Name: History of a Barometer
Uploaded: 2019-07-19
Duration: 4 min 12 s
Description: A barometer is a scientific instrument measuring air pressure in a certain environment. Do you want to know how it came into the picture? Watch this video.

Question

By a suitable mathematical expression, show that the buoyant force on a body placed in a liquid is equal to the weight of the liquid displaced by the body.

Accepted Answer

Consider a slid block inside the liquid. The liquid exerts pressure on the block from all sides. So

$Pressure exerted by the liquid on face AD = Pressure exerted by the liquid on face BC$

The top surface $AB$ is pressed by the liquid column $AEFB$ .

The bottom surface $CD$ is pressed upwards by the weight of the liquid column $DEFC$ .

If $a$ is the area of faces $AB$ and $CD$ $d$ is the density of liquid

then,

$T o t a l d o w n w a r d f o r c e o n f a c e A B = a h_{1} d g a n d t o t a l u p w a r d f o r c e o n f a c e C D = a h_{2} d g$

Since $h_{2}$ is greater than $h_{1}$ , the upward force on $CD$ is greater than the downward force on $AB$

then,

$Resultant upward force on on the block by the liquid = a h_{2} d g - a h_{1} d g = (h_{2} - h_{1}) a d g$

The quantity $(h_{2} - h_{1}) a$ is equal to the volume of the block $V$ . So the resultant upward force i=on the block is $V d g$ . The quantity $V d$ is equal to the mass of liquid displaced by the block.

$R e s u l t a n t u p w a r d f o r c e o n t h e b l o c k = (m a s s o f t h e d i s p l a c e d l i q u i d b y t h e b l o c k) g = W e i g h t o f t h e d i s p l a c e d l i q u i d b y t h e b l o c k$

The resultant upward force exerted by the liquid on the block is called the buoyant force or upthrust.

So $Buoyant force = weight of the liquid displaced by the body .$

By a suitable mathematical expression, show that the buoyant force on a body placed in a liquid is equal to the weight of the liquid displaced by the body.

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