1Week 13: Chapter 13Universal GravitationNewton’s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the distance between them G is the universal gravitational constant and equals 6.673 x 10-11Nm2/ kg2122gmmFGrClicker QuestionA planet has two moons of equal mass. Moon 1 is in a circular orbit of radius r. Moon 2 is in a circular orbit of radius 2r. What is the magnitude of gravitational force exerted by the planet on Moon 2 comparing that on Moon 1?A. 4 time as large as that on Moon 1B. Twice as largeC. EqualD. ½E. 1/4Finding the Value of G In 1789 Henry Cavendish measured G The two masses are fixed at the ends of a light horizontal rod Two large masses were placed near the small ones The angle of rotation was measured Law of Gravitation, cont This is an example of an inverse square law The magnitude of the force varies as the inverse square of the separation of the particles The law can also be expressed in vector form The negative sign indicates an attractive force1212 122ˆmmGrFrNotation is the force exerted by particle 1 on particle 2 The negative sign in the vector form of the equation indicates that particle 2 is attracted toward particle 1 is the force exerted by particle 2 on particle 112F21F2More About Forces The forces form a Newton’s Third Law action-reaction pair Gravitation is a field force that always exists between two particles, regardless of the medium between them The force decreases rapidly as distance increases A consequence of the inverse square law12 21FFGravitational Force Due to a Distribution of Mass The gravitational force exerted by a finite-size, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distribution were concentrated at the center The force exerted by the Earth on a particle of mass m near the surface of the Earth is 2EgEMmFGRG vs. g Always distinguish between G and g G is the universal gravitational constant It is the same everywhere g is the acceleration due to gravity g = 9.80 m/s2at the surface of the Earth g will vary by locationFinding g from G The magnitude of the force acting on an object of mass m in freefall near the Earth’s surface is mg This can be set equal to the force of universal gravitation acting on the object22EEEEMmmg GRMgGRg Above the Earth’s Surface If an object is some distance h above the Earth’s surface, r becomes RE+ h This shows that g decreases with increasing altitude As r , the weight of the object approaches zero2EEGMgRhVariation of g with Height3Johannes Kepler 1571 – 1630 German astronomer Best known for developing laws of planetary motion Based on the observations of TychoBraheKepler’s Laws Kepler’s First Law All planets move in elliptical orbits with the Sun at one focus Kepler’s Second Law The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals Kepler’s Third Law The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbitNotes About Ellipses F1and F2are each a focus of the ellipse They are located a distance c from the center The sum of r1and r2remains constant Use the active figure to vary the values defining the ellipse The longest distance through the center is the major axis a is the semimajor axisNotes About Ellipses, cont The shortest distance through the center is the minor axis b is the semiminor axis The eccentricity of the ellipse is defined as e = c /a For a circle, e = 0 The range of values of the eccentricity for ellipses is 0 < e < 1 The higher the value of e, the longer and thinner the ellipseNotes About Ellipses, Planet Orbits The Sun is at one focus Nothing is located at the other focus Aphelion is the point farthest away from the Sun The distance for aphelion is a + c For an orbit around the Earth, this point is called the apogee Perihelion is the point nearest the Sun The distance for perihelion is a – c For an orbit around the Earth, this point is called the perigeeKepler’s First Law A circular orbit is a special case of the general elliptical orbits Is a direct result of the inverse square nature of the gravitational force Elliptical (and circular) orbits are allowed for boundobjects A bound object repeatedly orbits the center An unbound object would pass by and not return These objects could have paths that are parabolas (e = 1) and hyperbolas (e > 1)4Orbit Examples Mercury has the highest eccentricity of any planet (a) eMercury= 0.21 Halley’s comet has an orbit with high eccentricity (b) eHalley’s comet= 0.97 Remember nothing physical is located at the second focus The small blue dotKepler’s Second Law Is a consequence of conservation of angular momentum The force produces no torque, so angular momentum is conservedUse the active figure to vary the value of e and observe the orbitP= x = M x = constantLrp rvKepler’s Second Law, cont. Geometrically, in a time dt, the radius vector rsweeps out the area dA, which is half the area of the parallelogram  Its displacement is given by| x d|rrd = dtrvKepler’s Second Law, final Mathematically, we can say The radius vector from the Sun to any planet sweeps out equal areas in equal times The law applies to any central force, whether inverse-square or notconstant2pdA Ldt MKepler’s Third Law Can be predicted from the inverse square law Start by assuming a circular orbit The gravitational force supplies a centripetal force Ksis a constant2Sun Planet Planet22233Sun24SGM M M vrrrvTTrKrGMKepler’s Third Law, cont This can be extended to an elliptical orbit Replace r with a Remember a is the semimajor axis Ksis independent of the mass of the planet, and so is valid for any planet2233Sun4STaKaGM5Kepler’s Third Law, final If an object is orbiting another object, the value of K will depend on the object being orbited For example, for the Moon around the Earth, KSunis replaced