Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton
II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds
13-1 Newton's Law of Gravitation The gravitational force o o o o o o o Holds us to the Earth Holds Earth in orbit around the Sun Holds the Sun together with the stars in our Galaxy Holds together the Local Group of galaxies Holds together the Local Supercluster of galaxies Attempts to slow the expansion of the Universe Is responsible for black holes Gravity is far-reaching and very important!
13-1 Newton's Law of Gravitation Gravitational attraction depends on mass of an object Earth has a large mass and produces a large attraction The force is always attractive, never repulsive Bodies attract each other through gravitational attraction Newton realized this attraction was responsible for maintaining the orbits of celestial bodies Newton's Law of Gravitation defines the strength of this attractive force between particles Between an apple & the Earth: ~0.8 N Between 2 people: < 1 µn
13-1 Newton's Law of Gravitation r F F m 1 m 2 The magnitude of the force is given by: Eq. (13-1) Where G is the gravitational constant: Eq. (13-2) The force always points from one particle to the other, so this equation can be written in vector form: Eq. (13-3)
13-1 Newton's Law of Gravitation M M M The shell theorem describes gravitational attraction for objects Earth is a nesting of shells, so we feel Earth's mass as if it were all located at its centre Gravitational force forms third-law force pairs (i.e. N3L) e.g. Earth-apple and apple-earth forces are both 0.8 N
13-1 Newton's Law of Gravitation Earth-apple and apple-earth forces are both ~0.8 N The difference in mass causes the difference in the apple:earth accelerations: ~10 m/s 2 vs. ~ 10-25 m/s 2
Consider the objects of various masses indicated below. The objects are each separated from another object by the distance indicated. In which of these situations is the gravitational force exerted on the two objects the largest? a) #1 b) #2 c) #3 d) #2 and #3 e) #1, #2, and #3
13-2 Gravitation and the Principle of Superposition The principle of superposition applies. i.e. Add the individual forces as vectors: Eq. (13-5) For a real (extended) object, this becomes an integral: Eq. (13-6) If the object is a uniform sphere or shell we can treat its mass as being at its centre instead M M M
13-2 Gravitation and the Principle of Superposition Example Summing two forces: Figure 13-4
Consider a system of particles, each of mass m. In which one of the following configurations is the net gravitational force on Particle A the largest? The horizontal or vertical spacing between particles is the same in each case. a) 1 b) 2 c) 4 d) 1 and 2 equally large e) 2 and 4 are equally large
13-3 Gravitation Near the Earth's Surface Combine F = GMm/r 2 and F = ma g : Eq. (13-11) Gives magnitude of gravitational acceleration at a given distance from the centre of the Earth Table 13-1 shows the value for a g for various altitudes above the Earth s surface
13-3 Gravitation Near the Earth's Surface The calculated a g will differ slightly from the measured g at any location on the Earth s surface Three reasons. The Earth. 1. mass is not uniformly distributed 2. is not a perfect sphere 3. rotates
13-3 Gravitation Near Earth's Surface Example Difference in gravitational force and weight due to rotation at the equator: N2L : F N ma g = m V 2, the centripetal acceleration. R Here V = ωr with ω the angular velocity, and F N = mg. Thus g = a g ω 2 R. Exercise for the student. Show this is about 0.034 m/s 2. Use R=6,400km and ω=2π radians/day. Question: do you weigh more or less at the equator than the Poles? Figure 13-6
13-5 Gravitational Potential Energy Gravitational potential energy for a two-particle system is written: Eq. (13-21) Note this value is negative and approaches 0 for r The gravitational potential energy of a system is the sum of potential energies for all pairs of particles Proof comes from integrating the force to obtain the work done. i.e. U = W = F dr and using F = GMm. r 2
13-5 Gravitational Potential Energy The gravitational force is conservative. The work done by this force does not depend on the path followed by the particles, only the difference in the initial and final positions of the particles. Since the work done is independent of path, so is the gravitational potential energy change Eq. (13-26) Figure 13-10
13-5 Gravitational Potential Energy For a projectile to escape the gravitational pull of a body, it must come to rest only at infinity (if at all). At rest at infinity: K = 0 and U = 0 (because r ) So K + U must be 0 at surface of the body to escape: This is the escape speed. The minimum value to escape. Rockets launch eastward to take advantage of Earth's rotational speed, to reach v escape more easily
13-5 Gravitational Potential Energy
You move a ball of mass m away from a sphere of mass M. 1. Does the gravitational potential energy of the system of the ball and sphere a) Increase, or b) Decrease. 2. Is the Work done by the gravitational force between the ball and the sphere a) Positive work, or b) Negative work r m M
In a distant solar system where several planets are orbiting a single star of mass M, a large asteroid collides with a planet of mass m orbiting the star at a distance r. As a result, the planet is ejected from its solar system. What is minimum amount of energy that the planet must receive in the collision to be removed from the solar system? a) b) r c) m M d) e)
2_A3: Retrograde Motion of the Planets
13-6 Planets and Satellites: Kepler's Laws The motion of planets in the solar system was a puzzle for astronomers, especially curious motions such as retrograde motion. Johannes Kepler (1571-1630) derived laws of motion using Tycho Brahe's (1546-1601) measurements Figure 13-11 Figure 13-12
13-6 Planets and Satellites: Kepler's Laws The orbit is defined by its semi-major axis a and its eccentricity e An eccentricity of zero corresponds to a circle Eccentricity of Earth's orbit is 0.0167
Kepler2L: Kepler s 2 nd Law Equal Areas in Equal Times
13-6 Planets and Satellites: Kepler's Laws Equivalent to Conservation of Angular Momentum See later in this course.
13-6 Planets and Satellites: Kepler's Laws The law of periods can be written mathematically as: Holds for elliptical orbits if we replace r with a, the semi-major axis.
13-6 Planets and Satellites: Kepler's Laws
A spacecraft is in low orbit of the Earth with a period of approximately 90 minutes. By which of the following methods could the spacecraft stay in the same orbit and reduce the period of the orbit? a) Before launch, increase the mass of the spacecraft to increase the centripetal force on it. b) Remove any unnecessary equipment, cargo, and supplies to reduce the mass and decrease its angular momentum. c) Fire rockets to increase the tangential velocity of the ship. d) None of the above methods will achieve the desired effect.
13-7 Satellites: Orbits and Energy Relating the centripetal acceleration of a satellite to the gravitational force, we can rewrite as energies: Eq. (13-38) Meaning that: Eq. (13-39) Therefore the total mechanical energy is: Eq. (13-40)
13-7 Satellites: Orbits and Energy Total energy E is the negative of the kinetic energy For an ellipse, we substitute a for r Therefore the energy of an orbit depends only on its semi-major axis, not its eccentricity All orbits in Figure 13-15 have the same energy Figure 13-15
13-7 Satellites: Circular Orbit Graph of variation in Energy for a circular orbit, radius r Note that: E(r) and U(r) are negative E(r) = -K(r) E(r), U(r), K(r) all 0 as r
13-7 Satellites: Orbits and Energy a) Which orbit (1, 2 or 3?) will the shuttle take when it fires a forward-pointing thruster so as to reduce its kinetic energy? b) Is the orbital period T then (i) greater than, (ii) less than or (iii) the same as, that of the circular orbit?
13 Summary The Law of Gravitation Superposition Eq. (13-1) Eq. (13-2) Gravitational Behavior of Uniform Spherical Shells The net force on an external object: calculate as if all the mass were concentrated at the centre of the shell Gravitational Acceleration Eq. (13-5) Eq. (13-11)
13 Summary Free-Fall Acceleration and Weight Earth's mass is not uniformly distributed, the planet is not spherical, and it rotates: the calculated and measured values of acceleration differ Gravitational Potential Energy Gravitation within a Spherical Shell A uniform shell exerts no net force on a particle inside Inside a solid sphere: Potential Energy of a System Eq. (13-19) Eq. (13-21) Eq. (13-22)
13 Summary Escape Speed Eq. (13-28) Kepler's Laws The law of orbits: ellipses The law of areas: equal areas in equal times The law of periods: Eq. (13-34) Energy in Planetary Motion Eq. (13-42) Kepler's Laws Gravitation and acceleration are equivalent