Physics 1501: Lecture 15 Today’s Agenda Midterm graded by next Monday (maybe …) Homework #6: Due Friday Oct. 14 @ 11:00 AM Topics Gravity (Chap.8) Review of Midterm I

Generalized Work Energy Theorem: Suppose FNET = FC + FNC (sum of conservative and non-conservative forces). The total work done is: WTOT = WC + WNC The Work Kinetic-Energy theorem says that: WTOT = K. WTOT = WC + WNC = K WNC = K - WC But WC = -U So WNC = K + U = E

Conservative Forces and Potential Energy We have defined potential energy for conservative forces DU = -W But we also now that W = FxDx Combining these two, DU = - FxDx Letting small quantities go to infinitessimals, dU = - Fxdx Or, Fx = -dU/dx

New Topic - Gravity Sir Isaac developed his laws of motion largely to explain observations that had already been made of planetary motion. Earth Sun Moon Note : Not to scale

Gravitation (Courtesy of Newton) Things Newton Knew, 1. The moon rotated about the earth with a period of ~28 days. 2. Uniform circular motion says, a = w2R 4. Acceleration due to gravity at the surface of the earth is g ~ 10 m/s2 5. RE = 6.37 x 106 m 6. REM = 3.8 x 108 m

Gravitation (Courtesy of Newton) Things Newton Figured out, 1. The same thing that causes an apple to fall from a tree to the ground is what causes the moon to circle around the earth rather than fly off into space. (i.e. the force accelerating the apple provides centripetal force for the moon) 2. Second Law, F = ma So, acceleration of the apple (g) should have some relation to the centripetal acceleration of the moon (v2/REM).

Moon rotating about the Earth : Calculate angular velocity :  = v / REM = 2 p REM / T REM = 2 p / T  = So  = 2.66 x 10-6 s-1. Now calculate the acceleration. a = 2R = 0.00272 m/s2 = .000278 g

Gravitation (Courtesy of Newton) Newton found that amoon / g = .000278 and noticed that RE2 / R2 = .000273 This inspired him to propose the Universal Law of Gravitation: |FMm |= GMm / R2 amoon g R RE G = 6.67 x 10 -11 m3 kg-1 s-2

Gravity... The magnitude of the gravitational force F12 exerted on an object having mass m1 by another object having mass m2 a distance R12 away is: The direction of F12 is attractive, and lies along the line connecting the centers of the masses. m1 m2 F12 F21 R12

Gravity... Compact objects: R12 measures distance between objects Extended objects: R12 measures distance between centers R12 R12

Gravity... Near the earth’s surface: R12 = RE Won’t change much if we stay near the earth's surface. i.e. since RE >> h, RE + h ~ RE. m Fg h M RE

Gravity... Near the earth’s surface...  =g So |Fg| = mg = ma a = g All objects accelerate with acceleration g, regardless of their mass! Where:

Example gravity problem: What is the force of gravity exerted by the earth on a typical physics student? Typical student mass m = 55kg g = 9.8 m/s2. Fg = mg = (55 kg)x(9.8 m/s2 ) Fg = 539 N Fg The force that gravity exerts on any object is called its Weight W = 539 N