Question

Explain Pascal’s law taking into consideration the effect of gravity. Explain the working of a hydraulic lift.

Answer 1

Now, we know that pressure in a fluid depends only upon depth, and any increase in pressure at the surface must be transmitted to every point in the field. This is known as Pascal’s law as it was first recoginsed by the French Scientist Blaise Pascal.

According to this law, pressure exerted on any fluid at any place is transmitted equally in all directions and is always normal to the container. This law is also called the law of transmission of pressure.

Proof of Pascal’s law: We can prove Pascal’s law by using two principles:

(i) The force on any layer of a fluid at rest is normal to the layer, and

(ii) Newton’s 1st law of motion.

We consider a small element ABCDEF in the form of a right angled prism inside a vessel containing fluid at rest.

The effect of gravity is same for all points in the element. We consider that fluid exerts pressure P_a, P_b and P_c on the forces BEFC, ADFC and ADEB respectively of this element and corresponding normal forces on these faces are F_a, F_b and F_c.

Let A_a, A_b and A_c be the respective areas of the three faces. In right angled  ∆ABC, let ∠ACB = θ

Now we consider different forces in equilibrium,

Horizontally, F_b sinθ = F_c ............(1)

Vertically F_b cosθ = F_a ..............(2)

Also, from the geometry of the figure, we get

A_b sinθ = A_c .........(3)

and A_b cosθ = A_a .........(4)

From equations (1), (2), (3) and (4)

This proves Pascal's law

1. Hydraulic Brakes: In hydraulic brakes, when a little force with the foot is applied on the pedal (see fig), the pressure so applied gets transmitted through brake oil to act on a larger area where pistons move the brake-shoes against the brake-drums. So on applying a little pressure on the pedal a large force is exerted on the wheel which produces deceleration.

2. Hydraulic Lift: One can see at motor service station that the cars and large trucks being raised to convenient heights so that mechanic can work under them. Here, a slight pressure is transmitted through the liquid to act on a large surface, thus, producing sufficient force to raise up the vehicle. If F₁ is the force acting on the piston of area A₁, then the pressure below that piston is \(P = \frac{F_1}{A_1}\)

According to the Pascal’s law, the pressure on any area element in the large piston will be same. If the cross-sectional area of this cylinder is A₂, then the total force acting on the larger piston will be

F₂ = P A₂ or P = \(\frac{F_2}{A_2}\)

So, \(\frac{F_1}{A_1} = \frac{F_2}{A_2}\)

The gain in force will be,

\(\frac{F_2}{F_1} = \frac{A_2}{A_1}\)

This ratio is also called mechanical advantage.