ICSE 7 Physics Measurement Basic Notes

1.0 Measurement of Density

In Class 6, we learned about basic measurements like length, mass, and time. In Class 7, we dive deeper into the concept of Density. Density is a characteristic property of a substance that tells us how "heavy" or "light" a material is for a given volume.

What is Density?

The Density of a substance is defined as its mass per unit volume. It indicates how closely the particles are packed within the substance.

SI Unit: $kg/m^3$ (kilogram per cubic metre)
CGS Unit: $g/cm^3$ (gram per cubic centimetre)

Density Formula

$$d = \frac{M}{V}$$

Where: $d$ = density, $M$ = mass of the substance, $V$ = volume of the substance

1.1 Relationship between SI and CGS Units

It is crucial to know how to convert density from one system of units to another, especially during numerical problem-solving.

⚠️ Unit Conversion Alert:

A common mistake is thinking $1\,g/cm^3$ is equal to $1\,kg/m^3$. In reality:

$$1\,g/cm^3 = 1000\,kg/m^3$$

To convert from CGS to SI, multiply by 1000. To convert from SI to CGS, divide by 1000.

Fig: Molecular Arrangement and Density

📝 Solved Example:

A block of iron has a mass of 15.6 kg and occupies a volume of 2000 $cm^3$. Calculate its density in $g/cm^3$.

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Solution:
1. Convert Mass to grams: $M = 15.6\,kg = 15.6 \times 1000 = 15600\,g$
2. Volume is given in $cm^3$: $V = 2000\,cm^3$
3. Apply Formula: $d = \frac{M}{V} = \frac{15600}{2000}$
4. Calculation: $d = 7.8\,g/cm^3$
Final Answer: The density of iron is $7.8\,g/cm^3$.

💡 Did You Know?

Water has its maximum density at 4°C. This unique property is why ice floats on water, allowing aquatic life to survive in frozen lakes!

2.0 Density of Irregular Solids

Calculating the density of a regular object like a cube is simple because we can calculate volume using a formula ($l \times b \times h$). However, for irregular objects like a stone or a piece of coal, we use the Displacement Method discovered by Archimedes.

2.1 Using a Measuring Cylinder

This method works on the principle that when a solid is completely immersed in a liquid, it displaces a volume of liquid equal to its own volume.

The Procedure

1. Find the mass ($M$) of the irregular solid using a beam balance.
2. Fill a measuring cylinder with water to a certain level ($V_1$).
3. Gently lower the solid into the cylinder using a fine thread.
4. Note the new water level ($V_2$).

Volume Calculation

$$Volume\,(V) = V_2 - V_1$$

Once you have Volume ($V$) and Mass ($M$), use: $d = M/V$

Fig: Measuring Volume of a Stone

⚠️ Precautions:

1. The solid must not be soluble in water.
2. The solid must not react chemically with the liquid.
3. Ensure no air bubbles are trapped while immersing the solid.

📝 Solved Example:

A stone of mass 75 g is lowered into a measuring cylinder. The water level rises from 30 ml to 55 ml. Find the density of the stone.

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Solution:
1. Mass (M): $75\,g$
2. Initial Volume ($V_1$): $30\,ml$
3. Final Volume ($V_2$): $55\,ml$
4. Volume of Stone ($V$): $V_2 - V_1 = 55 - 30 = 25\,cm^3$ (Note: $1\,ml = 1\,cm^3$)
5. Density ($d$): $M/V = 75/25 = 3\,g/cm^3$
Final Answer: The density is $3\,g/cm^3$.

💡 Did You Know?

If an irregular solid is lighter than water (floats), we use a sinker (a heavy metal piece) to pull it down so we can measure the displaced volume!

3.0 Density of Liquids

Unlike solids, liquids take the shape of their container, so we cannot measure their dimensions with a ruler. To find the density of a liquid, we typically use a specialized piece of glassware called a Density Bottle (also known as a Relative Density bottle).

3.1 The Density Bottle

A density bottle is a small glass bottle with a glass stopper. The stopper has a fine capillary tube (fine hole) running through it. This ensures that the bottle always holds exactly the same volume of liquid, as any excess liquid escapes through the hole when the stopper is inserted.

How to use it?

1. Find the mass of the empty, dry bottle ($M_1$).
2. Fill it with the liquid and find the mass ($M_2$).
3. The mass of the liquid is $M_2 - M_1$.
4. Since the volume ($V$) of the bottle is known (usually 25 ml or 50 ml), density can be calculated.

Liquid Density Formula

$$Density_{liquid} = \frac{M_2 - M_1}{Volume\,of\,Bottle}$$

Fig: Density Bottle (Specific Gravity Bottle)

⚠️ Common Error:

Never hold the density bottle by the body while weighing. The heat from your hand can cause the liquid to expand and escape through the capillary tube, leading to an incorrect mass reading. Always hold it by the neck.

📝 Solved Example:

An empty density bottle weighs 25 g. When filled with milk, it weighs 77 g. If the capacity of the bottle is 50 ml, find the density of milk.

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Solution:
1. Mass of empty bottle ($M_1$): $25\,g$
2. Mass of bottle + milk ($M_2$): $77\,g$
3. Mass of milk ($M$): $77 - 25 = 52\,g$
4. Volume of milk ($V$): $50\,ml = 50\,cm^3$
5. Density ($d$): $M/V = 52/50 = 1.04\,g/cm^3$
Final Answer: The density of milk is $1.04\,g/cm^3$.

💡 Did You Know?

A Lactometer is a special type of hydrometer used to check the purity of milk by measuring its density. If the milk is diluted with water, its density decreases, and the lactometer sinks deeper!

4.0 Relative Density (R.D.)

Sometimes, simply knowing the density of a substance isn't enough. We need to compare it with a standard substance to understand its behavior. In Physics, we use pure water at 4°C as our standard. This comparison is called Relative Density (also known as Specific Gravity).

What is Relative Density?

Relative Density is the ratio of the density of a substance to the density of water at 4°C. Since it is a ratio of two similar quantities, it has no units.

Relative Density Formula

$$R.D. = \frac{Density\,of\,Substance}{Density\,of\,Water\,at\,4°C}$$

Alternatively: $R.D. = \frac{Mass\,of\,any\,volume\,of\,substance}{Mass\,of\,same\,volume\,of\,water}$

4.1 Significance of R.D.

The Relative Density of a substance helps us predict whether it will sink or float in water:

• If R.D. > 1: The substance is denser than water and will sink (e.g., Iron, Gold).
• If R.D. < 1: The substance is less dense than water and will float (e.g., Cork, Wood, Kerosene).
• If R.D. = 1: The substance will float just submerged in water (e.g., certain plastics).

Fig: Relationship between R.D. and Flotation

⚠️ Exam Fact:

Numerical values of Density in $g/cm^3$ and Relative Density are identical. For example, if the density of Silver is $10.5\,g/cm^3$, its R.D. is simply $10.5$. However, in SI units ($kg/m^3$), the density would be $10,500\,kg/m^3$.

📝 Solved Example:

The relative density of mercury is 13.6. Calculate its density in (i) $g/cm^3$ and (ii) $kg/m^3$.

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Solution:
1. Density in CGS: Density ($g/cm^3$) = $R.D. \times 1\,g/cm^3$
Calculation: $13.6 \times 1 = 13.6\,g/cm^3$

2. Density in SI: Density ($kg/m^3$) = $R.D. \times 1000\,kg/m^3$
Calculation: $13.6 \times 1000 = 13,600\,kg/m^3$
Final Answer: Mercury has a density of $13.6\,g/cm^3$ or $13,600\,kg/m^3$.

💡 Did You Know?

The Dead Sea has a very high relative density (about 1.24) because of its high salt content. This makes the water so dense that humans can float effortlessly on its surface without swimming!

5.0 Variation of Density with Temperature

Density is not a fixed value for a substance; it changes whenever the temperature changes. This is because temperature affects the volume of a substance while its mass remains constant.

5.1 The General Rule

Most substances (solids, liquids, and gases) expand on heating and contract on cooling. Let's see how this affects density:

• On Heating: Volume increases $\rightarrow$ Density decreases (particles move apart).
• On Cooling: Volume decreases $\rightarrow$ Density increases (particles come closer).

Fig: Effect of Temperature on Particle Spacing and Density

5.2 Anomalous Expansion of Water

Water behaves differently than most liquids. Between 0°C and 4°C, water actually contracts on heating and expands on cooling. This is called the Anomalous Expansion of Water.

The 4°C Rule

At 4°C, water has its minimum volume and maximum density ($1000\,kg/m^3$ or $1\,g/cm^3$).

When water freezes into ice at 0°C, its volume increases, making ice less dense than water. This is why ice floats!

Density-Temperature Relation

Density $\propto \frac{1}{Volume}$

Since Mass is constant, Density is inversely proportional to Volume.

⚠️ Exam Caution:

When asked why a hot air balloon rises, do not just say "it's light." Use the correct physics terminology:

"Heating the air inside the balloon increases its volume, which decreases its density. Since the hot air is now less dense than the surrounding cool air, it rises due to buoyancy."

📝 Solved Example:

A sealed glass bottle completely filled with water is placed in a freezer. After some time, the bottle cracks. Why?

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Solution:
1. As water cools from 4°C to 0°C, it undergoes anomalous expansion.
2. Its volume increases as it turns into ice.
3. Since the glass bottle is rigid and cannot expand, the outward pressure exerted by the expanding ice cracks the glass.