Gravitation 3 PDF

5.3-4 Gravitation Inside Earth Learning Objectives 13.09 Identify that a uniform 13.10 Calculate the shell of material exerts no net gravitational force that is gravitational force on a exerted on a particle at a particle located...

5.3-4 Gravitation Inside Earth Learning Objectives 13.09 Identify that a uniform 13.10 Calculate the shell of material exerts no net gravitational force that is gravitational force on a exerted on a particle at a particle located inside it. given radius inside a nonrotating uniform sphere of matter. © 2014 John Wiley & Sons, Inc. All rights reserved.  5.3-4 Gravitation Inside Earth The shell theorem also means that: Forces between elements do not disappear, but their vector sum is 0 Let's find the gravitational force inside a uniform-density Earth a solid sphere, not a shell: Eq. (13-17) Both Mass (M) and distance to the centre (r) decrease. But r decreases faster and is inverse. So overall, F increases as we move inside. The reverse occurs as we move to the other side of the centre of the earth. (see next slide: inside the earth (if earth were uniform) Figure 13-7 © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-4 Gravitation Inside Earth The constant density is: Substitute in to Eq. 13-17: Eq. (13-19) If we write this as a vector equation, substituting K for the constants: Eq. (13-20) Object dropped through Earth oscillates (Hooke's law) © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-5 Gravitational Potential Energy Learning Objectives 13.11 Calculate the 13.13 Using the gravitational gravitational potential energy force on a particle near an of a system of particles (or astronomical body (or some uniform spheres that can be second body that is fixed in treated as particles). place), calculate the work done by the force when the 13.12 Identify that if a particle body moves. moves from an initial to a final point while experiencing a 13.14 Apply the conservation gravitational force, the work of mechanical energy done by that force (and thus (including gravitational the change in gravitational potential energy) to a particle potential energy) is moving relative to an independent of which path is astronomical body (or some taken. second body that is fixed in place). © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-5 Gravitational Potential Energy 13.15 Explain the energy 13.16 Calculate the escape requirements for a speed of a particle in particle to escape from an leaving an astronomical astronomical body body. (assumed to be a uniform sphere). © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-5 Gravitational Potential Energy Note that gravitational potential energy is a property of a pair of particles We cannot divide it up to say how much of it “belongs” to each particle in the pair We often speak as of the “gravitational potential energy of a baseball” in the ball-Earth system We get away with this because the energy change appears almost entirely as kinetic energy of the ball This is only true for systems where one object is much less massive than the other © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-5 Gravitational Potential Energy Eq. (13-21) Eq. (13-22) Figure 13-8 © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-5 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 © 2014 John Wiley & Sons, Inc. All rights reserved. 13-10 5.3-5 Gravitational Potential Energy Newton's law of gravitation can be derived from the potential energy formula by taking the derivative 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: Kinetic Energy K = 0 and gravitational potential energy U = 0 (because r → ∞) So K + U must be at least 0 at the surface of the body: Eq. (13-28) Rockets launch eastward to take advantage of Earth's rotational speed, to reach v more easily © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-5 Gravitational Potential Energy Table 13-2 Answer: (a) increases (b) negative work © 2014 John Wiley & Sons, Inc. All rights reserved.