PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 11 Last Lecture Angular velocity acceleration f i t t f ti f i t Rotational Linear analogy s r v t r x 0 v0 f vf a t t angle in radians at r 2 v 2 a r Centripetal acceleration cent r center to Newton s Law of Universal Gravitation m1m2 F G 2 r 3 m G 6 67 10 11 kg s2 Always attractive Proportional to both masses Inversely proportional to separation squared Gravitation Constant Determined experimentally Henry Cavendish 1798 Light beam mirror amplify motion Weight Force of gravity on Earth GM E m Fg RE2 But we know Fg mg GM E g 2 RE Example 7 14 Often people say astronauts feel weightless because there is no gravity in space This explanation is wrong What is the acceleration due to gravity at the height of the space shuttle 350 km above the earth surface 8 81 m s2 0 90 g Example 7 14 continued Correct explanation of weightlessness Everything shuttle people bathroom scale etc also falls with same acceleration No counteracting force earth s surface Accelerating Reference Frame Same effect would be felt in falling elevator Example 7 15a Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Carol Astronaut Ted whizzes around in a circular orbit at the same radius Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward and is at a radius R but hasn t Which astronauts hit the ground yet Bob experience Alice weightlessness A All 4 B Ted and Carol C Ted Carol and Alice Ted Example 7 15b Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Carol Astronaut Ted whizzes around in a circular orbit at the same radius Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward andastronaut is at a radius R but hasn t Assume each hit thew 180 ground yet weighs lbs on Earth Bob The gravitational force acting on Ted is A w B ZERO Ted Alice Example 7 15c Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Carol Astronaut Ted whizzes around in a circular orbit at the same radius Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward andastronaut is at a radius R but hasn t Assume each hit thew 180 ground yet weighs lbs on Earth Bob The gravitational force acting on Alice is A w B ZERO Ted Alice Example 7 15d Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Astronaut Ted whizzes around in a circular Carol orbit at the same radius Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward andastronaut is at a radius R but hasn t Assume each hit thew 180 ground yet weighs lbs on Earth Bob The gravitational force acting on Carol is A w B w 3 Ted C w 9 D ZERO Alice Example 7 15e Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Astronaut Ted whizzes around in a circular Carol orbit at the same radius Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward and is at a radius R but hasn t Which astronaut s hit the ground yet undergo an acceleration g 9 8 m s2 Bob Alice A Alice B Bob and Alice C Alice and Ted D Bob Ted and Ted Alice Kepler s Laws Tycho Brahe 1546 1601 Extremely accurate astronomical observations Johannes Kepler 1571 1630 Worked for Brahe Used Brahe s data to find mathematical description of planetary motion Isaac Newton 1642 1727 Used his laws of motion and gravitation to derive Kepler s laws Kepler s Laws 1 Planets move in elliptical orbits with Sun at one of the focal points 2 Line drawn from Sun to planet sweeps out equal areas in equal times 3 The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet Kepler s First Law Planets move in elliptical orbits with the Sun at one focus Any object bound to another by an inverse square law will move in an elliptical path Second focus is empty Kepler s Second Law Line drawn from Sun to planet will sweep out equal areas in equal times Area from A to B equals Area from C to D True for any central force due to angular momentum conservation next chapter Kepler s Third Law The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet R3 Constant 2 T The constant depends on Sun s mass but is independent of the mass of the planet Derivation of Kepler s Third Law Fgrav GMm 2 macent m R 2 R 2 T R 3 GM 2 2 T 4 m M Example 7 16 Data Radius of Earth s orbit 1 0 A U Period of Jupiter s orbit 11 9 years Period of Earth s orbit 1 0 years Find Radius of Jupiter s orbit 5 2 A U Example 7 17 ven The mass of Jupiter is 1 73x1027 kg and Period of Io s orbit is 17 days nd Radius of Io s orbit r 1 85x109 m Gravitational Potential Energy PE mgh valid only near Earth s surface For arbitrary altitude Mm PE G r Zero reference level is at r Example 7 18 You wish to hurl a projectile from the surface of the Earth Re 6 38x106 m to an altitude of 20x106 m above the surface of the Earth Ignore rotation of the Earth and air resistance a 9 736 m s a What initial velocity is required b What velocity would be required in order for the projectile to reach infinitely high b 11 181 m s I e what is the escape velocity c skip How does the escape velocity c 7 906 m s compare to the velocity required for a low earth orbit Chapter 8 Rotational Equilibrium and Rotational Dynamics Wrench Demo Torque Torque is tendency of a force to rotate object about some axis Fd F is the force d is the lever arm or moment arm Units are Newton meters Door Demo Torque is vector quantity Direction determined by axis of twist Perpendicular to both r and F Clockwise torques point into paper Defined as negative Counter clockwise torques point out of paper Defined as positive r r F F Non perpendicular forces Fr sin is the angle between F and r Torque and Equilibrium Forces sum to zero no linear motion Fx 0 and Fy 0 Torques sum to zero no rotation 0 Meter Stick Demo Axis of Rotation Torques require point of reference Point can be anywhere Use same point for all torques Pick the point …
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