FUNDAMENTAL AND DERIVED QUANTITIES

FUNDAMENTAL QUANTITIES AND UNITS: These are the fundamental dimensions which represent the basic physical quantities from which all other quantities can be derived.

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TYPES OF FUNDAMENTAL QUANTITIES

• Length (L): Measured in metres (m).
• Mass (M): Measured in kilograms (kg).
• Time (T): Measured in seconds (s).
• Electric Current (I): Measured in amperes (A).
• Temperature (Θ): Measured in kelvins (K).
• Amount of Substance (N): Measured in moles (mol).
• Luminous Intensity (J): Measured in candelas (cd).

DERIVED QUANTITIES AND THEIR UNITS: These are quantities derived from the combination of fundamental quantities.

TYPES OF DERIVED QUANTITIES AND THEIR DIMENSIONS

• Velocity (v)
• Formula: v=distance/time
• Dimension: [L][T]⁻¹.
• Acceleration (a)
• Formula: a = change in velocity/time
• Dimension: [L][T]⁻².
• Force (F)
• Formula: F = mass × acceleration
• Dimension: [M][L][T]⁻².
• Work (W)
• Formula: W = force × distance
• Dimension: [M][L]²[T]⁻².
• Energy (E)
• Formula: E = Work Done
• Dimension: [M][L]²[T]⁻².
• Power (P)
• Formula: P = Work done/time
• Dimension: [M][L]²[T]⁻³.
• Pressure (P)
• Formula: P=Force/Area
• Dimension: [M][L]⁻¹[T]⁻².
• Density (ρ)
• Formula: ρ=mass/volume
• Dimension: [M][L]⁻³.
• Momentum (p)
• Formula: p = mass × velocity
• Dimension: [M][L][T]⁻¹.
• Impulse (J)
• Formula: J = Force × time
• Dimension: [M][L][T]⁻¹.
• Kinetic Energy (KE)
• Formula: KE = 1/2 mv²
• Dimension: [M][L]²[T]⁻².
• Potential Energy (PE)
• Formula: PE = mgh (where h = height; g = acceleration due to gravity)
• Dimension: [M][L]²[T]⁻².
• Torque (τ)
• Formula: τ = Force × distance from pivot
• Dimension: [M][L]²[T]⁻².

1. CONVERSION OF UNITS

Dimensional analysis can be used to convert units.

For example, to convert 5 kilometres to metres:

1 km = 1000 m,

5 km = 5 × 1000 m = 5000 m. 

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          1.3     MEASUREMENT TOOL 

• Length: Metre rule.
• Mass: Beam balance.
• Temperature: Thermometer.
• Weight: Spring Balance.
• Caliper: Used for measuring diameter of cylinder, thickness of a coin.
• Micrometer Screw Gauge: Measures small dimensions accurately (e.g., thickness of wire).
• Precision: Up to 0.01 mm or 0.001 mm.

2.    PHYSICAL QUANTITIES

A physical quantity is a characteristic or property of an object or system that can be quantified through measurement. They are divided into two:

• Fundamental (base) quantities.
• Derived quantities.

TYPES OF PHYSICAL QUANTITIES

• Scalar Quantities – These have only magnitude (e.g., mass, temperature, distance).
• Vector Quantities – These have both magnitude and direction (e.g., force, velocity, acceleration).

1. DEFINITION OF TERMS
2. Distance: It is the total length of the path travelled by an object, regardless of the direction. It is a scalar quantity, meaning it only has magnitude and no direction.
3. Displacement: This is a vector quantity that refers to the shortest distance from the initial to the final position of an object, along with the direction.
4. Mass: It is a measure of the amount of matter in an object. It is an intrinsic property and does not change regardless of the object’s location. The standard unit of mass in the International System of Units (SI) is the kilogram (kg).
5. Weight: It is the force exerted by gravity on an object and is calculated as Weight = Mass × Gravitational Acceleration (W = mg). The gravitational acceleration on Earth is approximately 9.81 m/s².
6. Instantaneous Speed: It is the speed at a specific moment.
7. Inertia: It is the resistance of a body to changes in motion.
8. Power: Rate of doing work.
9. Energy: Capacity to do work.
10. Velocity: Rate of change of displacement.
11. Acceleration: Rate of change of velocity with time.

1. SCALARS AND VECTORS

SCALARS: Scalars are quantities that are fully described by a magnitude (numerical value) alone. They do not have a direction (e.g. Mass – 5kg, Temperature – 30°C, Energy – 50 J etc.).

VECTORS: Vectors are quantities that have both magnitude and direction. They are represented graphically by arrows (e.g Displacement 10 m to the north, Velocity 60km/h to the east, Force 5N at an angle of 30°etc.).

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OPERATIONS WITH VECTORS  

• Addition: Vectors can be added using the head-to-tail method or by using components.
• Subtraction: To subtract a vector, you can add its negative (reverse the direction).
• Multiplication
• Dot Product: A scalar quantity obtained from two vectors:

(A · B = |A| |B| cos(θ)).

• Cross Product: A vector quantity that results in a vector perpendicular to the plane formed by the two vectors:

                   (A × B = |A| |B| sin(θ) n).

1. IMPORTANT CONCEPTS
2. Unit Vectors: Vectors with a magnitude of one (1), used to indicate direction.
3. Resultant Vector: The vector that results from adding two or more vectors together.
4. Equilibrium: A state where the sum of all forces (vectors) acting on an object is zero.

1. RELATIVE VELOCITY

It is the velocity of one object as observed from another object. The velocity of an object A relative to another object B is the velocity that object A would appear to have to an observer moving with B

DIRECTION FOR RELATIVE VELOCITY

• If both objects move in the same direction, subtract their speeds.
• If they move in opposite directions, add their speeds.