Question

The pressure exerted by a liquid:
$(a)$ Increase with depth
$(b)$ Decrease with depth
$(c)$ Is constant
$(d)$ First increase then decrease

Answer 1

Hint – In this question use the concept that the pressure (p) exerted by the liquid is directly proportional to the square of the velocity (v) or speed (v) of the body in that liquid that is $p \propto {v^2}$but however using third law of motion we can directly claim that velocity is dependent upon height as $v = \sqrt {2gh} $. This will help commenting upon the effect on pressure due to variance in height.
Step-By-Step solution:
Let us consider an object is placed at a height h now we have to free fall this object so its initial velocity becomes zero and let its final velocity be (v). When it reaches the ground and we all know the acceleration of gravity is acting downwards therefore it is (a = g) m/s2
So from third equation of motion we have,
${v^2} = {u^2} + 2as$..................... (1), where v = final velocity, u = initial velocity, a = acceleration due to gravity and s = height of the particle.
Now substitute the values in equation (1) we have,
$ \Rightarrow {v^2} = {0^2} + 2\left( g \right)h$
$ \Rightarrow v = \sqrt {2gh} $
So the speed of the object falls freely from a height h is $\sqrt {2gh} $.
Now let us consider a tank having full of water and depth h and a hole at the bottom of the tank.
Now according to Torricelli's theorem the speed of the efflux (i.e. material that is flowing) from the bottom of the tank having depth (h) (i.e. having a hole at the bottom of the tank) is equal to the speed of the object having free fall from a height h.
Remember that height h of an object should be equal to the depth of the tank.
Therefore we use the h symbol for both the cases.
So the speed of the efflux from the bottom of the tank having depth h and having a sharp hole at the bottom of the tank is $\sqrt {2gh} $.
$ \Rightarrow v = \sqrt {2gh} $m/s............... (1)
So as we see that the speed of the fluid from the bottom of the tank is directly proportional to the square root of the depth (h).
So as we increased the depth the speed of the liquid also increased.
Now as we know that the pressure (p) exerted by the liquid is directly proportional to the square of the velocity (v) or speed (v) of the liquid.
$ \Rightarrow p \propto {v^2}$
Now from equation (1) we have,
$ \Rightarrow p \propto 2gh$
$ \Rightarrow p \propto h$
So pressure is directly proportional to the depth of the liquid.
So as we increase the depth pressure is also increasing.
Hence option (A) is the correct answer.

Note – It is very practical that the pressure increases as depth of water increases, this is the main reason that the submarines are made of up very strong material as their functioning is in very deep oceans so the walls of the submarine must be able to bear the pressure exerted by the fluid.