Chapter 13 Gravitation

Newton’s law of gravitation Any two (or more) massive bodies attract each other Gravitational force (Newton's law of gravitation) Gravitational constant G = 6.67*10 –11 N*m 2 /kg 2 = 6.67*10 –11 m 3 /(kg*s 2 ) – universal constant

Gravitation and the superposition principle For a group of interacting particles, the net gravitational force on one of the particles is For a particle interacting with a continuous arrangement of masses (a massive finite object) the sum is replaced with an integral

Chapter 13 Problem 5 Three uniform spheres of mass 2.00 kg, 4.00 kg and 6.00 kg are placed at the corners of a right triangle. Calculate the resultant gravitational force on the 4.00-kg object, assuming the spheres are isolated from the rest of the Universe.

Shell theorem For a particle interacting with a uniform spherical shell of matter Result of integration: a uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell's mass were concentrated at its center

Gravity force near the surface of Earth Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surface Thus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earth g = 9.8 m/s 2 This formula is derived for stationary Earth of ideal spherical shape and uniform density

Gravity force near the surface of Earth In reality g is not a constant because: Earth is rotating, Earth is approximately an ellipsoid with a non-uniform density

Gravitational field A gravitational field exists at every point in space When a particle is placed at a point where there is gravitational field, the particle experiences a force The field exerts a force on the particle The gravitational field is defined as: The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle

Gravitational field The presence of the test particle is not necessary for the field to exist The source particle creates the field The gravitational field vectors point in the direction of the acceleration a particle would experience if placed in that field The magnitude is that of the freefall acceleration at that location

Gravitational potential energy Gravitation is a conservative force (work done by it is path-independent) For conservative forces (Ch. 8):

Gravitational potential energy To remove a particle from initial position to infinity Assuming U ∞ = 0

Gravitational potential energy

Escape speed Accounting for the shape of Earth, projectile motion (Ch. 4) has to be modified:

Escape speed Escape speed: speed required for a particle to escape from the planet into infinity (and stop there)

Escape speed If for some astronomical object Nothing (even light) can escape from the surface of this object – a black hole

Chapter 13 Problem 30 (a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system, if it starts at the Earth’s orbit? (b) Voyager 1 achieved a maximum speed of km/h on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?

Kepler’s laws Three Kepler’s laws 1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus 2. The law of areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals 3. The law of periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit Johannes Kepler ( ) Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup ( )

First Kepler’s law Elliptical orbits of planets are described by a semimajor axis a and an eccentricity e For most planets, the eccentricities are very small (Earth's e is )

Second Kepler’s law For a star-planet system, the total angular momentum is constant (no external torques) For the elementary area swept by vector

Third Kepler’s law For a circular orbit and the Newton’s Second law From the definition of a period For elliptic orbits

Satellites For a circular orbit and the Newton’s Second law Kinetic energy of a satellite Total mechanical energy of a satellite

Satellites For an elliptic orbit it can be shown Orbits with different e but the same a have the same total mechanical energy

Chapter 13 Problem 26 At the Earth’s surface a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth.

Questions?

Answers to the even-numbered problems Chapter 13 Problem × 10 −7 m/s 2

Answers to the even-numbered problems Chapter 13 Problem kg and 2.00 kg

Answers to the even-numbered problems Chapter 13 Problem 10 (a) 7.61 cm/s 2 (b) 363 s (c) 3.08 km (d) 28.9 m/s at 72.9° below the horizontal

Answers to the even-numbered problems Chapter 13 Problem 24 (a) −4.77 × 10 9 J (b) 569 N down (c) 569 N up