Physical Quantities & Measurement Notes

Physical Quantities & Measurement

1. Measurement of Volume

Volume of an object is the amount of space occupied by it. The SI unit of volume is cubic metre (m³).

Volume of Regular Solids

Dimensions of regular solids can be measured using a metre rule.

Cube: V = (side)³
Cuboid: V = length × breadth × height
Sphere: V = ⁴⁄₃ πr³ (where r is radius)
Cylinder: V = πr²h (where r is radius and h is height)

Volume of Liquids

Liquids do not have specific shapes. We measure liquid volume using special devices like measuring cylinders. When liquid is poured into a cylinder, the reading of the upper surface gives the volume.

Volume of Irregular Solids

The displacement method is used. When an irregular solid is immersed in a liquid, it displaces a volume of liquid equal to its own volume.

Measuring Cylinder: Fill partially with water (volume V₁). Submerge the solid. The water level rises (volume V₂).
Volume of solid = V₂ - V₁
Other vessels used: Measuring beaker, Eureka can (holds liquid up to its spout).

2. Measurement of Density

Density of a matter is defined as the mass per unit volume present in it. It represents how closely the molecules are packed.

Density = Mass / Volume

Units of Density

CGS System: g/cm³ or g cm^-3
SI System: kg/m³ or kg m^-3

Density of an Irregular Solid

1. Find mass (M) using a physical/beam balance.
2. Find volume (V) using the displacement method.
3. Calculate using Density = M / V.

Density of Liquid (Using Density Bottle)

A density bottle is a specially designed 50 mL glass bottle with a capillary tube stopper, ensuring it holds an exact volume of liquid.

Mass of empty bottle = m₀ g
Mass of bottle + water = m₁ g
Mass of water = (m₁ - m₀) g
Since density of water is 1 g/cm³, Volume of water = Volume of bottle = (m₁ - m₀) cm³
Mass of empty bottle + unknown liquid = m₃ g

Density of liquid = (m₃ - m₀) / (m₁ - m₀) g cm^-3

Variation of Density with Temperature

Solids: Very little effect. Heating does not significantly change volume, so density remains relatively constant.
Liquids and Gases: Volume increases upon heating. Since volume is inversely proportional to density, an increase in temperature decreases density. This variation causes convection currents (e.g., warmer, less dense fluids rise while cooler, denser fluids sink).

3. Measurement of Area

Area is the amount of flat surface or region occupied by a closed figure. The SI unit is square metre (m²).

Conversion of Units of Area

Unit	Calculation	Equivalent in m² / cm²
1 km²	1 km × 1 km	10,00,000 m²
1 hectare	1 hm × 1 hm	10,000 m²
1 are	1 dcm × 1 dcm	100 m²
1 m²	1 m × 1 m	10,000 cm²

Measurement of Area using Graph Paper

To find the area of an irregular closed figure:

Place the figure on squared/graph paper (1 cm × 1 cm squares).
Make an outline of the figure.
Count the squares using the following rules:
- Full square: Take as 1 sq unit.
- More than half square: Take as 1 sq unit.
- Exactly half square: Take as 1/2 sq unit.
- Less than half square: Ignore it (0).

Formulae for Area of Regular Bodies

Square: (side)²
Rectangle: length × breadth
Circle: π × (radius)²
Triangle: 1/2 × base × height
Surface area of cylinder: 2π × radius × length
Surface area of sphere: 4π × (radius)²

4. Speed

Speed of a moving object is defined as the distance covered by it in unit time. It relates to the slowness or fastness of an object.

Speed = Total distance covered (d) / Total time taken (t)

Types of Motion

Uniform Motion: A vehicle moves with a constant speed along a straight path. Average speed = actual speed.
Non-uniform Motion: Speed keeps changing with time (e.g., due to traffic). In this case, we calculate average speed.

Measuring Devices in Vehicles

Speedometer: Measures the instantaneous speed of a vehicle in km/h.
Odometer: Records the total distance travelled by the vehicle in kilometres (km).

Fun Fact: Maglev trains are among the fastest trains, moving at an average speed of 450 km/h!