1. Introduction

Gases, just like liquids, are fluids and can exert forces on objects placed in them. Although air cannot be seen, it has mass, occupies space, and exerts pressure. When an object is placed in air, it experiences an upward force known as upthrust. This force explains why balloons rise, smoke moves upward, and parachutes slow down falling objects.

2. What Is Upthrust?

Upthrust (buoyant force) is the upward force exerted by a fluid (liquid or gas) on an object immersed in it.

In gases, upthrust occurs because air pressure increases with depth. The pressure acting on the bottom of an object is greater than the pressure acting on the top, resulting in a net upward force.

3. Archimedes’ Principle in Gases

Archimedes’ principle also applies to gases and states that:

An object wholly or partially immersed in a gas experiences an upthrust equal to the weight of the gas displaced.

This means the greater the volume of air displaced by an object, the greater the upthrust acting on it.

4. Mathematical Expression for Upthrust in Gases

The upthrust in gases can be calculated using the formula:$$U = \rho g V$$U=ρgV

Where:

• $U$U = upthrust (N)
• $\rho$ρ = density of the gas (kg/m³)
• $g$g = acceleration due to gravity (m/s²)
• $V$V = volume of gas displaced (m³)

5. Conditions for Floating, Rising, and Falling in Air

6. Illustrations of Upthrust in Gases

(a) Balloons

A helium or hot-air balloon rises because it displaces air that is denser than itself. The upthrust acting on the balloon is greater than its weight, causing it to rise upward.

(b) Parachutes

When a parachute opens, it increases the surface area and displaces more air. The upthrust (and air resistance) acting upward increases, reducing the downward motion of the parachutist and allowing a safe landing.

(c) Smoke and Hot Air

Hot air and smoke rise because they are less dense than the surrounding cooler air. The upthrust acting on them is greater than their weight.

7. Worked Examples

Example 1:

An object displaces 0.1 m³ of air. Given that its’ density of air = 1.2 kg/m³ and the acceleration due to gravity = 10 m/s². calculate upthrust in air.

Calculate the upthrust acting on the object.

Solution:$$U = \rho g V$$U=ρgV $$U = 1.2 \times 10 \times 0.1$$U=1.2×10×0.1 $$U = 1.2 \, \text{N}$$U=1.2N

Upthrust = 1.2 N

Example 2: Determining Motion of an Object in Air

A balloon has a weight of 0.9 N.
Upthrust acting on it is 1.2 N.

Since:$$\text{Upthrust} > \text{Weight}$$Upthrust>Weight

The balloon will rise.

Example 3: Volume of Air Required for Floating

An object has a weight of 2.4 N.
Density of air = 1.2 kg/m³, $g = 10 \, \text{m/s}^2$g=10m/s2

Calculate the minimum volume of air the object must displace to float.$$2.4 = 1.2 \times 10 \times V$$2.4=1.2×10×V $$V = \frac{2.4}{12}$$V=122.4​ $$V = 0.2 \, \text{m}^3$$V=0.2m3

Required volume = 0.2 m³

8. Problems Involving Upthrust in Gases

1. A balloon displaces 0.25 m³ of air.
   Density of air = 1.2 kg/m³, $g = 10 \, \text{m/s}^2$g=10m/s2.
   Calculate the upthrust acting on the balloon.
2. An object weighs 3 N and experiences an upthrust of 2 N in air.
   State whether the object will rise, float, or fall, giving a reason.
3. Explain why a parachutist falls slowly after opening a parachute.
4. A hot-air balloon displaces 1.5 m³ of air.
   Calculate the upthrust acting on the balloon.
5. State two applications of upthrust in gases in everyday life.

9. Conclusion

Upthrust in gases is an important concept that explains the behavior of objects in air. It depends on the density of the gas, the volume displaced, and gravity. Understanding upthrust helps explain real-life phenomena such as balloons rising, parachutes slowing descent, and hot air moving upward.

Related topics

• floating and sinking
• The law of floatation
• Upthrust in gases

Key matrices concepts include identity, determinant, inverse, and singular matrices, which help describe matrix properties and transformations.

Matrices are important tools in mathematics and science. They are used to solve systems of equations, describe transformations, and model real-life problems such as economics, physics, and computer graphics.
In this lesson, we will learn four important concepts related to square matrices:

• Identity Matrix
• Determinant of a Matrix
• Inverse of a Matrix
• Singular Matrix

2. Identity Matrices

Definition

An identity matrix is a square matrix that has:

• 1’s on the main diagonal
• 0’s everywhere else

It is similar to the number 1 in multiplication, because multiplying any matrix by the identity matrix gives the same matrix.

Notation of matrices

The identity matrix is denoted by I.

2 × 2 Identity Matrix

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

3 × 3 Identity Matrix

$$I_3$$I3​=​100​010​001​​

Illustration

Main diagonal →   1   1   1
Other entries →   0   0   0

Property

$$A \times I = I \times A = A$$

3. Determinant of a Matrix

Definition

The determinant of a square matrix is a single number that tells us important information about the matrix, such as:

• Whether the matrix has an inverse
• Whether the system of equations has a unique solution
• The scaling factor of a geometric transformation

Determinant of a 2 × 2 Matrix

For a matrix:$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

The determinant is:$$|A| = ad – bc$$

Example

$$A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$$A=[31​24​] $$|A| = (3 \times 4) – (2 \times 1) = 12 – 2 = 10$$∣A∣=(3×4)−(2×1)=12−2=10

The determinant represents area scaling:

• If |A| = 2 → area doubles
• If |A| = 0 → area collapses to a line

4. Inverse of a Matrices

Definition

The inverse of a matrix A is another matrix $A^{-1}$A−1 such that:$$A \times A^{-1} = I$$

Inverse of a 2 × 2 Matrix

For:$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$$$A^{-1} = \frac{1}{ad – bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

Note: The inverse exists only if the determinant is not zero.

Example

$$A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$$

Step 1: Find determinant:$$|A| = (2 \times 3) – (1 \times 5) = 6 – 5 = 1$$

Step 2: Find inverse:$$A^{-1} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$$

Illustration

Matrix A → changes a vector  
Inverse A⁻¹ → returns it back to original

5. Singular Matrix

Definition

A singular matrix is a matrix that has no inverse.

Condition

A matrix is singular if: ∣A∣=0

Example

$$A = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}$$A=[21​42​] $$|A| = (2 \times 2) – (4 \times 1) = 4 – 4 = 0$$∣A∣=(2×2)−(4×1)=4−4=0

Therefore, A is singular and cannot be inverted.

Illustration

Transformation squashes shape into a line → no way to reverse it

6. Summary Table

7. Practice Questions

Question 1

Find the determinant of:$$A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$$

Answer:
$|A| = (4×6) − (7×2) = 24 − 14 = 10$

Question 2

Write the 3 × 3 identity matrix.

Answer:$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Question 3

Determine whether the matrix is singular:$$A = \begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix}$$

Answer:
$|A| = (1×6) − (3×2) = 6 − 6 = 0$

Matrix is singular.

Question 4

Find the inverse of:$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

Answer:$$|A| = (1×4) − (2×3) = 4 − 6 = -2$$$$A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$$

8. Conclusion

Understanding identity, determinant, inverse, and singular matrices is important in solving systems of equations, physics, engineering, and computer science. These concepts help us understand when a matrix can be reversed and how it transforms.

Related Topics

• Operation on matrices
• Introduction to matrices
•  MATRICES AND TRANSFORMATIONS

The law of floatation is considered a special case of the Archimedes’ principle. The law states that: A floationg object displaces its own weight of the fluid in which it floats.

To investigate the law of floatation

Materials/apparatus

• Measuring cylinder
• water
• test-tube
• sand
• weighing balance

procedure

1. Half fill the measuring cylinder with water and record the level
2. Place a clean dry test tube into the cylinder and add some sand in it until it floats as shown. Record the new water level

3. Determine the volume of water displaced

4. Remove the test tube from the cylinder, dry it and determine it’s weight

5. Repeat the procedure five times, adding a little more sand each time and recording the volume of water displaced. Record the results in a table shown.

observations and conclusions

• The test tube sinks deeper every time some some is added
• Weight of the test tube and it’s content is equal to the weight of water displaced

Experiment 2

Materials Needed:

• Beaker or a transparent container (1–2 L)
• Water
• Objects of different densities and shapes (wood, plastic, metal, cork)
• Spring balance (optional, for measuring weight)
• Measuring cylinder or scale
• Graph paper (for recording observations)
• Ruler

Procedure:

1. Preparation:
   • Fill the beaker with water about 3/4 full.
   • Record the water level.
2. Observation of Floating and Sinking:
   • Gently place each object in water one by one.
   • Observe whether it floats, sinks, or partially floats.
3. Measurement (Optional for Quantitative Study):
   • Measure the weight of each object using a spring balance.
   • Observe how much of the object is submerged when it floats.
   • Record the depth of submersion.
4. Changing Variables (Shape & Density):
   • Take a piece of aluminum foil and make it into a flat sheet, then into a boat shape.
   • Place it in water and observe whether the shape affects flotation.
5. Record Observations:
   • Note which objects float and which sink.
   • For floating objects, note the fraction submerged.

Variables:

• Independent Variable: Type of object (density, material, shape)
• Dependent Variable: Whether the object floats or sinks, depth of submersion
• Controlled Variables: Volume of water, temperature of water, same container

Expected Observations:

• Objects denser than water will sink (e.g., metals like iron).
• Objects less dense than water will float (e.g., cork, wood).
• Changing the shape of an object can make it float even if it is denser than water (e.g., aluminum foil boat) because it displaces more water.

Conclusion:

• An object floats if the upthrust (buoyant force) is equal to its weight hence verifying the law of floatation.
• The fraction of the object submerged depends on its density relative to water.
• The shape can influence flotation by changing how much water is displaced.

Example problem

A ship of mass 250000kg floats on flesh water. If the ship enters the sea, determine the load that must be added to it so that it displaces the same volume of water as before. (Take density of fresh water as 1000Kgm^-3 and that of sea water as 1025kgm^-3)

solution

weight of the ship = 250000kg x 10Nkg^-1 = 25000000 N

from the law of floatation: weight of flesh water displaced = weight of the ship

Volume of sea water displaced when more load is added = 250m³

mass of sea water displaced = 250m³ x 1025kgm³ = 256,250kg

weight of the sea water displaced = 2,562,500 N

Extra load needed = weight of sea water to be displaced – weight of flesh water displaced

= 2,562,500N – 2,500,000N = 62, 500N

Related topics

• The Archimedes’ principle
• Physics digital

Fungi are a unique group of living organisms that are separate from plants, animals, and bacteria. They include molds, yeasts, and mushrooms. Unlike plants, fungi do not make their own food through photosynthesis. Instead, they absorb nutrients from organic material around them. Their cell walls are made of chitin, the same substance found in insect shells, which is one feature that distinguishes them from plants.

In terms of biology, fungi usually grow as long, thread-like structures called hyphae, which form a network known as mycelium. They reproduce both sexually and asexually, most commonly by producing spores that can spread through air, water, or living organisms. These spores allow fungi to survive in harsh conditions and colonize new environments. Some fungi, like yeast, are unicellular, while others are multicellular and can grow quite large.

Ecologically, fungi play a crucial role in maintaining balance in ecosystems. They are primary decomposers, breaking down dead plants and animals and recycling nutrients back into the soil. Many fungi also form symbiotic relationships. For example, mycorrhizal fungi live in association with plant roots and help plants absorb water and minerals, while lichens are partnerships between fungi and algae or cyanobacteria that can survive in extreme environments.

Fungi are important to human health in both positive and negative ways. Some fungi cause diseases, such as athlete’s foot or more serious infections in people with weak immune systems. However, fungi are also extremely beneficial. Penicillin, one of the first and most important antibiotics, was derived from a fungus. Other fungi are used to produce medicines, enzymes, and vitamins.

In food and industry, fungi are widely used and highly valuable. Edible mushrooms are a nutritious food source, providing protein, vitamins, and minerals. Yeast is essential in baking and brewing, as it ferments sugars to produce carbon dioxide and alcohol. Fungi are also involved in making cheese, soy sauce, and other fermented foods. Overall, fungi are essential organisms that support ecosystems, human health, and many everyday products.

examples of fungi plants

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Structure of fungi

The basic structural unit of most fungi is the hypha, a thin, thread-like filament. Hyphae grow and branch to form a complex network called the mycelium, which is usually hidden within soil, food, or other organic material. The cell walls of fungi are made of chitin, providing strength and protection. Some fungi, such as yeast, are unicellular, while others are multicellular and may form visible structures like mushrooms, which are actually reproductive parts.

Reproduction

Fungi reproduce in both asexual and sexual ways. Asexual reproduction is more common and occurs through methods such as spore formation, budding (in yeast), or fragmentation of hyphae. Sexual reproduction involves the fusion of specialized cells from two compatible fungi, followed by genetic recombination. In both cases, fungi usually produce spores, which are lightweight, easily dispersed, and capable of surviving unfavorable conditions.

Life Cycle

The fungal life cycle typically begins when a spore lands in a suitable environment and germinates into hyphae. These hyphae grow and form a mycelium that absorbs nutrients. Under favorable conditions, the mycelium produces reproductive structures that release new spores, continuing the cycle. In fungi that reproduce sexually, the life cycle includes stages of plasmogamy (fusion of cytoplasm), karyogamy (fusion of nuclei), and meiosis, leading to genetically diverse spores.

Fungi play a vital role in ecosystems by keeping nutrients circulating through the environment. They are key regulators of ecological balance because they break down complex organic materials that most other organisms cannot digest. Without fungi, dead plants and animals would accumulate, and essential nutrients such as carbon, nitrogen, and phosphorus would remain locked away instead of being reused by living organisms.

Fungi role in ecosystems

One of the most important ecological roles of fungi is decomposition. As decomposers, fungi release enzymes that break down dead leaves, wood, and animal remains into simpler substances. These nutrients are then returned to the soil, where they can be absorbed by plants and other organisms. Fungi are especially important in breaking down tough materials like cellulose and lignin found in plant cell walls, making them indispensable in forest and soil ecosystems.

Fungi also form symbiotic relationships with other organisms, meaning both partners benefit. A major example is mycorrhizae, a partnership between fungi and plant roots. The fungal hyphae extend far into the soil, greatly increasing the plant’s ability to absorb water and minerals such as phosphorus. In return, the plant provides the fungus with sugars produced during photosynthesis. This relationship improves plant growth, soil structure, and overall ecosystem productivity.

Another important symbiotic relationship is seen in lichens, which are formed by a fungus living together with an alga or cyanobacterium. The fungus provides protection, moisture, and support, while the alga or cyanobacterium produces food through photosynthesis. Lichens can survive in extreme environments such as bare rock, deserts, and polar regions, and they are often among the first organisms to colonize new or disturbed areas, helping to start soil formation and ecological succession.

Overall, through decomposition and symbiosis, fungi are essential for ecosystem health, stability, and sustainability.

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Floating and sinking describe how objects behave when placed in a fluid such as water or air. Whether an object floats or sinks depends on the balance between its weight and the upthrust (buoyant force) exerted by the fluid. Objects that float are supported by the fluid because the upthrust is equal to or greater than their weight, while objects sink when their weight is greater than the upthrust. This concept is closely related to density and helps explain many everyday phenomena, from ships staying afloat to stones sinking in water.

The Archimedes’ principle

Related topics

• Electric current and potential difference
• Buoyancy
• lab simulation