equation

(noun)

An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.

Examples of equation in the following topics:

  • Applications

    • There are four kinematic equations that describe the motion of objects without consideration of its causes.
    • Notice that the four kinematic equations involve five kinematic variables: $d$, $v$, $v_0$, $a$, and $t$.
    • Each of these equations contains only four of the five variables and has a different one missing.
    • Step two - Find an equation or set of equations that can help you solve the problem.
    • Choose which kinematics equation to use in problems in which the initial starting position is equal to zero

  • Particle Current

    • We can prove this simply by integrating the fourth equation over $d^3 {\bf p}$.
    • The first two terms yield the left-hand side of the equation above.
    • Let's define ${\bf V}=\langle {\bf v} \rangle$ and write out the equation above by components,
    • This is the continuity equation.
    • Because the fourth equation is consistent with the Lioville equation (seventh equation) and more generally with the Boltzmann equation (sixth equation) and $J^\mu_{;\mu}=0$ if particles are conserved, the Lioville and Boltzmann equations cannot hold if $\nabla_{\bf p} \cdot F \neq 0$ and particles are conserved.

  • Problem-Solving Techniques

    • Identify the problem and solve the appropriate equation or equations for the quantity to be determined.
    • Solve the appropriate equation or equations for the quantity to be determined (the unknown).
    • We cannot use any equation that incorporates t to find ω, because the equation would have at least two unknown values.
    • The equation $\omega 2=\omega 02+2$ will work, because we know the values for all variables except ω.
    • Taking the square root of this equation and entering the known values gives

  • Atomic Structure

    • The wavefunction evolves forward in time according to the time-dependent Schrodinger equation
    • If the Hamiltonian is independent of time we can solve this equation by
    • This realization allows us to write the equation that the wavefunction of an atom must satisfy
    • For most atomic states, these effects can be treated at perturbations.We can simplify these equations by using
    • This gives the following dimensionless equation

  • Problem Solving

    • The Ideal Gas Law is the equation of state of a hypothetical ideal gas.
    • Variations of the ideal gas equation may help solving the problem easily.
    • Substitute the known values into the equation.
    • Choose a relevant gas law equation that will allow you to calculate the unknown variable: We can use the general gas equation to solve this problem: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$.
    • Substitute the known values into the equation.

  • A General Approach

    • A next step is to use what is known to find the appropriate equation to find what is unknown.
    • While it is easiest to find an equation that leaves only one unknown, sometimes this is not possible.
    • In these situations, you can solve multiple equations to find the right answer.
    • Remember that equations represent physical principles and relationships, so use the equations and drawings in tandem.
    • You may then substitute the knowns into the appropriate equations and find a numerical solution.

  • Really Little Sound Waves

    • We can write $\displaystyle P' = (\partial P/\partial \rho)_s \rho'$ and rewrite the continuity equation to get
    • This is a wave equation with a sound speed of $c_s^2 = (\partial P/\partial \rho)_s$.
    • Let's take a solution to this equation for the pressure,

  • Application of Bernoulli's Equation: Pressure and Speed

    • This simply adds another term to the above version of the Bernoulli equation and results in
    • Bernoulli's equation can be applied when syphoning fluid between two reservoirs .
    • The Bernoulli equation can be adapted to flows that are both unsteady and compressible.
    • However, the assumption of inviscid flow remains in both the unsteady and compressible versions of the equation.
    • Adapt Bernoulli's equation for flows that are either unsteady or compressible

  • Mathematical Represenation of a Traveling Wave

    • is a solution of the wave equation for $c= \frac{\omega}{k}$.
    • is a solution to the wave equation.
    • To solve the equation 1, let's introduce new variables: $\phi = x-ct, \psi= x+ct$.
    • With the change of variables, the equation 1 becomes $\frac{\partial u_+}{\partial \phi} = 0$ for the equation with the "+" sign and $\frac{\partial u_-}{\partial \psi} = 0$ for the "-" sign.
    • Any function that contains "x+ct" or "x-ct" can be a solution of the wave equation.

  • Another velocity-dependent force: the Zeeman effect

    • So the vector differential equation governing this simple harmonic motion is:
    • In other words this one vector equation is equivalent to three completely separate scalar equations (using $\omega _0 ^2 = k/m$ )
    • The equations are uncoupled in the sense that each unknown ( $x,y,z$ ) occurs in only one equation; thus we can solve for $x$ ignoring $y$ and $z$ .
    • Plugging these into the equations for $x$ and $y$ gives the two amplitude equations
    • We can use the first equation to compute $x_0$ in terms of $y_0$ and then plug this into the second equation to get