Mechanical Energy Analog to Capacitance

Question

For starters I am a complete physics noob. I've been trying to understand basic fundamental ideas at a conceptual level I was drawn to the fact that work $(J) = FxD = CxV.$ I started trying to find mechanical and electrical analogs. For example moving a coulomb through static field to increase voltage seems analogous to moving a mass through a gravitational field to increase potential energy. If so a coulomb is the electric analog of a mass, and height (and the potential energy of each) would be analogous to electrical potential difference (aka voltage) etc. All was going well I was feeling I could picture the world of electrical energy (unintuitive) in terms of mechanical energy analogs (intuitive)… But then I hit capacitance. ..Is there any mechanical equivalent of capacitance?. And if not why not?. If C/V is capacitance is the mechanical version of that Force/ Distance or maybe Distance / Force. Is there a unit assigned to this mechanical capacitance I am imagining - if it exists? Does is question even make sense?

Answer 1

For a spring (spring constant $k$) mass ($m$) system with damping ($r\dot x$) proportional to the velocity ($\dot x$) the equation of motion can be written as $-kx - r\dot x = m \ddot x \Rightarrow m\ddot x +r \dot x + kx =0$ where $x$ is the displacement.
For an inductor $L$, resistor $R$ and capacitor $C$ series circuit Kirchhoff's voltage rule gives $L \dot I + RI + \frac QC=0$ where $I$ is the current $(= \dot Q)$ and $Q$ is the charge.
In terms of the charge $Q$ this equation can be written as $L\ddot Q + R\dot Q + \frac 1C Q = 0$.

This is where you can make a comparison between a mechanical system and an electrical system.

$m$ and $L$ can be thought of as being to do with the inertia of the systems and the kinetic energy of the systems $(\frac 12 m \dot x^2$ and $\frac 12 L\dot Q^2)$.
$r$ and $R$ can be though of as to do the dissipative part of the systems $(r \dot x^2$ and $R\dot Q^2)$.
$k$ and $\frac 1C$ can be thought of as being to do with the springiness of the systems and potential energy of the systems $(\frac 12 kx^2$ and $\frac 12 \frac 1C Q^2)$.
For the last couplet you have force $F = kx$ and potential difference $V = \frac 1 C Q$

Answer 2

A capacitor can be thought of as a flexible membrane in a water pipe. The water (charge) streaming into it from one side, makes it expand and "push" on the water on the other side (induced potential). But it doesn't pass through (no current flows through a capacitor, only to and from it). After being "filled" (fully charged), the flow stops.

With a pump (battery, voltage source), you can add pressure (potential) and thus cause more expansion (store more charge). When it has expanded enough to fully resist the push from the pump (voltage across the capacitor equals that across the battery), all flow stops.

Capacitance (a measure of "how much" charge you can store) is then analogous to the "stiffness" or "elasticity" of this membrane. It determines "how much" the membrane will expand at a certain push (how much charge that can be stored with a certain battery voltage).

In general, I find the water pipe analogy very fulfilling for most electronics topics and properties, for example:

• resistor (filter, pipe constriction),
• current (water flow rate),
• electric potential (pressure),
• potential difference/voltage (pressure difference),
• voltage source/battery (pump),
• inductor (turbine wheel),
• switch (manual valve),
• diode (ball valve).

Even the laws work to a large degree in a closed water pipe system (charge conservation is analogous to mass conservation etc.).

coulomb is the electric analog of a mass

For correct terminology, coulomb would be analogous to kilogram (analogous units), while charge would be analogous to mass (analogous properties).

Answer 3

my favorite mechanical analogue for capacitance is a coil spring. In this case, the effort variable (force) plays the role of voltage and flow variable (velocity) plays the role of current.

you cannot instantaneously impose a force on a spring, just as you cannot instantaneously impose a voltage upon a capacitor. you can instantaneously impose a velocity across the ends of a spring in the same way you can instantaneously impose a current on a capacitor. in this case, the spring deflects and responds with an opposing force. that force acts through a distance and hence performs work on the spring, which stores that work as potential energy- and you can use the relationships for this to develop the corresponding equations for the capacitor.

Answer 4

Here is a section from wikipedia regarding the Hydraulic analog of capacitor,

In the hydraulic analogy, charge carriers flowing through a wire are analogous to water flowing through a pipe. A capacitor is like a rubber membrane sealed inside a pipe. Water molecules cannot pass through the membrane, but some water can move by stretching the membrane. The analogy clarifies a few aspects of capacitors:

The current alters the charge on a capacitor, just as the flow of water changes the position of the membrane. More specifically, the effect of an electric current is to increase the charge of one plate of the capacitor, and decrease the charge of the other plate by an equal amount. This is just as when water flow moves the rubber membrane, it increases the amount of water on one side of the membrane, and decreases the amount of water on the other side. The more a capacitor is charged, the larger its voltage drop; i.e., the more it "pushes back" against the charging current. This is analogous to the more a membrane is stretched, the more it pushes back on the water. Charge can flow "through" a capacitor even though no individual electron can get from one side to the other. This is analogous to water flowing through the pipe even though no water molecule can pass through the rubber membrane. The flow cannot continue in the same direction forever; the capacitor experiences dielectric breakdown, and analogously the membrane will eventually break. The capacitance describes how much charge can be stored on one plate of a capacitor for a given "push" (voltage drop). A very stretchy, flexible membrane corresponds to a higher capacitance than a stiff membrane. A charged-up capacitor is storing potential energy, analogously to a stretched membrane.