\mu}\)
Rate of change of a some material's property (velocity, pressure etc.) as it flows through a particular point
A planet's orbit is an ellipse
\(r = \frac{p}{1+ \varepsilon \cos ( \theta )}\)
Lines from the planet to its sun captured at consistent time intervals partition the ellipse into triangles of equal area
A spherically symmetric body affects external objects gravitationally as though said body's mass were concentrated at its center.
A spherically symmetric shell has no net gravitational force on any internal object.