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Motion, Forces and Energy Gravitation: That Sinking Feeling

Motion, Forces and Energy Gravitation: That Sinking Feeling. Newton’s Law of Gravitation (1686): Every particle of matter in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely

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Motion, Forces and Energy Gravitation: That Sinking Feeling

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  1. Motion, Forces and EnergyGravitation: That Sinking Feeling Newton’s Law of Gravitation (1686): Every particle of matter in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. Fg Fg G is the gravitational constant = 6.67x10-11 Nm2kg-2. r Two particles separated by a distance r exert attractive gravitational forces of equal magnitude on each other.

  2. laser G Henry Cavendish first measured the value of G using a torsion balance (1798). mirror scale m1 Fg Modern versions use lasers and mirrors – the reflected laser beam is displaced from its original position as two large spheres m2 are brought close to the smaller spheres, m1. m2 m2 Fg m1 For m1=m2=1 kg, and a separation of 1 cm, the force between m1 and m2 is 6.67x10-7 N. The acceleration of each mass will be 6.67x10-7 ms-2.

  3. Weight We can develop a fundamental definition of g. Because the force acting on a mass near the Earth’s surface is mg, we can say: Free-fall Accelerations g(h) Altitude, h (km) g (ms-2) 1000 7.33 2000 5.68 5000 3.08 10000 1.49 20000 0.57 50000 0.13 For an object of mass m located a distance h above the Earth’s surface, we can write:

  4. Acceleration due to gravity on other planets, gp. Planet Mass (kg) Mean radius (m) gEq (ms-2) Mercury 3.24x1023 2.34x106 3.95 Venus 4.86x1024 6.10x10 8.72 Earth 5.97x1024 6.37x10 9.78 Mars 6.40x1023 3.32x106 3.84 Jupiter 1.89x1027 69.8x106 23.16 Saturn 5.67x1026 58.2x106 8.77 Uranus 8.67x1025 23.8x106 9.46 Neptune 1.05x1026 22.4x106 13.66 Pluto 6.60x1023 2.90x106 5.23 Moon 7.34x1022 1.74x106 1.62

  5. The Concept of Gravitational Field A mass creates a gravitational field around it. We can use a test mass as a detector of gravitational field by taking it to various points and measuring the gravitational force that acts on it and defining the field g as: dm g We can express the vectorial nature of the field as: M Here is the unit vector along the line joining M and dm.

  6. y L h dx x x Gravitational force between a particle and a bar The (red) segment of the bar of length dx has mass dM. The mass per unit length l is dM/dx or M/L.

  7. Analysis As L tends to zero, the force varies as 1/h2 as expected for two point masses. If h>>L, the force also varies as 1/h2; in other words, when two objects are separated by huge distances, they behave as point masses even though they may both be extended objects.

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