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 proportional to the square of the distance between them. G is the gravitational constant = 6.67x Nm 2 kg -2. r FgFg FgFg Two particles separated by a distance r exert attractive gravitational forces of equal magnitude on each other.

laser m2m2 m2m2 m1m1 m1m1 FgFg FgFg scale mirror G Henry Cavendish first measured the value of G using a torsion balance (1798). Modern versions use lasers and mirrors – the reflected laser beam is displaced from its original position as two large spheres m 2 are brought close to the smaller spheres, m 1. For m 1 =m 2 =1 kg, and a separation of 1 cm, the force between m 1 and m 2 is 6.67x10 -7 N. The acceleration of each mass will be 6.67x10 -7 ms -2.

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: For an object of mass m located a distance h above the Earth’s surface, we can write: Free-fall Accelerations g(h) Altitude, h (km)g (ms -2 )

Acceleration due to gravity on other planets, g p. PlanetMass (kg) Mean radius (m) g Eq (ms -2 ) Mercury 3.24x x Venus 4.86x x Earth 5.97x x Mars 6.40x x Jupiter 1.89x x Saturn 5.67x x Uranus 8.67x x Neptune 1.05x x Pluto 6.60x x Moon 7.34x x

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: Here is the unit vector along the line joining M and  m. We can express the vectorial nature of the field as: M g mm

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

Analysis As L tends to zero, the force varies as 1/h 2 as expected for two point masses. If h>>L, the force also varies as 1/h 2 ; in other words, when two objects are separated by huge distances, they behave as point masses even though they may both be extended objects.