Expression where a certain quantity is proportional to the force squared

[PainterGuy]

TL;DR Summary

Trying to find an expression where x ∝ Force² or x ∝ Power²

Hi,

I'm trying to find an expression or phenomenon where a certain quantity is proportional to force squared or power squared. For example, we have Force ∝ acceleration and Power ∝ I².

I'm not sure if my question is a legit. I was trying to understand something and this question came to my mind and wanted to clarify. Thank you!

 

[davenn]

Summary: Trying to find an expression where x ∝ Force² or x ∝ Power²

For example, we have Force ∝ acceleration and Power ∝ I².

but they are not directly proportional unless the other factor is known

for Force ∝ acceleration ... only when the mass doesn't change

for Power ∝ I² ... only when the resistance doesn't change

 

[willem2]

Summary: Trying to find an expression where x ∝ Force² or x ∝ Power²

The energy in a spring is proportional to the square of the force on it.

Power² gets harder. I can only find this:
Hook up a variable resistor R to a constant current source. The voltage across it is proportional to the power dissipated in the resistor. Also hook up a capacitor parallel to the resistor. The voltage across it is proportional to the power dissipated in the resistor, so the energy in it is proportional to the square of it.

 

[anorlunda]

The voltage across it is proportional to the power dissipated in the resistor

Not true, the voltage across it is proportional to the current in the resistor, not the power.

 

[willem2]

Not true, the voltage across it is proportional to the current in the resistor, not the power.

I specified a current source. If I is constant, Power is proportional to voltage

 

[PainterGuy]

Thank you.

I thought that I should provide more context because I'm still little confused.

In the integral, ∫cos(x)dx, cos(x) could represent a varying quantity such as force and dx could represent time. The integral gives us impulse with units N.s, Newton-seconds.

What could the integral, ∫cos(x)cos(x)dx, represent? I couldn't relate it to anything from physics. Could you please guide me with this?

Thanks a lot!

 

[willem2]

What could the integral, ∫cos(x)cos(x)dx, represent? I couldn't relate it to anything from physics. Could you please guide me with this?

Seeing squares of amplitude, speed, electric field, electric charge. voltage, current is common. You often integrate them over time or space to get power of energy. You don't really see the square of power itself much.
I can't really tell if an integral is valid without knowing where it came from.

 

[Ibix]

What could the integral, ∫cos(x)cos(x)dx, represent?

You can certainly come across the integral of $\cos^2$. For example a resistor of resistance $R$ subject to a voltage $V_0\cos(\omega t)$ will dissipate energy $(V_0^2/R)\int\cos^2(\omega t) dt$.

I can't think of a reason for integrating the square of power. Think about what integration is: it's the limit of a sum. We are saying that the energy dissipated by the resistor is the power it dissipates at any given time multiplied by a tiny time interval $dt$, summed over the range of times we're interested in. That's why the integral of power with respect to time is interesting - because power is energy per unit time and energy per unit time times time just gives us the total energy.

I can't think why you'd want to square power and then integrate with respect to time. You can certainly calculate it. It gives you something with units of energy squared per unit time, but I don't know why you'd want to know this. $E=\int Pdt$ is a bit like saying I charge £40/hr and it's going to take me three hours to do the job, so that'll be £120. Calculating $\int P^2dt$ is a bit like squaring my rate and multiplying by three hours. You can do it perfectly well, but why would you? It's not useful for anything.

I couldn't relate it to anything from physics.

That's exactly the point.

 

[PainterGuy]

Thanks a lot for your help!

@Ibix, your post was really helpful.

I also found another somewhat related thread where integral doesn't relate to anything physical.
https://www.physicsforums.com/threads/physical-meaning-of-time-integral-of-voltage.473967/