Physical Quantities and Measurement

A. Measurement of Surface Area:

• Regular Shapes: Square ($A = s^2$), Rectangle ($A = l \times b$), Triangle ($A = \frac{1}{2} b h$), Circle ($A = \pi r^2$).
• Irregular Lamina (Graph Paper Method):
  1. Place the irregular leaf or lamina on a $1\text{ cm}^2$ grid paper and trace its boundary with a sharp pencil.
  2. Count the number of fully enclosed squares ($N_f$).
  3. Count the number of squares that are half or more than half enclosed ($N_h$). Ignore squares less than half enclosed.
  4. $$\mathbf{\text{Approximate Area} = (N_f + N_h) \text{ cm}^2}$$

B. Measurement of Volume:

• SI Unit: Cubic meter ($\text{m}^3$). Practical units: Cubic centimeter ($\text{cm}^3$ or $\text{cc}$), Liter ($\text{L}$), Milliliter ($\text{mL}$). $$1\text{ m}^3 = 10^6\text{ cm}^3; \quad 1\text{ L} = 1,000\text{ mL} = 1,000\text{ cm}^3; \quad 1\text{ mL} = 1\text{ cm}^3$$
• Irregular Solid (Liquid Displacement Method):
  1. Fill a graduated cylinder with water to an initial level $V_1$.
  2. Tie the irregular insoluble solid (e.g., a stone) with a thin thread and gently submerge it completely without splashing.
  3. Read the new water level $V_2$ at the bottom of the meniscus.
  4. $$\mathbf{\text{Volume of Solid } V = V_2 - V_1}$$
• Eureka Can / Overflow Can Method: When the solid is too large for a measuring cylinder, fill a eureka can until water overflows through the spout. Place an empty measuring cylinder under the spout, gently lower the solid, and the overflow volume directly equals the solid volume!

A. Definition & Formula:

The Density ($d$ or $\rho$) of a substance is defined as its mass per unit volume:

• SI Unit: Kilogram per cubic meter ($\text{kg/m}^3$ or $\text{kg m}^{-3}$).
• CGS Unit: Gram per cubic centimeter ($\text{g/cm}^3$ or $\text{g cm}^{-3}$).

B. Master Unit Conversion Proof:

• To convert from $\text{g/cm}^3$ to $\text{kg/m}^3$: Multiply by $1,000$.
• To convert from $\text{kg/m}^3$ to $\text{g/cm}^3$: Divide by $1,000$.
• Example: Density of pure water at $4^\circ\text{C} = 1\text{ g/cm}^3 = \mathbf{1,000\text{ kg/m}^3}$.
• Example: Density of liquid mercury $= 13.6\text{ g/cm}^3 = \mathbf{13,600\text{ kg/m}^3}$.

A. Definition of Relative Density (RD):

The Relative Density of a substance is the ratio of the density of the substance to the density of pure water at $4^\circ\text{C}$:

• No Units: Because Relative Density is the ratio of two identical physical quantities (densities), it is a pure number having NO units.
• Numerical Equivalence: In the CGS system, since the density of water is $1\text{ g/cm}^3$, the numerical value of relative density is identical to its density in $\text{g/cm}^3$! (e.g., if RD of copper is $8.9$, its density is $8.9\text{ g/cm}^3 = 8,900\text{ kg/m}^3$).

B. Measurement of Relative Density using a Density Bottle:

A Specific Gravity Bottle (Pycnometer) is a specially blown glass flask with a ground glass stopper having a fine capillary bore to ensure a strictly constant volume $V$ of liquid.

1. Weigh the clean, dry empty bottle with stopper: $M_1$.
2. Fill the bottle completely with the experimental liquid, insert the stopper (excess liquid escapes through the capillary), wipe dry, and weigh: $M_2$. $$\text{Mass of liquid} = M_2 - M_1$$
3. Empty, rinse, and fill completely with pure water, insert stopper, wipe dry, and weigh: $M_3$. $$\text{Mass of equal volume of water} = M_3 - M_1$$
4. $$\mathbf{\text{Relative Density of Liquid (RD)} = \frac{M_2 - M_1}{M_3 - M_1}}$$

A. Speed ($S$):

Speed is the rate of change of distance with respect to time: $\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{D}{T}$.

• SI Unit: meters per second ($\text{m/s}$). Commercial Unit: kilometers per hour ($\text{km/h}$).
• Conversion: $1\text{ km/h} = \frac{1000\text{ m}}{3600\text{ s}} = \mathbf{\frac{5}{18}\text{ m/s}}$, and $1\text{ m/s} = \mathbf{\frac{18}{5}\text{ km/h} = 3.6\text{ km/h}}$.

B. Experimental Precautions:

• Meniscus Reading: For transparent wetting liquids (water), read the lowest point of the concave meniscus. For non-wetting convex liquids (mercury), read the highest point of the convex meniscus.
• Parallax Error: The observer's eye must be held exactly perpendicular and level with the liquid meniscus scale line.