User:AndreRD/Analogues in Physics

In Physics, there are many cases where two different phenomena that might seem unrelated share some similar underlying structure. When this happens, there is often a correspondence between the units and formulas of the two systems.

Translational Motion vs Rotational Motion

Translational Analog	Symbol	Unit	Rotational Analog	Symbol	Unit
Position	$\mathbf {x}$	m	Angle	$\theta$	rad
Time	$t$	s	Time	$t$	s
Linear Velocity	$\mathbf {v} ={\frac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}$	m s-1	Angular Velocity	${\boldsymbol {\omega }}={\frac {\operatorname {d} \theta }{\operatorname {d} t}}\cdot {\hat {\mathbf {u} }}$	rad s-1
Linear Acceleration	$\mathbf {a} ={\frac {\operatorname {d} \mathbf {v} }{\operatorname {d} t}}$	m s-2	Angular Acceleration	${\boldsymbol {\alpha }}={\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}$	rad s-2
Mass	$m$	kg	Moment of Inertia	$I$	kg m2
Linear Momentum	$\mathbf {p} =m\mathbf {v}$	kg m s-1 or N s	Angular Momentum	$\mathbf {L} =I{\boldsymbol {\omega }}$	kg m2 rad s-1 or N m s
Force	$\mathbf {F} ={\frac {\operatorname {d} \mathbf {p} }{\operatorname {d} t}}=m{\frac {\operatorname {d} \mathbf {v} }{\operatorname {d} t}}=m\mathbf {a}$	N = kg m s-2	Torque	${\boldsymbol {\tau }}={\frac {\operatorname {d} \mathbf {L} }{\operatorname {d} t}}=I{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}=I{\boldsymbol {\alpha }}$	kg m2 rad s-2 or N m
Translational Kinetic Energy	$E_{k}={\frac {1}{2}}mv^{2}={\frac {p^{2}}{2m}}$	J = kg m2 s-2	Rotational Kinetic Energy	$E_{r}={\frac {1}{2}}I\omega ^{2}={\frac {L^{2}}{2I}}$	J = kg m2 s-2
Translational Work	$W=\int \mathbf {F} \cdot \operatorname {d} \mathbf {x}$	J = N m = kg m2 s-2	Rotational Work	$W=\int \mathbf {\tau } \ \cdot \operatorname {d} \theta \$	J = N m = kg m2 s-2
Laws of rectilinear motion under constant acceleration	${\begin{aligned}\\&v_{f}=v_{i}+at\\&\Delta x={\tfrac {1}{2}}(v_{i}+v_{f})t\\&\Delta x=v_{i}t+{\tfrac {1}{2}}at^{2}\\&\Delta x=v_{f}t-{\tfrac {1}{2}}at^{2}\\&v_{f}^{2}=u_{f}^{2}+2a\Delta x\end{aligned}}$		Laws of circular motion under constant torque	${\begin{aligned}\\&\omega _{f}=\omega _{i}+\alpha t\\&\Delta \theta ={\tfrac {1}{2}}(\omega _{i}+\omega _{f})t\\&\Delta \theta =\omega _{i}t+{\tfrac {1}{2}}\alpha t^{2}\\&\Delta \theta =\omega _{f}t-{\tfrac {1}{2}}\alpha t^{2}\\&\omega _{f}^{2}=\omega _{i}^{2}+2\alpha \Delta \theta \end{aligned}}$

Electric Fields and Gravitational Fields

Electric Analog	Symbol	Unit	Gravitational Analog	Symbol	Unit
Electric charge	$q$	C (= A s)	Gravitational mass	$m$	kg
Coulomb's constant	$k_{e}={\frac {1}{4\pi \varepsilon _{0}}}=8.987\times 10^{9}\ \mathrm {N\ m^{2}\ C^{-2}}$	N m2 C-2	Gravitational constant	$G=6.67\times 10^{-11}\ \mathrm {N\ m^{2}\ kg^{-2}}$	N m2 kg-2
Electric field of a point charge at a given position	$\mathbf {E} ={\frac {k_{e}q}{r^{2}}}{\hat {\mathbf {r} }}$	N C-1	Gravitational field of a point mass at a given position	$\mathbf {g} ={\frac {Gm}{r^{2}}}{\hat {\mathbf {r} }}$	N kg-1 = m s-2
Electric force on point charge 2 from point charge 1	$\mathbf {F} =\mathbf {E} _{1}q_{2}={\frac {k_{e}q_{1}q_{2}}{r^{2}}}{\hat {\mathbf {r} }}$	N	Gravitational force on point mass 2 from point mass 1	$\mathbf {F} =\mathbf {g} _{1}m_{2}={\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {\mathbf {r} }}$	N
Work done across an electric field	$W=\int \mathbf {F} \cdot \operatorname {d} \mathbf {x} =q\int \mathbf {E} \cdot \operatorname {d} \mathbf {x} =qV$	J = N m = C V	Work done across gravitational field	$W=\int \mathbf {F} \cdot \operatorname {d} \mathbf {x} =m\int \mathbf {g} \cdot \operatorname {d} \mathbf {x} \ (=mg\Delta x)$	J = N m = kg m2 s-2
potential	Example	Example	Example	Example	Example
flux	Example	Example	Example	Example	Example
gauss's law	Example	Example	Example	Example	Example
Example	Example	Example	Example	Example	Example
Example	Example	Example	Example	Example	Example

Electricity like water in a pipe

File:Circuit analogy water pipes big.jpg

Since electrons cannot be seen and are intangible, a very popular model of explaining how electric circuit work is by the Water-in-Pipes model. This is a useful model to use because water flowing through pipes is a mechanical system that works in uch the same way as an electrical circuit, but with different units.

The main ideas are that:

• The pipe is like the wire in the electric circuit
• The pump is like the battery.
• The pressure generated by the pump drives water through the pipe; that pressure is like the voltage generated by the battery which drives electrons through the circuit.
• The seashells plug up the pipe and slow the flow of water, creating a pressure difference from one end to the other. In a similar way the resistance in the electric circuit resists the flow of electricity and creates a voltage drop from one end to the other. Energy is lost across the resistor and shows up as heat.

The power in the circuit equals the voltage times the current. The same power can be carried by a high voltage and a low current as is carried by a low voltage and a high current. The higher the current flow, however, the more energy is lost as heating of the wires. That's why high voltage and low current is used when transporting electrical energy along power lines.

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