Buoyancy-Definition, Equations, Practice problems,FAQs

Rohit conducts a small experiment in his bathtub: he drops a mug into the bathtub filled with water. He observes that the mug floats instead of sinking. Next, he takes a leather cricket ball and drops it into the tub, he observes that the ball sinks. Also, the instant when the mug or the ball was dropped into the tub, a small amount of water splashes off. Archimedes, a Greek philosopher, observed this phenomenon when he jumped into a tub full of water–he shouted “Eureka, Eureka!'' in sheer happiness. He concluded that whenever an object is fully immersed in a fluid, it displaces the amount of the fluid whose volume is equal to the volume of the body. This is the reason behind Rohit’s observation of the water getting splashed. Additionally, the fluid exerts an upward force on the submerged object, called “buoyant force”. On the other hand, if the weight of the object just balances the buoyant force exerted by the fluid, then it floats. In this article, we will explore buoyancy in detail.

Table of contents

Definition of buoyancy
Centre of buoyancy
Density and Relative Density
Apparent weight and real weight
Buoyancy in the case of accelerating fluids
Practice problems
FAQs

Definition of buoyancy

Buoyancy refers to the property of fluids that enable objects to float on their surfaces. The body could be partially or fully submerged. The upward force exerted by the fluid on the object is called the buoyant force. When a body is immersed in a fluid, it displaces some amount of fluid. The apparent weight of the body while immersed gets reduced by the amount equal to the weight of the fluid displaced. A steel ball sinks inside a fluid compared to a hollow plastic ball, because it is more dense, and the downward force exerted by it because of its own weight is more than the buoyant force exerted by the displaced fluid. The upward buoyant force exerted by a fluid which has less density than that of steel is not sufficient enough to balance the steel ball. Hence, the steel ball sinks while the hollow ball floats.

Note:

Buoyant force is also called upthrust.

Centre of buoyancy

Centre of buoyancy is the point where the net buoyant force acts on the body. In other words, it is the point where the centre of gravity of the displaced fluid acts.

Density and Relative Density

The density () of a substance is defined as the ratio of the mass of the substance to the volume of the substance. Higher the density, it means that the substance’s molecules are densely packed. So, the solids have higher density in comparison to liquids.

$ρ = \frac{M a s s}{V o l u m e}$

The SI unit of density is kg m-3. Its dimensional formula is $[M^{1} L^{- 3} T^{0}] .$

Relative density of a liquid is defined as the ratio of the density of the liquid to the density of another liquid taken as reference. If water at 40 C is used as the reference liquid, then the relative density becomes the specific gravity.

Specific gravity of a liquid is defined as the ratio of the density of the substance to the density of water at 4⁰ C.

$S p e c i f i c g r a v i t y o f a l i q u i d = \frac{D e n s i t y o f s u b s t a n c e}{D e n s i t y o f w a t e r a t 4^{0} C}$

It has no unit. It is just a ratio of two similar quantities. So, it is dimensionless.

Apparent weight and real weight

When a body is immersed in a liquid, then its weight appears reduced. This is due to the upthrust exerted by the fluid. The reduced weight is called the apparent weight (Wapp).

$W_{a p p} = A c t u a l w e i g h t - U p t h r u s t$

Let V be the volume of the body which is completely immersed in a fluid. Let s indicates the density of the solid and L indicates the density of the liquid. Then,

$W_{a p p} = V ρ_{s} g - V ρ_{L} g = V ρ_{s} g (1 - \frac{ρ_{L}}{ρ_{s}}) W_{a p p} = W_{a c t u a l} (1 - \frac{ρ_{L}}{ρ_{s}}) - - (i)$

g- acceleration due to gravity

Special cases

(i) If s<L, while fully immersed, the upthrust exerted is greater than the weight of the body. Hence, the net force on the body will be directed in the upward direction.

(ii) If s=L, the upthrust exerted is balanced by the weight of the body. In this case, the apparent weight from equation (i) becomes, Wapp=0

(iii) If s>L, the upthrust is less than the weight of the liquid.

Buoyancy in the case of accelerating fluids

Let us consider a body being placed inside a liquid of density L. The entire apparatus is placed in an elevator moving up with acceleration a. The upthrust/ buoyant force in this case would become, $F_{u p t h r u s t} = {V ρ}_{L} (g + a)$

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When the same elevator is moving downwards with an acceleration a, we would get

$F_{u p t h r u s t} = {V ρ}_{L} (g - a)$

Practice problems

Q. Density of ice is 900 kg m-3. A chunk of ice is floating in water (whose density is 1000 kg m-3). Calculate the fraction of the piece of ice inside the water.

A. Given,

$ρ_{i} = 900 k g / m^{3} ρ_{w} = 1000 k g / m^{3}$

Let Vi indicate the volume of the ice piece lying submerged inside the water, and V be the total volume of ice. We can understand that as the ice is less dense than the water. So it will float in the water.

If the ice piece is floating, then weight of the ice= Upthrust exerted by the water.

$V_{i} ρ_{w} g = V {ρ_{i}}_{g} \frac{V_{i}}{V} = \frac{ρ_{i}}{ρ_{w}} = \frac{900}{1000} = 0.9$

Q. A block of ice is floating in a glass vessel filled with water. How will the water level in the vessel start to change when the ice starts melting?

A. Let m indicate the mass of the ice that floats in water. If the ice is in equilibrium, then,

$m g = V_{i} ρ_{w} g, {\Rightarrow V}_{i} = \frac{m}{ρ_{w}} - - (i)$

Here, Vi indicates the portion of the ice that is immersed inside the water. Now, if V is the volume of water, once ice starts melting, mg=Vwg

$V_{=} \frac{m}{ρ_{w}} - - (i i)$

From equations (i) and (ii) we get

Vi=V the level of water does not change.

Q. An object floats with 40 % of its volume outside water. When the object is allowed to float in oil, 60% of its volume lies outside. The relative density of oil is,

(a) 0.9 (b) 1.0 (c) 1.2 (d) 1.5

A. d

Let s indicate the density of the solid, and V be its volume. Then 0.6 V is immersed in water. If w indicates the density of water, $V ρ_{s} g = 0.6 {V ρ}_{w} g - - (i)$

If L indicates the density of oil, then 0.4 V is the volume of the fluid immersed in oil.

$V ρ_{s} g = 0.4 {V ρ}_{L} g - - (i i)$

From (i) and (ii), we get

$0.6 {V ρ}_{w} g = 0.4 V ρ_{L} g R D_{L} = \frac{ρ_{L}}{ρ_{W}} = \frac{6}{4} = 1.5$

Q. An object having weight W and density is immersed in a liquid having density . Calculate the apparent weight.

$ρ - σ$ (b) $\frac{ρ - σ}{W}$ (c) $W (1 - \frac{σ}{ρ})$ (d) $W (1 - \frac{ρ}{σ})$

A. c

We know, $W_{a p p} = W_{a c t u a l} (1 - \frac{ρ_{L}}{ρ_{s}}), ρ_{s} = ρ, ρ_{L} = σ, W_{a c t u a l} = W$

Wapp=W1-

FAQs

Q. What factors affect buoyancy?
A. Buoyancy is affected by: the density of the fluid, the volume of the fluid displaced and the effective acceleration due to gravity. Because, when the container in which the fluid is placed is accelerating upwards with an acceleration a, the effective acceleration becomes g+a. Or, when the same elevator is moving downwards with an acceleration a, the effective acceleration becomes g-a.

Q. Does temperature affect buoyancy?
A. When the temperature increases, the molecules of the fluid get excited and start to move around rapidly, increasing the volume of the fluid and decreasing the density of the fluid. Since, the buoyant force is directly proportional to the density of the fluid, as temperature increases, the density decreases, buoyancy decreases as well.

Q. How does the shape of an object affect buoyancy?
A. Buoyant force depends upon the volume of the fluid displaced. It does not depend upon the surface area of the object. The shape of the body can only shift the point of action of the buoyant force.

Q. How is density related to the buoyant force?
A. Buoyant force is equal to the product of the volume of the object, the density of the fluid, and the acceleration due to gravity. Hence, higher the density of the fluid, higher is the buoyant force exerted on the body.