Circular Motion and Gravitation Centripetal Acceleration Recall linear acceleration! Δv! aavg t 3. Going around urve, at constant speed 1. Speeding up vi vi Δv a ac ac vi ac. Slowing down v velocity and acceleration vi hange of velocity can occur at constant speed - when direction changes! centripetal acceleration occurs Δv a centripetal means center seeking velocity is along a tangent (inertia) 1
Centripetal Acceleration and Force Centripetal acceleration - depends on speed (directly) - depends on radius of curve (inversely) v t r units: m s m m s Force causes acceleration -since F ma, F c m - entripetal force is not a new kind of force, but rather a role for any force to play in motion F c mv t r Examples centripetal acceleration car on curved road ball on string roller coaster loop moon in orbit electron in orbit centripetal force friction tension normal gravitation electrostatic Uniform Circular Motion Constant speed for circular motion - Distance for one revolution is circumference, πr - Time for one revolution is called period, T - Speed is distance divided by time v t πr T v t F c Centripetal acceleration and force in terms of period, T 4π r T F c 4π mr T Example: What is the centripetal acceleration at the Earth s equator? (radius of Earth is 6.37 x 10 6 m) 4π (6.37 10 6 m) (86, 400 s) 0.0337 m/s
Centripetal vs. Centrifugal Force centripetal force centrifugal force Reference frame of the observer watching the rotating can Reference frame of the observer inside the rotating can Living inside ircular rotating space station, the centrifugal force feels as real as the force of gravity, but is not an actual force! centripetal force comic Centripetal vs. Centrifugal Force In the reference frame of the observer outside the rotating wheel, the normal force on the feet is the centripetal force. In the reference frame of the observer inside the rotating wheel, the centrifugal force feels real, but is not an actual force, rather the result of a non-inertial reference frame. 3
Simulated Gravity 001 A Space Odyssey NASA drawing of a space colony Newton s Law of Gravitation (1686) Newton assumed that the laws of the universe apply to earthbound (terrestrial) and non-earth-bound (celestial) objects. Newton knew the force between two masses relates to the product of the masses. (After all, massive objects weigh more!) m m 1 m m 1 r proportional to Newton assumed that the force between masses relates inversely to the square of the distance between the masses. If force relates to an inverse square law, so does acceleration. (nd Law) 1 r Inverse Square Law image a 1 r 4
The Apple and the Moon aapple g 9.81 m/s found by Eratosthenes of Alexandria, 70 BC! Re 6.37 106 m amoon? Newton invents integral calculus to prove a central force found by Aristarchus of Samos, 40 BC! Rm 60Re Moon has centripetal acceleration as it orbits the Earth amoon v Rm 4π ( 60.33) ( 6.371 10 6 m ) 4π Rm m 0.0075 T (7.31 d 4 hr/d 3600 s/hr) s Moon s centripetal acc. vs. an apple s linear acc. amoon ra Re R amoon g e aapple rm (60Re ) 60 Re 9.81 1 m 0.0075 60 s Newton s Law of Gravitation universal gravitational constant gravitational force Fg masses Gm1m r G 6.67 10 11 Nm kg distance A gravitational constant (G, not g!) exists because force is already defined by the nd Law (where 1 newton 1 kg m/s) Newton is unable to determine the constant, although he makes a reasonable estimate. In 1798, Henry Cavendish measured the force between two masses, thereby determining the Gravitational constant, G. Henry Cavendish experiment 5
Applications of Law of Gravitation With the gravitational constant determined, Cavendish calculated the mass of the Earth, a value that had not yet been determined exactly, only approximated. weight force of gravitation mg GM E m R E Calculate the gravitational attraction between the Earth and the Moon. The Earth s mass is 5.97 x 10 4 kg and the Moon s mass is 7.36 x 10 kg. The average distance between the Earth and the Moon is 3.84 x 10 8 m. Gm 1m r 11 ( 6.67 10 ) 5.97 10 4 kg ( ) ( ) ( ) 6.37 106 M E gr E G 9.81 6.67 10 11 5.97 10 4 kg ( )( 7.36 10 kg) 1.99 10 ( 3.84 10 8 m) 0 N Honors: at what distance is the gravitational force from Earth and Moon balanced (called a LaGrange point) GmM E d GmM M (r d) 5.97 10 4 d 7.36 10 ( 3.84 10 8 d) d 3.458 108 m Newton and Satellite Motion Newton s Law of Gravitation predicts artificial satellites can orbit the earth with centripetal acceleration. Satellites have acceleration towards the center of Earth, but they also have tangential speed to keep them in orbit! Astronauts in orbit are often described as in a weightless environment. Not true! They continuously freefall around the Earth and still have weight but lack the familiar support (normal) force. satellite The International Space Station has an orbital period of about 9 minutes. What is the orbital speed of the ISS if it orbits at an altitude of 370 km above the Earth? v t πr T π (6.37 106 + 3.7 10 5 m) 9 60 s 7670 m/s (17160 mph!) 6
Applications of Law of Gravitation Newton proves Kepler s Laws of Planetary motion. Kepler s 3rd Law is easy to prove, since gravity is the centripetal force pulling a planet into circular motion. GmM R F c 4π mr T Use Kepler s 3rd Law to find the orbital period of the moon. Recall the Earth s mass and the moon s orbital radius from earlier examples. T 4π R 3 GM T orbital period (seconds) 4π (3.84 10 8 ) 3 (6.67 10 11 )(5.97 10 4 ) T.37 10 6 s (about 7.4 days) T 4π GM R3 M 4π R 3 GT orbital radius (meters) central body s mass (kg) 3rd Law Callisto, a moon of Jupiter, orbits once each 16.8 days. Its orbital radius is 1.88 x 10 9 m. Find Jupiter s mass. M 4π (1.88 10 9 ) 3 (6.67 10 11 )(16.8 4 60 60) M 1.87 10 7 kg (about 314 Earths!) Kepler s Laws of Motion 1. The paths of planets around the sun are elliptical with the sun at one focus. The planets sweep out equal areas in equal times. That is, they move faster when closer to the sun and slower when further from the sun 3. The square of the period of a planet is proportional to the cube of the distance from the sun (the mean distance, since the path is elliptical) 1st Law nd Law 3rd Law T cr 3 7