Gavitation AP/Honos Physics 1 M. Velazquez
Newton s Law of Gavitation Newton was the fist to make the connection between objects falling on Eath and the motion of the planets To illustate this connection in Pincipia Mathematica Newton descibed the act of thowing a pojectile hoizontally fom the top of a mountain Given a geat enough velocity, the pojectile would cicle the Eath completely befoe etuning to its stating point
Newton s Law of Gavitation To descibe how the gavitational foce acts between two masses, Newton deived the following equation: F G = G m 1m 2 2 m 1, m 2 = point masses (kg) = distance between the centes of mass (m) G = 6.67 10 11 N m 2 kg 2 F G F G m 1 m 2
12.4 m Newton s Law of Gavitation a) Calculate the gavitational foce between each of the thee pais of masses below b) Calculate the esultant gavitational foce acting on mass m 1 22.5 m m 1 152 kg m 3 425 kg m 2 368 kg
Gavitational Acceleation fo Spheical Bodies a = G M P R P 2 a = acceleation due to gavity M P = mass of the sphee o planet R P = adius of the sphee o planet G = 6.67 10 11 N m 2 kg 2 EXTRA CREDIT: Use the fomula above to calculate the acceleation due to gavity fo the following bodies Eath s Moon adius = 1,738 km mass = 7.35 10 22 kg Mas adius = 3,389 km mass = 6.39 10 23 kg Jupite adius = 69,912 km mass = 1.90 10 27 kg Uanus adius = 25,362 km mass = 8.68 10 25 kg
Exit Ticket, Pat 1 You will be estimating the gavitational foce of a fist bump. Fist, choose a patne. Then, stand face-to-face and fist bump. Use a mete-stick to measue the distance fom navel to navel (add an exta 8 cm fo each male and 7 cm fo each female in you pai). Then, convet each of you weights fom pounds to kilogams (1 lb = 0.4536 kg). Once this is done, you can use Newton s gavitation fomula: F G = G m 1m 2 2
Gavitational Potential Enegy Fo objects at the suface of the Eath (whee g is constant), potential enegy can be calculated using U = mgh. But as ou distance fom the suface incease, the value of g deceases. In addition, we want the potential enegy fom gavitation to become smalle the close we get to Eath s suface. It can be shown that the gavitational potential enegy of a system whee a mass m 1 is a distance fom the cente of the Eath (m 2 ) is: U = G m 1m 2 This potential enegy is zeo when the masses m 1 and m 2 ae infinitely sepaated fom each othe.
Consevation of Enegy By applying the laws of Consevation of Mechanical Enegy, we can descibe the total enegy of an object with mass m, speed v, and a distance away fom the cente of the Eath in the following way: E = K + U E = 1 2 mv2 GmM E These enegy equations can be used to calculate the speed and tajectoy of any moving objects in space, fom asteoids to comets to satellites.
Escape and Obit Velocities Suppose you wanted to send an object into space with enough velocity that it would eventually escape Eath s gavitational pull this efes to escape velocity. O pehaps you wanted to send the object into a pefectly cicula obit aound the Eath o obital velocity. These efe to specific magnitudes fo the speed of the object, and we can compute them in diffeent ways.
Escape and Obit Velocities The speed of the object will affect the type of motion o athe, which conic section can be used to epesent the motion.
Escape and Obit Velocities
Escape and Obital Velocities Fo an object to have a pefectly cicula obit aound Eath, the gavitational foce must be exactly equal to the centipetal foce. We use this fact to calculate obital speed (v o o v c ) Centipetal Foce mv o 2 = G mm E 2 Gavitational Foce v o 2 = G M E Obital Velocity v o = GM E 7,909 m s
Escape and Obital Velocities Fo an object to leave Eath s gavitational field, it must fist achieve escape velocity. We calculate this by assuming the object will stat with both kinetic and gavitational enegy, and end up with neithe. E 1 = E 2 K + U G = 0 1 2 m ov 2 e Gm om E v e 2 = 2 GM E = 0 Escape Velocity v e = 2 GM E 11,185 m s
Escape and Obital Velocities The only diffeence between escape velocity and obital velocity is the squae oot of 2. v e = v o 2
Exit Ticket, Pat 2: Escape and Obital Velocity Find the escape and obital velocity on Eath s moon. adius = 1,738 km mass = 7.35 10 22 kg v e = 2 GM m v o = GM m Poblem Set: Gavitation Pg. 378-379, #3, 6, 9, 12, 15, 18, 36, 39, 42, 45
Keple s Laws of Obital Motion Keple s Fist Law Planets follow elliptical obits, with the Sun at one focus of the ellipse.
Dawing Ellipses
Keple s Laws of Obital Motion Keple s Second Law As a planet moves in its obit, it sweeps out an equal amount of aea in an equal amount of time.
Keple s Laws of Obital Motion Keple s Thid Law The squae of the peiod, T 2, of any planet is popotional to the cube of the semimajo axis (mean distance), 3, of its obit. O: T 2 3 T 2 = constant 3 T 2 = 4π2 GM 3 Read you book (Chapte 12) fo poof of this constant