One of the most important facts about fluid pressure is that a change in pressure at one part of the liquid will be transmitted without any change to other parts. This was put forward by Blaise Pascal (1623 - 1662), a French mathematician and physicist. This rule is known as Pascal's law.

*Pascal's law*

*One of the most important
facts about fluid pressure is that a change in pressure at one part of the
liquid will be transmitted without any change to other parts. This was put
forward by Blaise Pascal (1623 - 1662), a French mathematician and physicist. This
rule is known as Pascal's law.*

*Pascal's law states that if
the effect of gravity can be neglected then the pressure in a fluid in
equilibrium is the same everywhere.*

Consider any two points A and B inside the
fluid. Imagine a cylinder such that points A and B lie at the centre of the
circular surfaces at the top and bottom of the cylinder (Fig.). Let the fluid
inside this cylinder be in equilibrium under the action of forces from outside
the fluid. These forces act everywhere perpendicular to the surface of the
cylinder. The forces acting on the circular, top and bottom surfaces are
perpendicular to the forces acting on the cylindrical surface. Therefore the
forces acting on the faces at A and B are equal and opposite and hence add to
zero. As the areas of these two faces are equal, we can conclude that pressure
at A is equal to pressure at B. This is the proof of Pascal's law when the
effect of gravity is not taken into account.

*Pascal's law and effect of gravity*

When gravity is taken into account, Pascal's law is to be modified.
Consider a cylindrical liquid column of height *h* and density ρ in a vessel as shown in the Fig..

If the effect of gravity is neglected, then pressure at M will be
equal to pressure at N. But, if force due to gravity is taken into account,
then they are not equal.

As the liquid column is in equilibrium, the forces acting on it are
balanced. The vertical forces acting are

(i) Force P1A acting vertically down on the top surface.

(ii) Weight mg of the liquid column acting vertically downwards.

(iii) Force P2A at the bottom surface acting vertically upwards.
where P1 and P2 are the pressures at the top and bottom faces, A is the area of
cross section of the circular face and m is the mass of the

cylindrical liquid column.

At equilibrium, P1A + mg - P2A = 0 or P1A + mg = P2A

*P2 = P1 + mg/A*

*m = Ahρ*

*P2 = P1 + Ah g/A*

*P2 = P1 + hρg*

This equation proves that the pressure is the
same at all points at the same depth. This results in another statement of *Pascal's law* which can be stated as *change in pressure at any point in an
enclosed* *fluid at rest is transmitted
undiminished to all points in the fluid and act in all directions.*

*Applications of Pascal's law*

*(i) Hydraulic lift*

An important application of Pascal's law is the hydraulic lift used
to lift heavy objects. A schematic diagram of a hydraulic lift is shown in the
Fig.. It consists of a liquid container which has pistons fitted into the small
and large opening cylinders. If *a _{1}*
and

*F/ a _{1} = W/a_{1}*

*W = Fa _{2}/a_{1}*

_{}

*This is the load that can be
lifted by applying a force F on A. In the above equation 2 a _{1}/a_{2
} is called mechanical advantage of
the hydraulic lift. One can see such a lift in many automobile service stations*

*(ii) Hydraulic brake*

When brakes are applied suddenly in a moving
vehicle, there is every chance of the vehicle to skid because the wheels are
not retarded uniformly. In order to avoid this danger of skidding when the
brakes are applied, the brake mechanism must be such that each wheel is equally
and simultaneously retarded. A hydraulic brake serves this purpose. It works on
the principle of Pascal's law.

Fig. shows the schematic
diagram of a hydraulic brake system. The brake system has a main cylinder
filled with brake oil. The main cylinder is provided with a piston P which is
connected to the brake pedal through a lever assembly. A *T* shaped tube is provided at the other end of the main cylinder.
The wheel cylinder having two pistons P_{1} and P_{2} is
connected to the *T* tube. The pistons
P_{1} and P_{2} are connected to the brake shoes S_{1}
and S_{2} respectively.

When the brake pedal is pressed, piston P is
pushed due to the lever assembly operation. The pressure in the main cylinder
is transmitted to P_{1} and P_{2}. The pistons P_{1}
and P_{2} push the brake shoes away, which in turn press against the
inner rim of the wheel. Thus the motion of the wheel is arrested. The area of
the pistons P_{1} and P_{2} is greater than that of P.
Therefore a small force applied to the brake pedal produces a large thrust on
the wheel rim.

The main cylinder is connected to all the
wheels of the automobile through pipe line for applying equal pressure to all
the wheels .

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11th 12th std standard Class Physics sciense Higher secondary school College Notes : Pascal's law and effect of gravity |

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