1.0 Measurement of Density
Measurement is the heart of Physics. While we are familiar with measuring length and mass, Density is a derived physical quantity that tells us how "compact" a substance is. It explains why an iron nail sinks in water while a massive wooden log floats.
What is Density?
Density of a substance is defined as its mass per unit volume. It is a characteristic property of a material and does not change with the shape or size of the object.
- S.I. Unit: $kg/m^3$ (kilogram per cubic metre)
- C.G.S. 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 Unit Conversion & Relationships
In numerical problems, you will often need to convert between S.I. and C.G.S. units. The relationship is derived as follows:
$1\,g/cm^3 = 1000\,kg/m^3$
For example, the density of water is $1\,g/cm^3$, which is equal to $1000\,kg/m^3$.
Always ensure that mass and volume are in the same system of units before calculating. If Mass is in $kg$, Volume must be in $m^3$. If Mass is in $g$, Volume must be in $cm^3$. Mixing $kg$ with $cm^3$ will lead to an incorrect answer!
A block of iron has a mass of 15.6 kg and a volume of 0.002 m³. Calculate its density in both S.I. and C.G.S. units.
Solution:
1. Given: Mass ($M$) = $15.6\,kg$, Volume ($V$) = $0.002\,m^3$.
2. S.I. Density: $d = \frac{15.6}{0.002} = 7800\,kg/m^3$.
3. Conversion to C.G.S: Since $1000\,kg/m^3 = 1\,g/cm^3$,
Density in C.G.S = $\frac{7800}{1000} = 7.8\,g/cm^3$.
Final Answer: $7800\,kg/m^3$ or $7.8\,g/cm^3$.
The Dead Sea is so dense due to its high salt content that humans can float on its surface without any effort! The density of the water there is much higher than the average density of the human body.
2.0 Measurement of Density of Objects
Calculating density for a regular cube is easy—you just measure its sides. But in a laboratory, we often deal with Irregular Solids (like a stone) and Liquids. To find their density, we must use specific instruments like the Measuring Cylinder and the Density Bottle.
2.1 Density of an Irregular Solid
For a solid that doesn't have a mathematical shape, we use the Displacement Method (Archimedes' Principle). Since the solid is insoluble in water, the volume of water it displaces is exactly equal to its own volume.
Volume by Displacement
$$V = V_2 - V_1$$
Where: $V_1$ = Initial water level, $V_2$ = Water level with submerged solid.
2.2 Density of a Liquid (Density Bottle)
A Relative Density Bottle is a small glass flask with a glass stopper having a fine capillary tube. It is designed to hold a fixed volume of liquid at a given temperature. By weighing the bottle empty, with water, and with the experimental liquid, we can find the liquid's density.
Steps for Liquid Density:
- Find mass of empty bottle ($M_1$).
- Find mass of bottle + Liquid ($M_2$).
- Mass of Liquid = $M_2 - M_1$.
- Density = $(M_2 - M_1) / Volume_{bottle}$.
When using a measuring cylinder, always read the lower meniscus of the water at eye level. Reading from an angle causes parallax error, leading to incorrect volume measurements.
A measuring cylinder is filled with water to the 30 ml mark. When a stone of mass 50 g is immersed, the water level rises to 42 ml. Find the density of the stone.
Solution:
1. Initial Volume ($V_1$): $30\,ml$
2. Final Volume ($V_2$): $42\,ml$
3. Volume of Stone ($V$): $42 - 30 = 12\,ml = 12\,cm^3$.
4. Mass ($M$): $50\,g$.
5. Density ($d$): $M / V = 50 / 12 \approx 4.17\,g/cm^3$.
Final Answer: $4.17\,g/cm^3$.
The Density Bottle has a hole in the stopper so that when you insert it, any excess liquid overflows. This ensures the volume of the liquid inside is always exactly the same, every single time!
3.0 Speed: Distance and Time
While density tells us about the "matter" in an object, Speed tells us about its "motion." In Physics, speed is a scalar quantity that describes how fast an object is moving, regardless of its direction.
What is Speed?
Speed is defined as the distance travelled by a body in unit time.
- S.I. Unit: $m/s$ (metre per second)
- Common Unit: $km/h$ (kilometre per hour)
- Nature: Scalar quantity (has magnitude but no direction).
The Speed Formula
$$v = \frac{s}{t}$$
Where: $v$ = speed, $s$ = distance, $t$ = time
3.1 Unit Conversion: The Magic Fraction
In ICSE numericals, you will frequently need to switch between $km/h$ and $m/s$. Instead of doing a long calculation every time, use these shortcuts:
- To convert $km/h$ to $m/s$: Multiply by $\frac{5}{18}$
- To convert $m/s$ to $km/h$: Multiply by $\frac{18}{5}$
If a body travels different distances in different time intervals, we calculate Average Speed. Do not simply average the two speeds; use the formula:
$$Average\,Speed = \frac{Total\,Distance}{Total\,Time}$$
A train travels at a speed of 90 km/h. How much distance will it cover in 20 seconds? Express the answer in metres.
Solution:
1. Convert Speed to S.I.: $v = 90 \times \frac{5}{18} = 5 \times 5 = 25\,m/s$.
2. Given Time ($t$): $20\,s$.
3. Find Distance ($s$): From formula $v = s/t$, we get $s = v \times t$.
4. Calculation: $s = 25 \times 20 = 500\,m$.
Final Answer: The train covers $500\,m$.
The Peregrine Falcon is the fastest animal on Earth. When it performs its signature hunting stoop (dive), it can reach speeds of over 320 km/h (89 m/s)!