#### Transcript Lecture-17-10-31 - University of Virginia

Gravity Newton’s Law of Universal Gravitation Newton’s insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit. Universal Gravitation m1m2 F12 G 2 rˆ12 force on m1 due to m2 r12 Nature is nice! m1m2 2 12 r 1.00000001 1 m 2.0000001 12 r 1.00000001 2 m The gravitational force is always attractive, and points along the line connecting the two masses: The two forces shown are an action-reaction pair. G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth). Ex. How strong is gravitational attraction between you and the person next to you? Solution: Estimate: m1 m2 60 kg ; r12 F 6.67 10 11 1m 60 60 12 3 10 7 N If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces. This is called the principle of superposition. Gravitational Attraction of Spherical Bodies Gravitational force between a point mass and a sphere*: the force is the same as if all the mass of the sphere were concentrated at its center. a consequence of 1/r2 (inverse square law) *Density of sphere must be radial symmetric Gravitational Force at the Earth’s Surface The center of the Earth is one Earth radius away, so this is the distance we use: g The acceleration of gravity decreases slowly with altitude... ...until altitude becomes comparable to the radius of the Earth. Then the decrease in the acceleration of gravity is much larger: In the Space Shuttle a) they are so far from Earth that Earth’s gravity doesn’t act any more Astronauts in the b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward space shuttle c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path float because: d) their weight is reduced in space so the force of gravity is much weaker In the Space Shuttle a) they are so far from Earth that Earth’s gravity doesn’t act any more Astronauts in the b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward space shuttle c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path float because: d) their weight is reduced in space so the force of gravity is much weaker Astronauts in the space shuttle float because they are in “free fall” around Earth, just like a satellite or the Moon. Again, it is gravity that provides the centripetal force that keeps them in circular motion. Follow-up: How weak is the value of g at an altitude of 300 km? Satellite Motion: FG and acp Consider a satellite in circular motion*: Gravitational Attraction: Necessary centripetal acceleration: • Does not depend on mass of the satellite! • larger radius = smaller velocity smaller radius = larger velocity Relationship between FG and acp will be important for many gravitational orbit problems * not all satellite orbits are circular! A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground. These satellites are used for communications and weather forecasting. How high are they? RE = 6378 km ME = 5.87 x 1024 kg Averting Disaster a) it’s in Earth’s gravitational field b) the net force on it is zero The Moon does not c) it is beyond the main pull of Earth’s gravity crash into Earth d) it’s being pulled by the Sun as well as by Earth because: e) none of the above Averting Disaster a) it’s in Earth’s gravitational field b) the net force on it is zero The Moon does not c) it is beyond the main pull of Earth’s gravity crash into Earth d) it’s being pulled by the Sun as well as by Earth because: e) none of the above The Moon does not crash into Earth because of its high speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity providing the centripetal force. Follow-up: What happens to a satellite orbiting Earth as it slows? Two Satellites Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? a) 1/8 b) ¼ c) ½ d) it’s the same e) 2 Two Satellites Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? Using the Law of Gravitation: we find that the ratio is . a) 1/8 b) ¼ c) ½ d) it’s the same e) 2 Note the 1/R2 factor Gravitational Potential Energy Gravitational potential energy of an object of mass m a distance r from the Earth’s center: (U =0 at r -> infinity) Very close to the Earth’s surface, the gravitational potential increases linearly with altitude: Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign. Energy Conservation Total mechanical energy of an object of mass m a distance r from the center of the Earth: This confirms what we already know – as an object approaches the Earth, it moves faster and faster. Escape Speed Escape speed: the initial upward speed a projectile must have in order to escape from the Earth’s gravity from total energy: If initial velocity = ve, then velocity at large distance goes to zero. If initial velocity is larger than ve, then there is non-zero total energy, and the kinetic energy is non-zero when the body has left the potential well Maximum height vs. Launch speed Speed of a projectile as it leaves the Earth, for various launch speeds Kepler’s Laws of Orbital Motion Johannes Kepler made detailed studies of the apparent motions of the planets over many years, and was able to formulate three empirical laws 1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse. Elliptical orbits are stable under inverse-square force law. You already know about circular motion... circular motion is just a special case of elliptical motion Only force is central gravitational attraction - but for elliptical orbits this has both radial and tangential components Kepler’s Laws of Orbital Motion 2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time. Apogee r v Δt Perigee This is equivalent to conservation of angular momentum L mrp v p mra va rp v p t ra va t Ap Aa Kepler’s Laws of Orbital Motion 3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power. This can be shown to be a consequence of the inverse square form of the gravitational force. Orbital Maneuvers Which stable circular orbit has the higher speed? How does one move from the larger orbit to the smaller orbit? Binary systems If neither body is “infinite” mass, one should consider the center of mass of the orbital motion 2 r1 2 r2 T1 T2 v1 v2 CM : m1r1 m2 r2 Four equations in four unknowns Guess My Weight If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value Guess My Weight If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value The weight that a scale reads is the normal force exerted by the floor (or the scale). At the equator, you are in circular motion, so there must be a net inward force toward Earth’s center. This means that the normal force must be slightly less than mg. So the scale would register something less than your actual weight. Earth and Moon I a) the Earth pulls harder on the Moon Which is stronger, b) the Moon pulls harder on the Earth Earth’s pull on the c) they pull on each other equally Moon, or the d) there is no force between the Earth and the Moon Moon’s pull on Earth? e) it depends upon where the Moon is in its orbit at that time Earth and Moon I a) the Earth pulls harder on the Moon Which is stronger, b) the Moon pulls harder on the Earth Earth’s pull on the c) they pull on each other equally Moon, or the d) there is no force between the Earth and the Moon Moon’s pull on Earth? e) it depends upon where the Moon is in its orbit at that time By Newton’s Third Law, the forces are equal and opposite. Principle of Equivalence You’re standing at rest on a scale. The display shows that it is exerting a force on you of F = mg = 60 * 9.81 = 589 N You’re now at rest in outer space, far from any star or planet. You fire your thruster which exerts a force of 150 N on you (m = 60 kg). At what rate do you accelerate? F = ma so a = F/m = 150/60 = 2.5 m/s2 m and m represent two different concepts. Why can we treat them interchangeably? General Relativity More complete theory of gravity. Replaces “spooky” action-at-a-distance with curvature of space, an idea that is just about as “spooky” as action-at-a-distance. Required to make GPS work! Black holes If an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light – light itself will not be able to escape the surface. This is a black hole. The light itself has mass (in the mass/energy relationship of Einstein), or spacetime itself is curved Gravity and light Light will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found.