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This is a closed container, which is filled with liquid (blue). We give the piston a horizontal force (red). Will there be a greater force on the slope of the piston? Why? According to Pascal's law, it seems that a greater force will be obtained. But I'm not sure. enter image description here

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Pushing the piston increases the pressure in the liquid. This pressure will be the same throughout the liquid if we neglect the increase in pressure with depth due to gravity. The liquid exerts a force at right angles to the surface on all surfaces with which it is in contact. The force it exerts per unit area of surface is equal to the pressure.

Let the sloping part (area A') of the piston be at an angle $\theta$ to the end face (area A) of the piston if it were shaped as usual. Then $$A'= \frac{A}{\cos \theta}$$So the force on the sloping part of the piston will be greater than it would be on the end of the piston if shaped as usual, since $pA'>pA$ simply because $A'>A.$.

The component of this force parallel to the axis of the piston is $$pA' \cos \theta = p \times \frac{A}{\cos \theta} \times \cos \theta =pA$$ So this force component is exactly the same as if the piston were shaped as usual.

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  • $\begingroup$ That is to say, a small force can produce a large force? $\endgroup$ – enbin zheng May 5 at 0:37
  • $\begingroup$ You could say that, but you don't need a specially-shaped piston in order to make this happen. The walls of the container itself, being of greater area than the piston (conventional or sloping), will have a greater force on them than the force you are applying. Recommend you read about Pascal's principle and the 'hydraulic press'. $\endgroup$ – Philip Wood May 6 at 20:30

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