Chapter 7: Rotational Motion and the Law of Gravity

Objectives Be able to distinguish between a rotation and a revolution. Be able to distinguish between frequency and period. Be able to calculate tangential speed. Understand the concept of a centripetal acceleration.

Circular Motion revolution: object moving in a circular (or elliptical) path around an axis point rotation: object spinning around its axis period (T): time required for one complete cycle frequency (f): number of cycles per unit time hertz (hz): cycles/second

Uniform Circular Motion r What is the tangential speed (in m/s) of a palm tree on the equator? What is it for a Ponderosa pine in Polson? R earth = 6380 km tangential speed v

Centripetal Acceleration vivi vfvf – v i vfvf vv  v = v f – v i = v f + ( – v i )   d r r centripetal acceleration (a c ): a center-seeking change in velocity

Objectives Understand the concept of centripetal force. Be able to identify or give examples of forces acting as centripetal forces. Be able to solve centripetal force problems.

Centripetal Force centripetal force: any center-seeking force that results in circular motion v FcFc v F c is unbalanced: it causes a change in velocity. F c and v are perpendicular: no net work is done by F c so the KE (and speed) remains constant. v

Centripetal Forces Forces acting as centripetal forces: hammer throw motorcycle cage car turning on road moon orbiting earth e- orbiting nucleus (tension) (normal force) (friction) (gravity) (electromagnetic)

Centripetal Force At what maximum speed that a car make a turn of radius 12.3 meters if the coefficient of friction between the tires and the road is 1.94? What is the magnitude of the F c if the mass of the car is 1383 kg?

Twirl-O Problem On the popular Twirl-O, a passenger is held inside a large spinning cylinder. If the radius of the ride is 4.0 m, with what rotational period must the ride rotate in order for the passenger to not fall? The  s between the wall and the passenger is 0.60.

Objectives Understand how Newton’s third law relates to the concept of a “centrifugal” force. Explain how simulated gravity could be achieved on a spacecraft. Be able to solve simulated gravity problems.

“Centrifugal Force” The force equal-and-opposite to a centripetal force is known as a centrifugal force. can on bug (F C is F N ) bug on can (~ F W ) From the bug’s point of view, it feels like the normal force exerted upward by the ground.

Simulated Gravity F N = F C A simulated gravity can be produced by adjusting r and T. If r = 95 m, what does T need to be ? simulated weight (F W ) = F C = m · a C = m · g FWFW

Centripetal Force Extra-Credit At what minimum height will a Hot Wheels car make it around the loop-the-loop without falling? Hint: at the top of the loop the only force acting is F w (= F c ) h = ? find the equation r

Objectives Explain the factors that affect the force of gravity between two objects. Understand the concept of the universal gravitational constant, G. Be able to solve gravitation problems.

Universal Gravitation 1660s: Isaac Newton first realized that gravity keeps the moon in orbit around the earth (F G = F c ) gravity: an attractive force between two masses What factors affect the strength of the force? F G ~ m 1 · m 2 F G ~ 1 / r 2

Universal Gravitational Constant “Big G” was first measured by Cavendish in 1797 G = 6.67 x Nm 2 /kg 2

Mass of the Earth The earth has a radius of 6380 km. If a 1.0 kg mass weighs 9.81 N, what is the mass of the earth?

Universal Gravitation Problem How much gravitational force does the sun (150 million km away = 1 AU) exert on a 65 kg person? M sun = 2.00 x kg.

Objectives Be familiar with Kepler’s third law. Understand how his law can be derived. Perform calculations related to the law.

Ptolemy, Aristotle, and the Catholic Church: geocentric model Aristarchus, Aryabhata, Copernicus: heliocentric model Galileo: moons orbit Jupiter Kepler develops 3 laws of orbital motion A Brief History of Astromony

Kepler’s Third Law Johannes Kepler (1619): r 3 /T 2 = 1 for all planets in our solar system r = # AU and T = # yrs Planet T (yrs) r (AU) T2T2 r3r3 Mercury Venus Earth1.00 Mars Jupiter Saturn

Kepler’s 3 rd Law Proof For any pair of satellites orbiting the same star/planet. What is the orbital period of Jupiter if r = 5.2 AU?

Objectives Be able to explain why the same side of the moon always faces the earth. Be able to explain how the force of gravity relates to ocean tides. Understand the concept of a black hole.

The Moon’s Orbit center of mass ≠ center of gravity as it orbits, the same side of moon must face the earth rotational T = orbital T

The Tides F G ~ 1/r 2, so F A > F B > F C, tidal bulges form (not to scale!) two high, two low tides daily (polar view)

Tidal Forces F g of the sun is 180 X greater than the moon but F g from moon has 2X greater difference: SUN on EARTH Near side: x10 22 N Far side: x10 22 N Difference: x10 22 N MOON on EARTH Near side: x10 22 N Far side: x10 22 N Difference: x10 22 NTwice as much!

Tides full moon quarter moons new moon (most extreme)

Extreme Tides The tides are most extreme (higher and lower) at higher latitudes 15 m at Bay of Fundy, Nova Scotia