मराठी
CISCEISC (Science) ISC Class 11
प्रश्नपत्रिका१७
संकल्पना स्पष्टीकरण आणि व्हिडिओ२०९
अभ्यासक्रम

System of Units - Derived Quantities and Units

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Topics

  • Definition: Derived units
  • Formation of Derived Units
  • Supplementary Units: The Special Cases
  • Angle Conversions: Degrees ↔ Radians
  • Key Examples of Derived Quantities
  • Example
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Definition: Derived Quantities

Derived quantities are physical quantities that depend on and can be calculated using fundamental quantities.

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Formation of Derived Units

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Supplementary Units: The Special Cases

1. Plane Angle (θ): The amount of rotation or turning between two straight lines meeting at a point. 

  • When you open a door, the angle between the door's initial and final positions is a plane angle.
  • Mathematical Definition: Plane angle (θ) = Arc length (s) ÷ Radius (r)
  • Formula: θ = s/r
  • Unit: Radian (rad)
  • Key Fact: One complete circle = 2π radians = 360°

Fig 1.1 (a): Plane angle dθ.

2. Solid Angle (Ω): A 3D version of a plane angle - it measures how much space an object occupies when viewed from a point. 

  • When you look at the Moon from Earth, the solid angle tells you how much of your total view is occupied by the Moon.
  • Mathematical Definition: Solid angle (Ω) = Surface area (A) ÷ (Radius)²
  • Formula: Ω = A/r²
  • Unit: Steradian (sr)
  • Key Fact: A complete sphere = 4π steradians

 Fig 1.1 (b): Solid angle dΩ

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Angle Conversions: Degrees ↔ Radians

Key Relationships: 

  • π radians = 180°
  • 1 radian = 180°/π = 57.297°
  • 1° = π/180 = 0.01745 radians 

Useful Conversions:

Degrees Radians Common Description
0° 0 rad No rotation
90° π/2 rad ≈ 1.57 rad Quarter turn
180° π rad ≈ 3.14 rad Half turn
360° 2π rad ≈ 6.28 rad Full turn

Smaller Units: 

  • 1° = 60 minutes (60')
  • 1' = 60 seconds (60'')
  • 1' = 2.91 × 10⁻⁴ rad
  • 1'' = 4.847 × 10⁻⁶ rad
Maharashtra State Board: Class 11

Key Examples of Derived Quantities

Everyday Analogy: Think of fundamental quantities as basic ingredients (like flour, eggs, milk) and derived quantities as recipes that combine these ingredients to make something new (like a cake or bread).

Derived Quantity Formula Fundamental Quantities Used SI Unit
Speed/Velocity Distance ÷ Time Length, Time m/s or m s⁻¹
Momentum Mass × Velocity Mass, Length, Time kg⋅m/s or kg m s⁻¹
Area Length × Width Length m²
Density Mass ÷ Volume Mass, Length kg/m³ or kg m⁻³
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Example

Worked Example: Moon's Solid Angle 

Problem: Calculate the solid angle subtended by the Moon at any point on Earth. 
Given: Moon's diameter = 3,474 km, Earth-Moon distance = 3.84 × 10⁸ m 

Solution Steps: 

Step 1: Find the Moon's radius 
Radius = Diameter ÷ 2 = 3,474 ÷ 2 = 1,737 km = 1.737 × 10³ m 

Step 2: Calculate Moon's cross-sectional area (as seen from Earth) 
Area = π × r² = π × (1.737 × 10³)² = π × 3.017 × 10⁶ m² 

Step 3: Apply solid angle formula 
Ω = A/R² = (π × 3.017 × 10⁶) ÷ (3.84 × 10⁸)² 
Ω = (π × 3.017 × 10⁶) ÷ (1.474 × 10¹⁷) 
Ω = 6.425 × 10⁻⁵ sr 

What this means: The Moon occupies about 0.000064 steradians of your total field of view!

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