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Let's consider the following arrangement:

enter image description here

I know that because of conservation of energy the force on the car has to be amplified. The work $W$ done is

$$W = \text{Force} \cdot \text{distance} \, ,$$

which in turn is the amount of energy transferred.

How is this force amplified at the molecular level. For example, if we put sand in place of water and suppose that the molecules are connected with springs then can we show that the springs will get compressed in such a way so as to amplify the force on the car?

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from the figure: liquid that cannot be compressed... so forget compression of springs.

Molecularly, particles in a liquid have much less space to move around (short mean free paths) compared to particles in a gas. The result is that any applied pressure to these particles is instantaneously transferred through the fluid. The mechanism by which this happens is the collision of particles which exchange kinetic energy/momentum. The amplification of the force occurs because the area at position 2 is much larger so many more particles can impose a force there compared to the piston at position 1.

Imagine you impose a pressure $p$ at position $1$, by conservation of mechanical energy (disregard effects of gravity), this must be the same pressure at position $2$. So we can state: $$p=\frac{F_1}{A_1}=\frac{F_2}{A_2}\rightarrow\frac{F_2}{F_1}=\frac{A_2}{A_1}$$ If we assume $A_2/A_1\gg1$ like in the figure it follows, $F_2/F_1\gg1$, i.e. a small force at position $1$ translates to a large force at position $2$.

By conservation of energy the amount of work $W$ at position $1$ must be the same as at position $2$, i.e.: $$W=F_1\Delta x_1=F_2\Delta x_2\rightarrow\frac{\Delta x_2}{\Delta x_1}=\frac{F_1}{F_2}\ll1$$ We find that displacement of the piston at position $2$ is only a fraction of the displacement of the pistion at position $1$.

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  • $\begingroup$ But what happens at the molecular level? I am looking for this type of explanation: physics.stackexchange.com/a/22994/93864 I know that the energy is transferred in someway which amplifies the force. But what is that way? $\endgroup$ – user31782 Nov 17 '15 at 13:59
  • $\begingroup$ That answer deals with solids which usually has particles in lattices structures; this could be modeled with springs. Fluids cannot (generally) be modeled this way because they have much more freedom than particles in solids. The mechanism you are looking for is the collision of particles which exchange kinetic energy/momentum. The amplification occurs because the area at position $2$ is much larger so many more particles can impose a force there compared to the piston at position $1$. $\endgroup$ – nluigi Nov 17 '15 at 14:15
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    $\begingroup$ ...where "instantaneously" means "at the speed of sound in the fluid." $\endgroup$ – Solomon Slow Nov 17 '15 at 14:25
  • $\begingroup$ Yes that is what I was looking for. The number of particles are more at 2. A lot of particles get K.E from 1 and reach at 2 and strike simultaneously. Thanks for explaining. $\endgroup$ – user31782 Nov 17 '15 at 14:29
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    $\begingroup$ @user - you are overthinking this; kinetic energy is a scalar quantity, it has no direction and can't be decomposed in different contributions. Fluid particles also are in constant random motion (speed of sound in air ~300 m/s) even if there is no bulk motion of the fluid. So their velocity vector at position $2$ is not in any preferred direction. The small 'solid' particles will have to be small enough to have the same freedom of movement as the fluid particles. $\endgroup$ – nluigi Nov 18 '15 at 10:04

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