Physical Quantities and Measurement Notes

Physical Quantities and Measurement

Measurement of Volume

• Volume: The amount of space occupied by a substance.
• SI unit of volume: cubic metre (m³).
• Submultiples of m³: decimetre cube (dm³) and centimetre cube (cm³).
• Volume of a liquid is measured in litres (L) and millilitres (mL).
• Relationships between units:
  • $1 m^3 = 1 m \times 1 m \times 1 m = 100 cm \times 100 cm \times 100 cm = 1,000,000 cm^3$
  • $1 mL = 1 cm^3$
  • $1 L = 1000 mL = 1000 cm^3$
  • $1 m^3 = 1000 L$
  • $1 L = \frac{1}{1000} m^3 = 0.001 m^3$

Volume of Solids

• The volume of a solid (cube or cuboid) can be calculated using formulae.

Volume of a Cuboid

• Formula: Volume = length × breadth × height

Volume of a Cube

• Formula: Volume = length × length × length (since length = breadth = height)

Volume of Liquids

• Liquids are stored in containers, and the volume of liquid a container can hold is its capacity (inner volume).
• Measuring devices include:
  • Measuring cylinder
  • Devices for petrol, diesel, kerosene oil
  • Devices for milk and liquid foods
  • Pipettes and burettes (chemistry laboratory)
  • Tumbler measure (chemist in a dispensary)

Measuring Volume Correctly

• Liquids in containers form a curved surface.
• Concave meniscus (e.g., water, kerosene): curve appears downwards.
  • Read the level at the bottom of the curve.
• Convex meniscus (e.g., mercury): curve appears upwards.
  • Read the level at the top of the curve.

Experiment: Volume of an Irregular Solid (Stone)

• Materials: Measuring cylinder, stone, water
• Procedure:
  1. Partially fill a measuring cylinder with water and note the initial volume, $V_1$ mL.
  2. Suspend the stone in the water, ensuring it's fully immersed and doesn't touch the sides or bottom.
  3. Note the new water level, $V_2$ mL.
• Observation:
  • Initial volume of water = $V_1$ mL
  • Volume of water with the stone = $V_2$ mL
• Calculation:
  • Volume of the stone = $(V<em>2 - V</em>1)$ mL = $(V<em>2 - V</em>1)$ cm³
    *Tips: Every matter in this universe occupies space and has mass. It means that every matter has some volume associated with it.

Measurement of Area

• Area: The amount of surface covered by a closed shape.
• SI unit of area: square metre (m²).
• Larger units:
  • Acre
  • Hectare
  • Square kilometre
• $1 acre = 100 m^2$
• $1 hectare = 10,000 m^2$
• $1 km^2 = 1,000,000 m^2 = 100 hectares$
• Smaller units:
  • Square centimetre
  • Square millimetre
• $1 cm^2 = 1 cm \times 1 cm = (\frac{1}{100}) m \times (\frac{1}{100}) m = (\frac{1}{10,000}) m^2 = 0.0001 m^2$
• $1 mm^2 = 1 mm \times 1 mm = (\frac{1}{1000}) m \times (\frac{1}{1000}) m = (\frac{1}{1,000,000}) m^2 = 0.000001 m^2$

Area of Regular Shapes

• Use mathematical formulae for definite shapes.

• Area of a square = side $\times$ side

• Area of a rectangle = length $\times$ breadth

Area of Irregular Shapes

• Use a graph paper.

Experiment: Area of an Irregular Shape (Leaf)

• Materials: Graph paper, leaf, pencil, pen
• Procedure:
  1. Place the leaf on the graph paper and trace its outline.
  2. Count the number of complete squares within the outline.
  3. Count the number of incomplete squares that are half or more than half inside the shape.
  4. Ignore squares less than half within the outline.
  5. Add up all the squares (total number of squares counted in the step above) and find the total area of the leaf (approx).
• Observation:
  • Approximate area of leaf = (No. of full squares $\times$ 1) cm² + (No. of half or more than half squares) cm²
• Calculation: The area of an irregular shape (a leaf) can be measured using a square centimetre grid with the help of the formula given above.

Measurement of Density

• Density: The mass of a substance per unit volume.
• Example: Iron is denser than wood because 1 m³ of iron has more mass than 1 m³ of wood.
• Formula: $Density = \frac{mass}{volume}$
• Density can be increased by increasing mass or decreasing volume.
• SI unit of density: kg/m³ (also written as kg m⁻³).
• CGS unit of density: g/cm³ (also written as g cm⁻³).

Determining the Density of a Solid

• Formula: $density = \frac{mass}{volume}$
• Volume of regular solids:
  • Rectangular block: length × breadth × height
  • Cube: length³
• Volume of irregular solids: Use the method described earlier.

Example 1

• A piece of metal has a mass of 89 g. When it is put in a measuring jar containing water, the level of water rises from 60 cm³ to 70 cm. What is the density of the metal?
• Solution:
  • Volume (V) = 70 cm³ - 60 cm³ = 10 cm³
  • $D = \frac{M}{V} = \frac{89}{10} = 8.9 g/cm^3$

Example 2

• The mercury contained in the bulb of a thermometer weighs 6.8 g. Calculate its volume if the density of mercury is 13.6 g/cm³.
• Solution:
  • $V = \frac{M}{D} = \frac{6.8}{13.6} = 0.5 cm^3$

Speed

• Rest: A body does not change its position with respect to its surroundings.
• Motion: A body changes its position with respect to its surroundings.
• Speed: The distance covered by a body in unit time.
• Formula: $Speed = \frac{Distance travelled}{Time taken}$
• If a body travels a distance d in time t, then its speed is given as, $Speed = \frac{d}{t}$
• Speed can also be defined as the rate of change of distance.
• SI unit of speed: metre/second (m/s or ms⁻¹).
• Other unit: kilometre/hour (km/h or kmh⁻¹).
• Types of speed:
  • Uniform or constant speed: equal distances in equal intervals of time.
  • Non-uniform or variable speed.

Example 1

• A person crosses a 20 m wide street in 5 seconds. What is his speed?
• Solution: $Speed = \frac{distance}{time} = \frac{20}{5} = 4 m/s$

Example 2

• An aeroplane covers a certain distance at a speed of 240 km/h in 5 hours. Calculate the distance covered by the aeroplane.
• Solution: Distance = speed × time = 240 × 5 = 1,200 km