Chapter Notes: Physical Quantities and Measurement

Table of Contents
1. Introduction
2. Density
3. Vessels for Measuring Volume
4. Density Bottle
5. Relative Density
6. Floating and Sinking
7. Points To Remember
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Introduction

Density

• Each body has a certain mass and a volume.
• Every object has a mass, which is how heavy it is, and a volume, which is how much space it takes up.
• For example, if the mass of an object increases, its density also increases.
• Similarly, if the volume of an object increases, its density decreases, if the mass stays the same.
• Different materials have different densities; for example, the density of iron is more than the density of wood.
• Iron has a higher density than wood because, for the same volume, iron is much heavier than wood.
• For example, if we take a piece of iron and a piece of wood of the same volume, the iron will be much heavier.
• This means iron is more closely packed than wood.
• In simple words, iron is denser than wood because its particles are packed more tightly.
• Similarly, if we compare water and oil, water is denser than oil because water has more mass for the same volume.
• Density is a property of a substance; it tells us how much mass is packed in a certain volume.

The formula for density is:
Density (d) = Mass of the substance (M) ÷ Volume of the substance (V)
So, d = M ÷ V

Unit of density

Unit of density = Unit of mass ÷ Unit of volume

• In the SI system, the unit of mass is kg, and the unit of volume is m³.
• So, the SI unit of density is kg/m³.
• Another common unit of density is g/cm³ (grams per cubic centimeter).

Relationship between kg/m³ and g/cm³:
1 kg/m⁻³ = 10⁻³ g/cm³
Or, 1 g/cm³ = 10³ kg/m³

Examples:
Example 1: Find the density of an iron cube with a mass of 1 kg and a volume of 100 cm³.
Sol: We know, density (d) = Mass (M) ÷ Volume (V)
Given: Mass (M) = 1 kg = 1000 g, Volume (V) = 100 cm³
So, d = 1000 g ÷ 100 cm³ = 10 g/cm³

Example 2: The mass of a copper cube is 0.89 kg, and its volume is 100 cm³. Find its density.
Sol: Given: Mass (M) = 0.89 kg = 890 g, Volume (V) = 100 cm³
So, density (d) = 890 g ÷ 100 cm³ = 8.9 g/cm³

Do You Know?

• The density of a substance does not change with its shape or size.
• Almost all substances expand on heating and shrink on cooling.
• So, the density of a substance decreases when it is heated because its volume increases.
• Similarly, the density of a substance increases when it is cooled because its volume decreases.
• Water is an exception: its density increases from 0°C to 4°C but decreases above 4°C.
• So, the density of water is maximum at 4°C, which is 1 g/cm³.

Determination of Density of a Regular Solid

To find the density of a regular solid, we need to measure its mass and volume:

• Step 1: Measure the mass (M) of the given regular solid using a beam balance.
• Step 2: Find the volume (V) of the given regular solid using the correct formula.
  For a cube, the formula for volume is:
  Volume of cube = (one side)³
  For a cuboid, the formula for volume is:
  Volume of cuboid = length × breadth × height
  For a sphere, the formula for volume is:
  Volume of sphere = 4/3 × π × (radius)³
  (where π = 3.14)
  The side of a cube, or the length, breadth, and height of a cuboid, or the radius of a sphere, can be measured using a metre ruler.
• Step 3: Calculate the density (d) of the regular solid using the formula:
  d = M ÷ V
  For Example: If the mass of an iron cube is 210 g and one side of the cube is 3 cm, find its density.
  Sol: Given: Mass (M) = 210 g, Side of cube = 3 cm
  Volume of cube (V) = (one side)³ = (3)³ = 27 cm³
  So, density (d) = M ÷ V = 210 g ÷ 27 cm³ = 7.78 g/cm³

Vessels for Measuring Volume

Some of the vessels used for measuring liquids are given below:

• Measuring cylinder: It is made of glass or plastic and is graduated (has markings to show volume).
• Measuring beaker:It is made of glass, plastic, or metal (like aluminium) and is used to measure liquids (like milk, oil, etc.).
  • A measuring beaker is available in different capacities, such as 50 mL, 100 mL, 200 mL, 500 mL, 1 litre, etc.
  • The capacity of a measuring beaker is marked on it.
• Eureka can: A Eureka can is a glass or metal container with a spout (or side opening) near its mouth, used to measure the volume of liquids.
• When we pour liquid into the Eureka can, any excess liquid overflows through the spout.

Determination of Density of an Irregular Solid

To find the density of an irregular solid, we need to measure its mass and volume:

• The mass of the irregular solid can be measured using a beam balance.
• To measure the volume of an irregular solid, we use the displacement method.
• In the displacement method, a solid, when put in a liquid, pushes out (displaces) the same volume of liquid as its own volume.
  • Calculate volume: Volume = V₂ - V₁.
  • Calculate density: ρ = m / V.

Examples:
Example 1: A stone has a mass of 180 g. In a measuring cylinder, the water level rises from 40 cm³ to 60 cm³ when the stone is submerged.
Sol: Volume = 60 - 40 = 20 cm³.
Density = 180 / 20 = 9 g/cm³.

Example 2: A metal piece (mass 250 g) raises the water level from 30 cm³ to 55 cm³.
Sol: Volume = 55 - 30 = 25 cm³.
Density = 250 / 25 = 10 g/cm³.

Determination of Density of a Liquid

To find the density of a liquid (like milk, oil, etc.), we need to measure its mass (M) and volume (V).

• The mass (M) of the liquid is measured using a beam balance.
  Calculate the mass of the liquid: m₂ - m₁.
• The volume (V) of the liquid is measured using a measuring cylinder.
• Then, the density (d) of the liquid is calculated using the formula: d = M ÷ V

Examples:
Example 1: An empty measuring cylinder weighs 120 g. After adding 50 cm³ of oil, it weighs 160 g.
Sol:  Mass of oil = 160 - 120 = 40 g.
Density = 40 / 50 = 0.8 g/cm³.

Example 2: A cylinder (mass 150 g) with 200 cm³ of a liquid weighs 350 g.
Sol: Mass of liquid = 350 - 150 = 200 g.
Density = 200 / 200 = 1 g/cm³.

Density Bottle

• A density bottle is a special glass bottle used to find the density of a liquid.
• It is a small glass bottle with a glass stopper that has a narrow hole through it.
• The bottle can store a fixed volume of liquid, generally 25 mL or 50 mL.
• When the bottle is filled with liquid and the stopper is inserted, any extra liquid rises through the hole and comes out.
• So, the bottle always contains the same volume of liquid each time it is filled.
• For Example: A 50 cm³ density bottle is used to measure the density of alcohol, ensuring the exact volume is always 50 cm³ when filled.

Determination of Density of a Liquid Using Density Bottle

• To find the density of a liquid using a density bottle, we need to measure the mass of the liquid and the mass of water taken in the density bottle.
• We use a beam balance to measure the mass.
• The mass of water in the density bottle gives us the volume of the liquid.

Steps:

• Dry the density bottle thoroughly and measure its mass (m₁).
• Fill the bottle completely with the liquid, ensuring no air bubbles, and insert the stopper (excess liquid escapes through the stopper's hole).
• Measure the mass of the bottle with the liquid (m₂).
• Calculate the mass of the liquid: m₂ - m₁.
• Use the known volume of the bottle (V).
• Calculate density: ρ = (m₂ - m₁) / V.

Examples:
Example 1: A 25 cm³ density bottle has an empty mass of 20 g. When filled with a liquid, it weighs 45 g.
Sol: Mass of liquid = 45 - 20 = 25 g.
Density = 25 / 25 = 1 g/cm³.

Example 2: A 50 cm³ density bottle (empty mass 30 g) weighs 80 g when filled with oil.
Sol: Mass of oil = 80 - 30 = 50 g.
Density = 50 / 50 = 1 g/cm³.

Relative Density

• The relative density of a substance is the ratio of the density of the substance to the density of water.
• So, Relative Density (R.D.) = Density of the substance ÷ Density of water
• For example, if the density of iron is 7.8 g/cm³ and the density of water is 1 g/cm³, then:
  Relative density of iron = Density of iron ÷ Density of water
  = 7.8 g/cm³ ÷ 1 g/cm³ = 7.8
  But since 1 g/cm³ is the density of water, we can also write relative density as:
  Relative density of a substance = Mass of 1 cm³ of the substance ÷ Mass of 1 cm³ of water
  Or, Relative density of a substance = Mass of any volume of the substance ÷ Mass of the same volume of water
  So, if the relative density of iron is 7.8, it means that a piece of iron of any volume has a mass 7.8 times the mass of an equal volume of water.

Unit of Relative Density

• Relative density is just a number; it has no unit.
• It is a dimensionless quantity because it is a ratio of two densities.

Measurement of Relative Density of a Liquid

• Measure the mass of the empty, dry density bottle (m₁).
• Fill the bottle with water, insert the stopper, and measure the mass (m₂).
• Dry the bottle, fill it with the liquid, and measure the mass (m₃).
• Calculate the mass of water: m₂ - m₁.
• Calculate the mass of the liquid: m₃ - m₁.
• Since the bottle's volume is the same for both, relative density = (Mass of liquid) / (Mass of water) = (m₃ - m₁) / (m₂ - m₁).

Examples:
Example 1: A 50 cm³ density bottle has an empty mass of 25 g, weighs 75 g with water, and 65 g with oil.
Sol: Mass of water = 75 - 25 = 50 g.
Mass of oil = 65 - 25 = 40 g.
Relative density = 40 / 50 = 0.8.

Example 2: Empty bottle (20 g), with water (70 g), with liquid (90 g).
Sol: Mass of water = 70 - 20 = 50 g.
Mass of liquid = 90 - 20 = 70 g.
Relative density = 70 / 50 = 1.4.

Did You Know?

• A density bottle is used to measure a liquid's relative density.
• Given that water has a density of 1 g cm⁻³, the density of a substance in g cm⁻³ equals its relative density.
• Since water's density is 1000 kg m⁻³, a substance's density in kg m⁻³ is 1000 times its relative density.
• Today, relative density is no longer referred to as specific gravity.

Density of a Substance in Different States

• A substance can be a solid, liquid, or gas.
• For example, ice (solid), water (liquid), and steam (gas) are three states of the same substance.
  • In solids, particles are very close together, so solids are the most dense.
  • In liquids, particles are a little farther apart, so liquids are less dense than solids.
  • In gases, particles are very far apart, so gases are the least dense.

Density and Relative Density of Some Common Substances

• Cork is less dense than water because its density (0.25 g cm⁻³) is less than water's density (1.0 g cm⁻³).
• Iron is denser than water because its density (7.8 g cm⁻³) is more than water's density.
• Ice has a density of 0.917 g cm⁻³, which is less than water's density of 1.0 g cm⁻³, so ice floats on water.

Floating and Sinking

When a piece of cork and an iron nail are placed on water, the cork floats while the nail sinks. This happens because the cork's density is lower than water's, but the iron nail's density (made of iron) is higher than water's. Therefore, an object floats in a liquid if its density is less than the liquid's, and sinks if its density is greater.

Examples:

• A solid iron ball with a density of 7.86 g cm⁻³ will sink in water (density 1.0 g cm⁻³) but float in mercury (density 13.6 g cm⁻³).
• A small piece of cork floats on water because its density is less than water's, while an iron nail sinks in water since iron's density exceeds that of water.

Principle of Floating

When an object is put in a liquid, two forces act on it:

• First force is the weight of the object (W), which pulls the object downwards and makes it sink.
• Second force is the buoyant force (F_B), which pushes the object upwards. The buoyant force is equal to the weight of the liquid displaced by the part of the object that is inside the liquid. Because the buoyant force pushes the object up, it is also called upthrust.

Depending on these forces, three things can happen to the object:

Case 1: The weight of the body W is greater than the buoyant force F_B

• In this case, the total force on the object is (W - F_B), which acts downwards.
• This makes the object sink to the bottom of the liquid.
• This happens when the density of the object is greater than the density of the liquid.
• For example, a stone sinks in water because its density is more than the density of water.

Case 2: The weight of the body W is equal to the buoyant force F_B

• In this case, the total force on the object is zero (W = FB).
• This means the apparent weight of the object is zero.
• The object will float just inside the surface of the liquid, not fully above or below it.
• This happens when the density of the object is equal to the density of the liquid.
• For example, an egg floats just at the surface of salty water if their densities are the same.

Case 3: The weight of the body W is less than the buoyant force, if the body is completely immersed in liquid

• In this case, the total force acts on the object upwards because F_B is greater than W.
• The object will float partly above the surface of the liquid.
• Only a small part of the object stays inside the liquid, just enough so that the weight of the liquid displaced (F_B) becomes equal to the weight of the object (W).
• This happens when the density of the object is less than the density of the liquid.
• For example, a piece of wood floats on water because its density is less than the density of water.
• When the object floats, F_B = W, so the apparent weight of the object is zero.

Summary Table

Law of Floating

When an object floats in a liquid, the weight of the liquid displaced by the submerged part equals the object's total weight.

Some Applications of Floating

(I) Floatation of an Iron Ship

• A ship made of iron can float on water even though iron is denser than water.
• The ship is hollow, so it has a lot of empty space inside.
• This empty space makes the ship's overall density less than water's density.
• The ship displaces a lot of water, and the weight of this displaced water equals the ship's weight, so it floats.

(II) Floatation of Man

• A person can float more easily in saltwater than in freshwater.
• Saltwater has a higher density than freshwater because it has salt dissolved in it.
• The higher density of saltwater makes it easier for a person to displace enough water to float.
• In freshwater, a person might sink because the density of freshwater is less, so less water is displaced.

(III) Floatation of Ice on Water

• Ice floats on water because its density is less than water's density.
• The density of ice is 0.917 g cm⁻³, while water's density is 1.0 g cm⁻³.
• When ice floats, only a small part (1/10th) of it is underwater, and 9/10th is above water.
• The weight of the water displaced by the underwater part of the ice equals the ice's total weight.

(IV) Submarine

• A submarine can dive underwater or rise to the surface of the water.
• It has special tanks that can be filled with water or air.
• When the tanks are filled with water, the submarine becomes heavier and sinks because its density increases.
• When the tanks are filled with air, the submarine becomes lighter and rises because its density decreases.

(V) Icebergs

• Icebergs are large pieces of ice floating on seawater.
• They are dangerous for ships because most of the iceberg (9/10th) is underwater, and only a small part (1/10th) is visible above the water.
• The density of ice is 0.917 g cm⁻³, and the density of seawater is 1.02 g cm⁻³, so ice floats on seawater.
• An iceberg floats with a large portion submerged because the weight of the displaced seawater equals the iceberg's weight.

(VI) Whales

• Whales are sea animals that can float and dive in water.
• They have a special organ called the swim bladder in their body.
• When the swim bladder is filled with air, the whale's density decreases, and it rises to the surface of the water.
• When the swim bladder is emptied of air, the whale's density increases, and it sinks in the water.

(VII) Balloons

• A balloon filled with hydrogen or helium rises in the air.
• Hydrogen and helium are gases that are much less dense than air.
• Because the balloon's density is less than the air's density, it experiences an upward force called buoyant force.
• This buoyant force makes the balloon rise in the air.

Points To Remember

• Equal masses of different substances have different volumes because their densities are different.
• Equal volumes of different substances have different masses because their densities are different.
• If a body has mass M and volume V, its density is found by the formula: Density = Mass / Volume.
• The SI unit of density is kg m⁻³, and the CGS unit is g cm⁻³.
• In the SI unit, 1 g cm⁻³ = 1000 kg m⁻³.
• The density of a substance does not change when its size or shape changes.
• Density increases when a substance is compressed and decreases when it expands.
• In gases, density increases when the temperature decreases and decreases when the temperature increases.
• In solids and liquids, density usually decreases when heated and increases when cooled, except for water.
• Water is an exception: its density increases when heated from 0°C to 4°C and decreases when heated above 4°C.
• A density bottle is a specially designed bottle used to measure the density of a liquid.
• Relative density of a substance is defined as the ratio of the density of the substance to the density of water.
• Relative density can also be calculated as the ratio of the mass of any volume of the substance to the mass of an equal volume of water.
• Relative density is just a number and has no unit because it is a ratio of the same quantities.
• The buoyant force is an upward force on an object in a liquid due to the liquid displaced by the object.
• When a body is in a liquid, the buoyant force depends on the volume of the body in the liquid and the density of the liquid.
• If the density of the body is more than the density of the liquid, the body sinks.
• If the density of the body is less than or equal to the density of the liquid, the body floats.
• When a body floats, only a portion of it is submerged in the liquid, depending on the densities of the body and the liquid.
• According to the law of floatation, the weight of a floating body is equal to the weight of the liquid displaced by the submerged part of the body.
• For a body in different liquids, the portion of the body submerged depends on the density of the liquid: more dense liquids cause less submersion, and less dense liquids cause more submersion.