Upthrust, Viscosity and Terminal Velocity

Upthrust

Objects float if they are less dense than, or of equal density with, the fluids (liquids and gases) they occupy. For example, while steel is denser than water, a steel ship can be made to float by introducing air-filled chambers which reduce the density of the ship relative to that of water. Floating bodies usually have part (or all) of their bodies in the liquid and as a result experience an upward force called upthrust (or buoyant force). Consider for instance an object of mass m, cross-sectional area A and height h fully submerged in a liquid of density ρ.

The upward pressure the liquid exerts on the object at X is less than pressure the liquid exerts on the object at Y which is equal to  ρgh. Also, the pressure the object exerts downwards on the liquid at X is less than that it exerts on the liquid at Y which is equal to mg/A . For the body not to sink below point Y, the upward force and hence pressure must be equal to the downward force (hence pressure) at that point. Thus, 

                                                                               (i)

Where V is the volume of liquid displaced which is equal to the volume of the submerged part of the object. Hence;

                                                         (ii)

But,

                                                                       (iii)

Where V is the volume of liquid displaced which is equal to the volume of the submerged part of the object. Hence;

                                                                (iv)

Given that;

  (v)

then;

                                                                 (vi)

Equation (vi) shows that when an object floats in a liquid (or fluid in general), the weight of the liquid displaced is equal to the weight of the submerged section of the object.  This is known as the law of floatation. 

The terms on the left-hand sides of equations (v) and (vi) represent upthrust (U) which is the upward force the liquid exerts on the floating object, while the right-hand side represents the weight of the part of the object submerged, that is;

                                        (vii)

It follows from equation (vii) that; 

(i) If upthrust is greater than the weight of an object, the object floats with part of it partially submerged

(ii) If upthrust is equal to the weight of the object, the object floats while fully submerged but close to the surface.

(iii) If upthrust is less than the weight of the object, the object sinks.

(iv) The upthrust experienced by a floating object is proportional to the density of the liquid it is immersed in. It is for this reason that it is easier to float in the ocean (salty) than in a lake (fresh) as salty water is denser than fresh water. 

(v) The upthrust experienced by a floating object is also proportional to the volume of the section of the object submerged. This means that the larger the body submerged, the greater the upthrust.

Viscosity and Terminal Velocity

The frictional force (force that opposes motion) that a fluid (gas or liquid) exerts on an object moving through it is called viscous drag. Liquids that flow easily for example water offer less resistance to objects moving through them compared to fluids that do not flow easily (viscous fluids) for example glycerine. The coefficient of viscosity is a measure of resistance to the flow of a fluid; the higher the coefficient of viscosity, the higher the resistance to flow.

Consider a small metallic ball, mass m, radius r, volume V and density ρ falling freely from rest through a long column of a viscous fluid such as glycerine of density σ. Three forces act on the ball:

(1) Gravitational force W (weight) acting vertically downwards where; 

                                                                                 (i)

But

and

                                                                               (ii)

hence;

                                                                   (iii)

(2) Upthrust U due to buoyancy which acts vertically upwards: Upthrust is equal to the weight of the liquid displaced, while volume of the liquid displaced is equal to the volume V of the ball. If m_L be the mass of the water displaced and σ the density of the liquid, it follows that;

                                                                          (iv)

                                                                          (v)

(3) Viscous drag F which acts vertically upwards. Viscous drag is the resistance the fluid offers to the motion of the ball. Like all forms of friction, viscous drag acts in a direction opposite the direction of motion. It can be shown that:

                                                                              (vi)

where η is the coefficient of viscosity of the liquid r the radius of the ball and v its velocity. Equation (vi) shows that viscous drag increases with coefficient of viscosity of the liquid, size and velocity of the ball.

Initially, the resultant upward force U+F on the ball is less than the downward force W hence the ball accelerates downwards. As the velocity increases, viscous drag also increases leading to an increase in the resultant upward force. A point is eventually reached when the resultant upward force equals the downward force, in which case the net force on the body falls to zero. Beyond this point, the ball moves downwards with a constant velocity. The constant velocity is referred to as terminal velocity. The motion of a ball falling through a viscous fluid can be represented graphically as:

Examples

Example 1: KCSE 2021

(1) Figure 10 shows a wooden block of volume 90cm³ floating with a third of its body submerged in water of density 1gcm^-3 . (𝑔 = 10𝑁𝑘𝑔⁻¹) 

Determine:

(a) the weight of the block

Solution

weight of the body = weight of the liquid displaced.

Given that

then the mass of liquid displaced;

volume (V) of the liquid displaced = volume of submerged section;

Mass of liquid displaced; 

weight of the body = weight of the liquid displaced 

(b) the weight of a metal block that can be placed onto the block so that its top surface is on the same level as the water surface. (3marks)

Solution

If fully submerged, 

Mass of the liquid displaced

Weight object=weight of liquid displaced 

(2) Figure 11 shows a solid metal suspended in oil using a thread.

(a) Other than upthrust, list two other forces acting on the sphere. (2 marks).

• Tension
• Weight

(b) The oil is carefully and gradually drawn from the beaker. State the effect on each of the two forces in (b). (2 marks)

• The body is under 3 forces; tension and upthrust upwards and weight downwards. As the oil is drawn from the beaker, the section of the body submerged reduces hence upthrust reduces. Hence tension gradually increases. 
• Weight remains constant since it is a function of mass and acceleration due to gravity, both of which remain unchanged.

KCSE 2020

(1) An object placed on the surface of water in a beaker starts to sink immediately. It is observed that it stops sinking when half of its volume is below the water surface. State the reason for this observation. (1 mark)

Initially, weight of the body is greater than the upthrust (weight of the liquid displaced). As more of the body goes under the liquid, more liquid is displaced hence upthrust increases. The body come to rest when upthrust becomes equal to the weight.

KCSE 2019

(1) Initially, weight of the body is greater than the upthrust (weight of the liquid displaced). As more of the body goes under the liquid, more liquid is displaced hence upthrust increases. The body come to rest when upthrust becomes equal to the weight.

But g=10m/s² hence;

Mass of water displaced = 650 g

Volume of water displaced = volume of the object= 800 cm³

Hence

(2) Figure 8 shows the graph of velocity against time for a small steel ball falling in a viscous liquid.

(a) Describe the motion of the steal ball as represented

part OA; force downwards is greater than total force upwards hence the body is accelerating.

(b) Explain why the velocity between A and B is constant. (3 marks)

As the ball falls through the fluid, the viscous drag increases until the sum of the viscous drag and the upthrust becomes equal the weight of the steel ball, hence the net force and consequently acceleration falls to zero and the body moves at a constant velocity called terminal velocity.

KCSE 2018

(1) A wooden cube of side 0.5m floats in water fully submerged. Determine the weight of the cube. (Density of the water = 1g.cm^-3). (2 marks)

Since object is fully submerged, then the weight of the cube equals the weight of the water displaced.

Volume of the object equals V the volume of liquid displaced. 

KCSE 2017

(1) In an experiment to determine the density of Liquid R. a student obtained the followed data: 

• Mass of an empty density bottle = 55.0 g 
• Mass of the density bottle + water = 80.0 g 
• Mass of the density bottle + Liquid R = 70.0 g. 

Determine the density of Liquid R. (density of water is 1000 kgm^-3 (3 marks)

Density is defined as mass of a substance divided by its volume, that is;

The density (D_R of liquid R is therefor given by:

 Mass of R is easy to obtain since;

Working with mass in kg (SI units);

 Volume of R on the other hand cannot be obtained in such a straight forward manner. We however know that the density bottle will hold a liquid that is equal to its volume hence:

The volume of the density bottle is not provided in the question. We however know that;

From definition of density, it follows that;

Hence;

(2) State the law of flotation. (1 mark)

a floating body displaces a fluid equal to the weight of part of the body submerged and is equal to upthrust.

(3) Figure 6 shows two solids W and X made of the same material and immersed in water.

(a) State with a reason which one of the containers experiences a greater upthrust? (2 marks)

X because the larger the body submerged the greater the upthrust.

(b) Solid W weighs 12N in air. 2N in water and 4 N in another liquid. Determine the density of the other liquid. (3 marks)

To find density of the other liquid, we need to find its mass and volume. Now; 

But 

Taking g=10 m/s² it follows that;

 Now, 

Given the density of water as 1000 kg/m³ then;

(4) Figure 7 shows two identical wooden blocks each of mass 0.2 kg suspended in water by two strings M and N.

Given that the upthrust on each block is 3.2N, determine the tension in string: 

(i) M. (2 marks)

String M is preventing first block from moving up hence tension T_N is acting downwards. Weight of first block is downwards while upthrust is upwards:

Upward forces are equal to downward forces hence; 

(ii) N. (2 marks)

For second block, the tension T_M and upthrust are acting upwards while the tension of the second string T_N and weight are acting downwards.

(5) State any one application of hydrometers. (1 mark)

A hydrometer is an instrument for measuring relative density of liquids. A hydrometer when dipped in a denser liquid for example glycerine will displace less liquid (will not go as deep) while when dipped in a less dense liquid (for example paraffin), it will displace more liquid (will go deeper). It can be used to determine the purity of a liquid, for example test the purity of milk given that the density of pure milk is different from the density from that of ‘watered’ milk.

Practice Questions

KCSE 2016

(1)    Figure 1 shows a change in volume of water in a measuring cylinder when an irregular solid is immersed in it.

Given that the mass of the solid is 567 g, determine the density of the solid in g/cm3 (Give your answers correct to 2 decimal places). (3 marks) 

(2) Figure 4 shows a uniform light bar resting horizontally on corks floating on water in two beakers A and B.    

  Explain why the bar tilts towards side A when equal amount of heat is supplied to each beaker (2 mark                                    

KCSE 2015

(1) ) Figure 14 shows a block floating in water.

When the water is heated; it is observed that the block sinks further. Explain this observation. (2 marks)

KCSE 2014

(1) Figure 11 shows a test—tube whose cross-sectional area is 2 cm^z partially filled with lead shot floating vertically in water.

(Take gravitational acceleration as IO ms² and density of water of as I g cm³) 

(a)Determine the:

(i) volume of the water displaced; (2 marks)

l(ii) weight of water displaced. (3 marks)

(b) State the combined weight of the test—tube and the lead shot. (1 mark)

(c) Determine the length of the test—tube that would be submerged in a liquid of density 0.8 g cm³. (4 marks)

(d) The set up in figure 11 can be used as a hydrometer to measure densities of liquids. State how such a hydrometer would be improved to measure small differences in densities of liquids. (1 mark) 

KCSE 2013

(1) Figure 12 shows a weighing balance on which a beaker containing some water is placed. The reading on the balance is 2.80 N. A metal block weighing 2.7 N is suspended from a spring balance.

(a) State what is observed on the spring balance and the weighing balance, as the metal block is gradually lowered into the water.

(i) Observation on spring balance. (1 mark)

(ii) Observation on weighing balance. (1 mark)

(b) Explain the observation made on the spring balance in (I). (2 marks)

(c|) When the metal block is fully immersed in the water, the reading on the spring balance is found to be 2.46 N. Determine the:

(i) reading on the weighing balance. (2 marks)

(ii) density of the metal. (3 marks)

(2) Figure 13 shows a hydrometer with a thin stem floating in water in a beaker.

State with a reason what is observed on the hydrometer when the temperature of the water is raised. (2 marks)

KCSE 2012

(1) Figure 13 shows a log of wood of mass 20 kg submerged in water in a pond and held in position by a string fixed to the bottom of the pond.



Given that the density of water is 1000 kgm^-3 and that of wood is 800 kgm^-3, determine the:

(a) The volume of the log. (3 marks)

(b) Upthrust of the log. (2 marks)

(c )Tension in the string. (2 marks

KCSE 2011

(1) State the condition necessary for a body to float in a fluid. ( 1 mark)

(2) A ship made of steel is observed to float on water yet the density of steel is approximately eight times that of water. Explain this observation. (2 marks)

(3) Figure 17 shows three stages of an experiment to determine relative density of cork which normally floats on water. To make it sink, a sinker is hung below the cork.

In (I) a spring balance is used to measure the weight W of the cork in air.

in (ll) the spring balance is used to measure the apparent weight W 1, when only the sinker is submerged in water.

In (III) the spring balance is used to measure the apparent weight W2 when both the cork and the sinker are submerged.

The following observations were made.

• W = 0.08N
• W‘ = O.60N
• W2 = O.28N

Use this information to determine the:

(i) upthrust on cork. (2 marks)

(ii) relative density of cork. (3 marks) 

(4) Figure 18 shows parts of a simple submarine, a ship that can travel both on Water and under water. To do this water is pumped in or out of the ballast tanks.

Explain how the tanks are used to change the depth of the submarine. (2 marks)

KCSE 2010

(1)The weight of a solid in air is 5.0N. When it is fully immersed in a liquid of density 500kgm ³, its weight is 4.04N.  Determine:

      (a) the upthrust in the liquid     (1 mark)

      (b) the volume of the solid      (2 marks)

(2)  In an experiment to determine the density of sand using a density bottle, the following measurements were recorded:

• Mass of empty density bottle = 43.2g
• Mass of density bottle full of water = 66.4g
• Mass of density bottle with some sand = 67.5g
• Mass of density bottle with the sand filled up with water  = 82.3g

Use the above data to determine the:

(a) mass of water that completely filled the bottle;    (2 marks)

(b) volume of the water that completely the bottle;        (1 mark)

(c) volume of the density bottle;      (1 mark)

(d) mass of sand;             (1 mark)

(e) mass of water that filled the space above the sand;     (1 mark)

(f) volume of the sand;      (3 marks)

(g) density of the sand.       (2 marks

(3) Figure 8 shows a stone of mass 4.0kg immersed in water and suspended from a spring balance with a string. The beaker was placed on a compression balance whose reading was 85N. The density of the stone was 3000kgm^-3 while the density of the liquid was 800kgm^-3.

Determine:

(a) Volume of the liquid displaced;      (2 marks)

(b) upthrust on the stone;                    (4 marks)

(c) reading of the spring balance;        (2 marks)

(d) reading of the compression balance when the stone was removed from the water.        (2 marks)

Dr. Margaret W. Chege