Mechanical Energy Analog to Capacitance

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?

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 $V = \frac 1 C Q$

• Interesting food for thought for me since much is over my head, esp re the need to include an inductor and resistor to find a capacitance analog in Mech Energy systems. From my uber ignorant position it seems like the "meat" of what you are saying is that the spring constant k is the mechanical analog to capacitance. If not, is there a SI unit for Mech energy that is an analog to the Farad. That is, I suppose, the meat of my question since that is a simple "yes or no" (and if yes, the name of the unit), once I have that I can go try to make the connections you presented so thoroughly Commented Aug 26, 2018 at 5:00
• @AAnderson As I have pointed out the mechanical equivalent of reciprocal capacitance $\frac 1C$ is the spring constant. Reciprocal capacitance has the units $\rm F^{-1} = \frac {\rm V}{\rm C}$ to be compared with $\frac {\rm N}{\rm m}$ for the spring constant Commented Aug 26, 2018 at 5:11

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).

• Exactly the type of clarity I am trying to get at. So whats the Mech energy (ME) unit akin to the Farad? And what is the ME property akin to Capacitance? Stated a wee bit differently, charge has a ME counterpart, mass, and the unit of charge, C has a ME counterpart, kg. What is the ME counterpart of capacitance, and what is the ME counterpart of Farad. Is there a word for ME capacitance? Maybe it is unlike charge which has counterpart mass, but rather it is like current which has the ME counterpart “current” and which has no unit -unless there is a unit of mechanical current, is there? ) Commented Aug 26, 2018 at 16:38
• @AAnderson The unit would then be the unit for stiffness or elastic modulus: Pascal (or Newton-per-square-metre). The property would be stiffness... I explained that in the answer, no? Commented Aug 26, 2018 at 19:49
• @AAnderson I am not sure what you mean with "mechanical current". Do you mean something like water current or flow rate? Then it definitely has units: cubic-metres-per-second (a good analogy, since the ampere is coulomb-per-second). You could maybe even stretch it and call the water current flow rate unit: kilograms-per-second, if we choose to consider mass and not volume. Commented Aug 26, 2018 at 19:50

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.

• This is great to visualize it. But there was the 2nd half of my question… beyond the conceptual analogs, is there an ACTUAL recognized SI analog to capacitance (with a unit akin to the Farad) in Mech energy systems? In the same way that I think it is pretty universally understood that the Newton is the “push” in the mechanical world and the Volt is the “push” in the Electrical world, Friction is the Mechanical energy’s version of resistance, what is the Mech energy version of Capacitance and is there an SI unit for it? Commented Aug 26, 2018 at 5:15
• I do not know if there is an "official" SI-unit designator for mechanical compliance, but a quick way to get at it is this: for any linear spring, (force) = -k x (meters) where k is the spring constant which means k has the units of (force)/(length) = newton/meter, which would be the units of mechanical compliance in SI. Commented Aug 26, 2018 at 6:51

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.

• "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." This dosen't make sense to me. A rubber membrane is a balloon. If I put a balloon in a pipe I stop the flow unless/unil it breaks. At which point it has no impact. This sounds like a switch that is triggered by a compoenets failure (like a fuse - that starts flow rather than stops it). But I dont think capacitors are fuses or vice versa. What am I missing? Commented Aug 26, 2018 at 16:54
• That breaking stress corresponds to dielectric strength.Example:For air it is around 3 millions volts, if Potential difference goes about it,it starts conducting current from one plate to another plate via ionized form of itself. Commented Aug 28, 2018 at 9:18