1. Gravitation
- Concept: Gravitational force causes objects to fall towards Earth, planets to orbit the Sun, and the Moon to orbit Earth.
- Newton’s Insight: The same force (gravity) attracts an apple to Earth and the Moon to Earth; objects fall towards the center (Fig. 9.1).
- Activity 9.1: Whirl a stone tied to a thread; it moves in a circular path due to centripetal force (gravity for Moon). Released, it flies tangentially.
- Universal Law of Gravitation:
- Statement: Every object attracts every other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers (Fig. 9.2).
- Formula: \( F = G \frac{M \times m}{d^2} \), where \( G = 6.673 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2} \).
- SI Unit of G: \( \text{N m}^2 \text{kg}^{-2} \), found by Henry Cavendish.
- Inverse-Square Law: If distance \( d \) increases by 6, force decreases by \( \frac{1}{36} \).
- Applications: Explains Earth’s pull, Moon’s orbit, planetary motion, and tides.
- Example 9.1: Earth (\( 6 \times 10^{24} \, \text{kg} \)) and Moon (\( 7.4 \times 10^{22} \, \text{kg} \)), distance \( 3.84 \times 10^8 \, \text{m} \); \( F = 2.02 \times 10^{20} \, \text{N} \).
- Questions:
- 1. Universal Law: Force \( \propto \frac{M \times m}{d^2} \).
- 2. Earth-Object Force: \( F = G \frac{M \times m}{R^2} \), where \( R \) is Earth’s radius.
Activity: Whirl a stone on a string to feel centripetal force!
2. Free Fall
- Definition: Objects falling towards Earth under gravitational force alone.
- Acceleration Due to Gravity (g):
- Causes velocity increase during fall; \( g = 9.8 \, \text{m/s}^2 \).
- Formula: \( g = G \frac{M}{R^2} \), where \( M = 6 \times 10^{24} \, \text{kg} \), \( R = 6.4 \times 10^6 \, \text{m} \).
- Varies: Greater at poles than equator due to Earth’s shape.
- Force: \( F = m g \); weight of object.
- Motion Equations (with \( a = g \)):
- \( v = u + g t \)
- \( s = u t + \frac{1}{2} g t^2 \)
- \( v^2 = u^2 + 2 g s \)
- Sign: \( g \) is positive downward, negative upward.
- Activity 9.3: Drop paper and stone; paper falls slower due to air resistance. In vacuum, both fall at same rate (Galileo’s experiment).
- Examples:
- 9.2: Car falls 0.5 s, \( g = 10 \, \text{m/s}^2 \); speed = 5 m/s, average speed = 2.5 m/s, height = 1.25 m.
- 9.3: Object rises 10 m; initial velocity = 14 m/s, time to peak = 1.43 s.
- Questions:
- 1. Free Fall: Motion under gravity alone.
- 2. Acceleration Due to Gravity: Acceleration caused by Earth’s gravitational pull.
Activity: Drop objects to observe free fall!
3. Mass and Weight
- Mass: Measure of inertia; constant everywhere (kg).
- Weight:
- Force due to gravity: \( W = m g \); SI unit: Newton (N).
- Varies with location due to changes in \( g \).
- Weight on Moon:
- Moon’s mass (\( 7.36 \times 10^{22} \, \text{kg} \)), radius (\( 1.74 \times 10^6 \, \text{m} \)).
- \( W_m = \frac{1}{6} W_e \); Moon’s gravity is \( \frac{1}{6} \) Earth’s.
- Examples:
- 9.4: 10 kg object on Earth; \( W = 98 \, \text{N} \).
- 9.5: 10 N on Earth; Moon weight = 1.67 N.
- Questions:
- 1. Mass vs. Weight: Mass is constant, weight varies with \( g \).
- 2. Moon Weight: \( \frac{1}{6} \) Earth’s due to lower mass and smaller radius.
Activity: Compare weight on Earth and Moon!
4. Thrust and Pressure
- Thrust: Force perpendicular to a surface (e.g., pushing a pin, Fig. 9.3).
- Pressure:
- Thrust per unit area: \( P = \frac{F}{A} \); SI unit: Pascal (Pa) = N/m².
- Smaller area, higher pressure (e.g., nail’s sharp tip).
- Example 9.6: 5 kg block, dimensions 40×20×10 cm; thrust = 49 N.
- 20×10 cm: \( P = 2450 \, \text{N/m}^2 \).
- 40×20 cm: \( P = 612.5 \, \text{N/m}^2 \).
- Applications: Wide tyres, broad foundations reduce pressure; sharp tools increase pressure.
- Fluid Pressure: Fluids exert pressure in all directions, transmitted undiminished.
Activity: Press a pin to feel thrust and pressure!
5. Buoyancy
- Buoyancy: Upward force by a fluid on an immersed object.
- Activity 9.4: Plastic bottle floats; pushing it down feels upward force (buoyancy).
- Activity 9.5: Iron nail sinks, cork floats; depends on density.
- Float: Object’s density < liquid’s density (cork < water).
- Sink: Object’s density > liquid’s density (nail > water).
- Questions:
- 1. Thin strap: High pressure on shoulder due to small area.
- 2. Buoyancy: Upward force by fluid on immersed object.
- 3. Float/Sink: Depends on density relative to liquid.
Activity: Test objects in water to see if they float!
6. Archimedes' Principle
- Statement: Buoyant force equals the weight of fluid displaced by the immersed object.
- Activity 9.7: Stone on spring balance; weight decreases in water due to buoyancy (Fig. 9.6).
- Explanation: Buoyant force reduces net force on string; no further decrease when fully immersed.
- Applications: Ship/submarine design, lactometers, hydrometers.
- Questions:
- 1. Mass on Scale: 42 kg is mass (constant, not weight).
- 2. Cotton vs. Iron: Iron is heavier; cotton’s lower density gives same mass for larger volume.
Activity: Weigh an object in air and water!
7. Exercises
- Exercise Questions:
- 1. Distance halved: \( F \) becomes 4 times (inverse-square).
- 2. Equal fall: \( g \) is mass-independent.
- 3. 1 kg on Earth: \( F = 9.8 \, \text{N} \).
- 4. Equal force: Newton’s third law; act on different bodies.
- 5. Earth doesn’t move: High mass, low acceleration.
- 6. Force changes:
- (i) Double mass: \( F \) doubles.
- (ii) Double distance: \( F \) becomes \( \frac{1}{4} \); triple: \( \frac{1}{9} \).
- (iii) Double both masses: \( F \) quadruples.
- 7. Importance: Explains Earth’s pull, orbits, tides.
- 8. Free fall acceleration: \( g = 9.8 \, \text{m/s}^2 \).
- 9. Gravitational force: Weight (\( W = m g \)).
- 10. Gold at poles: Less weight at equator (lower \( g \)).
- 11. Paper vs. ball: Air resistance slows flat paper.
- 12. 10 kg object: Earth \( W = 98 \, \text{N} \), Moon \( W = 16.33 \, \text{N} \).
- 13. Ball 49 m/s up: Max height = 122.5 m, total time = 10 s.
- 14. Stone 19.6 m: Final velocity = 19.6 m/s.
- 15. Stone 40 m/s up: Max height = 80 m, displacement = 0 m, distance = 160 m.
- 16. Earth-Sun: \( F = 3.57 \times 10^{22} \, \text{N} \).
- 17. Stones meet: At 44.1 m after 2.45 s.
- 18. Ball 6 s: (a) 29.4 m/s, (b) 44.1 m, (c) 19.6 m above ground.
- 19. Buoyant force: Upward.
- 20. Plastic floats: Density < water, buoyant force > weight.
- 21. 50 g, 20 cm³: Density = 2.5 g/cm³, sinks.
- 22. 500 g, 350 cm³: Density = 1.43 g/cm³, sinks; displaces 350 g water.
Activity: Solve a gravitation problem!
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