Measuring Matter: Mass, Volume, & Density

• P - Point → In the above paragraph, I point out to the theory of: The concept of density as the relationship between mass and volume, and to Archimedes' principle of water displacement.

• E - Evidence → That supports the theory is: Archimedes discovered the principle while testing the purity of a crown. The water displacement method continues to be used to determine the volume of irregularly shaped objects.

• E - Explanation → Now I explain my point: Density evolved into a precise scientific concept expressed by the formula density = mass/volume, while measurement methods improved from simple marked containers to standardised metric units such as litres and millilitres.

• L - Link → I link my knowledge of physical quantities and measurement to: How fundamental quantities like mass and volume combine to define a core property of matter.

1. Density of a substance = $\frac{\text{Mass}}{\text{Volume}}$

2. A wooden log floats in water because the density of wood is less than that of water, while a metal nail sinks because the density of the metal (iron) is more than that of water.

3. A measuring cylinder is used to measure the volume of irregular solids, by the water displacement method.

4. If an object's relative density is less than 1, it means the object is lighter than water and will float on water.

5. The law of flotation states that an object floats only when the weight of the water it pushes aside (displaces) becomes equal to its own weight.

1. Cubic centimeter
   Reason — Gram, Kilogram and Tonne are units of mass, whereas cubic centimetre is a unit of volume.

2. Temperature
   Reason — Displacement, volume and density are directly involved in determining the density of a substance in this chapter, whereas temperature is not one of the quantities measured in this process.

3. Iron
   Reason — Wood, Cork and Ice have a relative density less than 1 and float on water, whereas iron has a relative density more than 1 and sinks in water.

4. Speed
   Reason — Mass, Volume and Density are related to one another (density = mass/volume), whereas speed is a quantity related to the motion of a body.

5. Current
   Reason — Relative density, Float and Sink are related to the floating and sinking of bodies, whereas current is not related to this concept.

1. Density is the mass per unit volume of a substance.

2. SI unit of mass — kilogram (kg); SI unit of volume — cubic metre (m³).

3. Specific gravity is another name for relative density — the ratio of the density of a substance to the density of water at 4°C.

4. Hydrometer.

5. A body sinks in a liquid when its density is greater than the density of the liquid.

1. A body or an object having definite volume has a defined mass.

2. Eureka can is a cylindrical glass or plastic beaker which has a side opening present near its mouth.

3. To determine the density of a liquid, we need to measure its mass and volume.

4. A relative density (RD) bottle (pycnometer) is a small glass bottle with a glass stopper.

5. Buoyant force is equal to the weight of the liquid displaced by the immersed part of the body.

1. True.

2. True.

3. False.
   Corrected Statement — A balloon filled with helium gas rises up in air because the density of helium is less as compared to that of air.

4. False.
   Corrected Statement — If the relative density of a substance is greater than 1, it is heavier than water and hence, it will sink in water.

5. False.
   Corrected Statement — A solid usually has a higher density than a liquid, but not always — for example, ice (a solid) is less dense than water (a liquid), so ice floats on water.

1. An iron nail sinks in the seawater as the density of iron is more than that of water but a ship made of iron floats.

2. The formula for relative density is Density of substance ÷ Density of water at 4°C.

3. The seawater has higher density because of the salts present in it whereas river water has lower amount of salts and hence, less density than the seawater.

4. Measuring cylinders are used to measure volume.

5. Relative density has no unit (it is the ratio of two similar quantities).

Density is defined as the mass per unit volume of a substance (ρ = $\frac{\text m}{\text V}$), so it tells us how much mass is packed into a given volume. If the molecules of a substance are tightly packed, a given volume contains more mass and the density is high; if the molecules are loosely packed, the same volume contains less mass and the density is low. Hence, density is a measure of how much mass is contained in a given volume.

Relative density compares the density of a substance to the density of water. It tells us if an object will float or sink because:

• If Relative Density is less than 1: The substance is lighter (less dense) than water, so it floats.

• If Relative Density is more than 1: The substance is heavier (more dense) than water, so it sinks.

When a body floats in a liquid, it sinks just enough to push aside (displace) a part of the liquid. The body keeps floating only when the upward buoyant force balances its downward weight, and this happens when the weight of the displaced liquid becomes equal to the weight of the body. Hence, a floating object displaces a weight of liquid equal to its own weight.

An irregular solid has no definite shape, so its volume cannot be calculated by formula. When it is fully immersed in water, it displaces water equal to its own volume. The increase in water level (V₂ − V₁) in a measuring cylinder gives the volume of the solid by water displacement method.

Ice has many empty spaces between its particles, which makes its density less than that of water. Therefore, ice floats on water. It sinks only up to the point where the weight of the water displaced by the immersed part becomes equal to the weight of the ice. Hence, about one-tenth of ice remains above water and nine-tenths stays inside water.

To determine the density of a liquid, we measure its mass and its volume. First, take an empty, dry beaker and find its mass (m₁) using a beam balance. Then pour a known volume (V) of the liquid, measured with a measuring cylinder, into the beaker and find the new mass (m₂). The mass of the liquid is (m₂ − m₁).

The density of the liquid is then calculated as:

Density = $\frac{\text{Mass}}{\text{Volume}} = \frac{\text{m}_2 − \text{m}_1}{\text{V}}$

The law of flotation states that when a body floats in a liquid, the total weight of the body is equal to the weight of the liquid displaced by its immersed part. In other words, the apparent weight of a floating body is zero.

When a body is immersed in a liquid, two forces act on it:

• The weight of the body (W), which acts vertically downwards due to gravity.
• The buoyant force or upthrust (F_B), which acts vertically upwards.

A submarine has special ballast tanks that control its average density. When the tanks are filled with water, its density becomes greater than that of seawater, so it sinks. When the water is pumped out, its density becomes less than that of seawater, causing it to rise to the surface.

A pycnometer (specific gravity bottle) is used to determine the relative density of a liquid. It is a glass bottle of fixed volume with a stopper having a fine capillary tube. When completely filled, excess liquid flows out through the capillary, ensuring the bottle always contains the same volume of liquid.

The different vessels used for measuring the volume of liquids are:

• Measuring Cylinder: A graduated glass or plastic container used to measure the volume of liquids in millilitres (mL).

• Measuring Beaker: A container made of glass or plastic used to hold and measure the approximate volume of liquids.

• Eureka Can (Overflow Can): A container used to determine the volume of irregular solid objects by collecting the water displaced when the object is completely immersed.

To determine the density of a solid, we need to measure its mass and its volume.

• Measure the mass of the solid using a beam balance.
• Determine its volume:
  • For a regular-shaped solid, calculate the volume using the appropriate geometric formula.
  • For an irregular-shaped solid, find the volume by the water displacement method using a measuring cylinder.
• Calculate density using the formula ρ = $\frac{\text m}{\text V}$.

The relative density of a liquid is determined using a relative density (RD) bottle (pycnometer). The procedure is:

• Take an empty, clean and dry RD bottle and find its mass (m₁) using a physical or beam balance.
• Fill the bottle completely with distilled water and insert the stopper so that the excess water drains out. Dry it from outside and find its mass (m₂).
• Empty and dry the bottle, then fill it completely with the liquid whose relative density is to be found. Insert the stopper, dry it and find its mass (m₃).
• Now calculate the mass of water = (m₂ − m₁) and the mass of liquid = (m₃ − m₁).

Relative density = $\frac{\text{Mass of liquid}}{\text{Mass of water}} = \frac{m_3 - m_1}{m_2 - m_1}$

When a body is immersed in a liquid, two forces act on it: its weight (W) downward and buoyant force (F_B) upward. Based on their magnitude, three cases arise:

• Case I (F_B < W): The body sinks because weight is greater than upthrust. (Density of body > density of liquid)

• Case II (F_B = W): The body floats just below the surface as forces are balanced. (Density of body = density of liquid)

• Case III (F_B > W initially): The body rises upward. As it emerges partly from the liquid, the displaced volume decreases until the buoyant force becomes equal to its weight. It then floats partially above the surface. (Density of body < density of liquid)

Five applications of flotation are:

• Balloons: A balloon filled with helium rises in air because helium is less dense than air, so the buoyant force on the balloon is greater than its weight.
• Flotation of ice on water: Ice floats on water with about one-tenth of its part outside the water, as the density of ice is less than that of water.
• Flotation of an iron ship: A ship made of iron floats because its hollow design contains air, which makes its average density less than that of water.
• Submarines: A submarine sinks or rises by filling its tanks with water to increase its density, or by forcing the water out to decrease its density.
• Flotation of man: It is easier for a man to swim in seawater than in river water, because seawater has a higher density due to the salts present in it.

1. Air-filled bottle floats because air makes its average density less than water. Sand-filled bottle sinks because sand increases its average density above water.

• If density of body < water → it floats
• If density of body > water → it sinks

2. Nakul can first find: The mass (m) of the stone using the beam balance. Then he can pour water in the measuring cylinder, note the initial level (V₁), immerse the stone fully and note the new level (V₂). The volume of the stone is (V₂ − V₁). The density is then calculated as ρ = m/(V₂ − V₁).

3. If the relative density of an object is more than 1, it means the object is heavier (denser) than water, and so it will sink in water.

4. According to the law of flotation, an object floats when the weight of the water displaced by its immersed part is equal to its own weight.

5. Ships are not solid metal; they have a hollow design with large empty spaces filled with air. This makes the average density of the ship less than that of water, so the ship displaces water equal to its own weight and floats, even though it is made of heavy metal.

1. We can guess the density by seeing how heavy a substance is for its size. If something is heavy for its size, it has a high density; if it is light for its size, it has a low density.

2. Both have the same weight, so they feel equally heavy when lifted. But cotton occupies more space because it is less dense, while iron occupies less space because it is more dense.

3. Density is the mass per unit volume of a substance. So the more mass a substance has in a given volume, the higher its density; and the less mass it has in the same volume, the lower its density.

Although the coin and plastic cap have the same size, they have different densities. The coin is denser than water, so it sinks, while the plastic cap is less dense than water, so it floats. Thus, floating or sinking depends on density, not size.

When the bottle is empty and sealed, it contains air, so its average density is less than water and it floats. When filled with sand, its mass increases while volume remains the same, so its average density becomes greater than water and it sinks. Thus, floating or sinking depends on average density.

Mass of the object = 300 g
Volume = 150 cm³

$$\begin{aligned}\text{Density} = \frac{\text{Mass}}{\text{Volume}}\\[1em] = \frac{300}{150}\\[1em] = 2 \text{ g cm}^{-3}\end{aligned}$$

Since the density of the object (2 g cm⁻³) is greater than the density of water (1 g cm⁻³), the object will sink in water.

Mass of the solid = 2 kg = 2000 g

Volume of the solid = 200 mL = 200 cm³ (1 mL = 1 cm³)

$$\begin{aligned}\text{Density of the solid} = \frac{\text{Mass}}{\text{Volume}} \\[1em] = \frac{2000}{200} \\[1em] = 10 \text{ g cm}^{-3}\end{aligned}$$

$$\begin{aligned}\text {Relative density} = \frac{\text{Density of solid}}{\text{Density of water}} \\[1em] = \frac{10}{1} \\[1em] = 10\end{aligned}$$

Since the relative density (10) is greater than 1, the object will sink in water.

To test which materials float and which sink:

• Take a large bowl and fill it with water.
• Gently place each object — the pebbles and the rubber balls — one by one on the surface of the water.
• Observe which objects float on the surface and which sink to the bottom.
• Record the result for each object. An object floats if its average density is less than that of water and sinks if its average density is greater than that of water.

This shows that objects with density less than water float, while those with density more than water sink.

A ship is made of metal, which is dense, but it is hollow inside and filled with air. This increases its volume without adding much mass, so its average density becomes less than water.

When the ship is placed in water: It displaces a large amount of water. The buoyant force (upthrust) becomes equal to the ship’s weight.

Even though metal is heavy, a ship floats because its overall (average) density is less than water due to its hollow structure, and it displaces enough water to balance its weight.

1. The bottle with water felt heavier because water has greater density than oil. Since both bottles had the same volume, the water-filled bottle had more mass, so it felt heavier.

2. The property that relates mass and volume is called density (density = mass/volume).

3. Whether an object floats or sinks depends on its density compared to water. If its density is less than water, it floats, and if it is more than water, it sinks.

4. Metal has more density than wood. That is why a metal key sinks in water, while a piece of wood floats.

5. Density is called a characteristic property of matter because it is fixed for a given substance at fixed temperature and does not depend on the amount of substance.

1. Solids generally have the highest density. This is because the particles (molecules) in a solid are very tightly packed with the smallest spaces between them, so a given volume contains the most mass.

2. The volume of an irregular solid like a stone is measured by the water displacement method. The stone is fully immersed in water in a measuring cylinder, and the rise in the water level (V₂ − V₁) gives the volume of the stone.

3. A measuring cylinder is more accurate than a beaker. This is because a measuring cylinder is narrow and has fine, closely spaced graduation marks in millilitres, which allow the volume of the liquid to be read more precisely.

4. The mass of air is very difficult to measure because air has a very low density (about 0.00127 g cm⁻³). So a given volume of air contains very little mass, which is almost negligible compared with that of solids and liquids.

5. The differences in density arise from how tightly the particles are packed. In solids, the particles are very tightly packed with the smallest spaces, so solids are the most dense. In liquids, the particles are a little farther apart, so liquids are less dense than solids. In gases, the particles are very far apart with the largest spaces, so gases are the least dense.

1. The two measurements required are the mass of the object and its volume, because density = mass/volume.

2. When an object is fully immersed in water in a measuring cylinder, the water level rises. The increase in water level gives the volume of the object because it displaces an equal volume of water.

$$\begin{aligned}\text {Density} = \frac{\text{Mass}}{\text{Volume}} \\[1em] = \frac{300}{100} \\[1em] = 3 \text{ g cm}^{-3}\end{aligned}$$

4. When oil is poured into water, the oil does not mix with the water but forms a separate layer above the water.

5. Oil forms a layer above water because oil is less dense than water. A liquid with lower density floats over a liquid with higher density, so the oil stays on top of the water.

1. The law of flotation states that a body floats in a liquid when the weight of the displaced liquid equals the weight of the body.

2. A steel ship is hollow and contains large empty spaces filled with air. Because of this hollow shape, its average density becomes less than that of water, and it displaces a large volume of water whose weight equals the weight of the ship. So the ship floats, even though steel is denser than water.

3. An object that just floats on water has a relative density equal to 1, that is, its density is equal to the density of water.

4. Changing the shape of an object changes the volume of water it can displace. If an object is given a hollow, wide shape (like a boat), it displaces more water and its average density becomes less than that of water, so it can float even if the same material would sink as a solid lump.

5. Knowledge of relative density helps in designing floating objects. Boats are made hollow so their average density is less than water, and life jackets use light materials or air to reduce the wearer’s average density so they float easily.

1. The mass per unit volume of a substance can be defined as density.

2. A relative density (RD) bottle (pycnometer) is a special bottle used for determining the relative density of a liquid.

3. The instrument used to measure the volume of an irregular solid is called a measuring cylinder.

4. The relative density of a substance is equal to the ratio of its density to the density of water.

5. A body will float in a liquid if its density is less than that of the liquid.

1. False.
   Corrected Statement — Relative density has no unit, as it is the ratio of two similar quantities.

2. False.
   Corrected Statement — A hydrometer is used to measure the relative density (specific gravity) of liquids.

3. True.

4. False.
   Corrected Statement — A wooden block floats in water because its relative density is less than 1.

5. True.

Density is the mass per unit volume of a substance (Density = Mass/Volume). Its SI unit is kilogram per cubic metre (kg m⁻³).

The law of flotation states that when a body floats in a liquid, the total weight of the body is equal to the weight of the liquid displaced by its immersed part.

A piece of iron sinks because the density of iron is greater than that of water. A ship made of iron floats because it is hollow and contains air, which makes its average density less than that of water. Due to this, the ship displaces enough water to balance its weight, so it floats.

• The relative density of a substance tells us whether it will float or sink in water, which is useful in designing ships, boats and life jackets.
• A hydrometer is used to measure the relative density (specific gravity) of liquids, which can also help compare the concentration or purity of suitable liquids.

To determine the density of a liquid, we measure its mass and its volume. Take an empty, dry beaker and find its mass (m₁). Then pour a known volume (V) of the liquid, measured with a measuring cylinder, into the beaker and find the new mass (m₂). The mass of the liquid is (m₂ − m₁), so the density of the liquid = (m₂ − m₁)/V.

Density is defined as the mass per unit volume of a substance. So, for a given volume, the density is greater if the mass is greater, and it is smaller if the same volume contains less mass.

If a substance has a mass m and occupies a volume V, then:

$\text {Density} (ρ) = \frac{\text{Mass}}{\text{Volume}}$

For example, if a body of mass 100 g occupies a volume of 50 cm³, its density = $\frac{100}{50}$ = 2 g cm⁻³.

To find the density of an irregular solid (such as a stone) using a Eureka can:

• First, find the mass (m) of the solid using a beam balance.
• Fill the Eureka can with water up to its spout, and place an empty measuring beaker just below the spout.
• Tie the solid with a thread and lower it gently into the Eureka can until it is fully immersed, taking care that it does not touch the base of the can.
• The water displaced by the solid overflows through the spout and is collected in the measuring beaker. This volume of water gives the volume (V) of the solid.
• The density of the solid is then calculated using the formula ρ = $\frac{\text m}{\text V}$.

Application of density: It helps us compare how heavy different substances are for the same volume (for example, iron is denser than wood).

Application of relative density: Relative density can be used to check the purity or concentration of suitable substances.

The principle of flotation is based on two forces acting on a body in a liquid: weight (W) downward and buoyant force (F_B) upward. A body floats in equilibrium when the buoyant force is equal to its weight; therefore, the weight of the displaced liquid equals the weight of the body. If the buoyant force is initially greater than the weight, the body rises until this equilibrium condition is reached. This occurs when the average density of the body is less than or equal to the density of the liquid.

Two real-life examples:

• Iron ship: A ship made of iron floats because its hollow shape makes its average density less than that of water, so the buoyant force balances its weight.
• Flotation of ice on water: Ice floats on water because the density of ice is less than that of water, so it displaces water equal to its own weight while floating.

Weight of the solid in air = 120 g

Mass of solid = 120 g

Weight of the solid in water = 90 g

Loss in weight in water = (120 − 90) g = 30 g

This loss in weight is equal to the weight of the water displaced, so the weight of water displaced = 30 g.

Mass of water displaced = 30 g

Since the density of water = 1 g cm^-3, the mass of water displaced = 30 g
Then the volume of water displaced

$$\begin{aligned}=\frac{\text{Mass of water displaced}}{\text {Density of water}}\\[1em] =\frac{30}{1}\\[1em] = 30 \text{ cm}^3\end{aligned}$$ = 30 cm³. This is also the volume of the solid, so V = 30 cm³.

$$\begin{aligned}\text{Density of the solid} =\frac{\text{Mass}}{\text{Volume}} \\[1em] = \frac{120}{30}\\[1em] = 4 \text{ g cm}^{-3}\end{aligned}$$

$$\begin{aligned}\text{Relative density} = \frac{\text{Mass of solid}}{\text{Mass of displaced water}}\\[1em] = \frac{120}{30}\\[1em] = 4\end{aligned}$$

Hence, the density of the solid is 4 g cm⁻³ and its relative density is 4.