Chapter 13 Gravitation Newtons law of gravitation Any

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 125 000 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 Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup (1546 -1601) Johannes Kepler (1571 -1630) 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

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 0. 00167)

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 2 2. 67 × 10− 7 m/s 2

Answers to the even-numbered problems Chapter 13 Problem 4 3. 00 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 × 109 J (b) 569 N down (c) 569 N up