1. Gravitation Attractive force between two masses (m 1,m 2 ) r = distance between their centers.

2. 1. What is the Gravitational Force on an object at the Surface of the Earth? Object has mass (m) Radius of the Earth: R E =6.4*10 6 m Mass of the Earth: M E =6.0*10 24 kg Big G: G = 6.67*10 -11 Nm 2 /kg 2

3. 2. Kepler’s Laws of Planetary Motion (consequence of Newton’s Laws) 1.Planets move in elliptical orbits with the Sun at one focus. 2.A line from the sun to a planet sweeps out equal areas in a given period of time. 3.The larger the radius of a planet’s orbit, the larger the period (year) of the planet. Kepler animation *You are not responsible for laws 2 and 3.

4. 3. Planet in Elliptical Orbit about the Sun. Where does a planet have greatest kinetic energy? Where does a planet have greatest potential energy? Where is the planet moving fastest? (HINT: Total Energy is Conserved)

5. Circular Motion Circular motion occurs about an axis –Rotation: object spins about an internal axis Earth rotates about its polar axis once a day. –Revolution: object moves about an external axis Earth revolves once about the sun each year.

6. Speeds involved in Circular Motion Linear speed (v): distance covered per unit time by a point on the object. (m/s) (Also called tangential speed) Rotational speed: amount of angle swept out per unit time. (revolutions per minute) On a rigid rotating object: –Is rotational speed everywhere the same? –Is linear speed everywhere the same?

7. For a Rigid Rotating Object Linear velocity is proportional to rotational velocity v ~ r ω r = distance from axis of rotation. “~” means “proportional to”

8. Suppose you get a flat tire while driving You put on a “toy spare” tire that came with the car. The toy spare tire is smaller than your other tires. How does this affect your driving?

9. What causes Circular Motion? Suppose I swing an object at constant speed in a circle. (“uniform circular motion”) Does the object have constant velocity? Does the object accelerate? Does the object feel a force? If so, what causes the force? In what direction is the force? How does the object move if I cut the rope?

10. Centripetal Force To keep in object in circular motion, we must constantly exert a force –Perpendicular to the object’s velocity –Directed inward toward the center of the orbit. This direction is called the centripetal direction. The force is called the centripetal force. Examples of centripetal force: –Tension in string, keeping ball in orbit. –Sun pulling on Earth, keeping it in orbit. –Earth pulling on Moon, keeping it in orbit.

11. Fictitious Forces: Centrifugal Force When a car turns left (inward, “centripetal”), why do you feel pushed to the right (outward, “centrifugal”)? Do you feel like you can stand on the wall of the car? Can we simulate gravity by standing on the wall of a rotating cylinder?

12. Fictitious Force: Coriolis Effect A force we see due to the rotation of the Earth and how things on Earth move. –Foucault Pendulum: proved that Earth Rotates –Affects projectile motion –Affects flight plans of pilots. Coriolis Effect in Action: –Movie From NasaMovie From Nasa –Wiley Animation and DiscussionWiley Animation and Discussion –Wiley discussion2Wiley discussion2 –Effects on Airplane FlightEffects on Airplane Flight –Coriolis Effect on Wind DirectionCoriolis Effect on Wind Direction

13. Rotational Mechanics: Torque Torque causes things to rotate about an axis (just as ________ causes things to ________________). Types of Torque we see everyday: –Torsion or twisting: Torque applied about the length of an object. –Bending: Torque applied about an axis perpendicular to the object’s length.

14. What makes up a Torque? Do we need a force? Do we need a net force? Do we need anything else? OR (put another way ) Can I get a torque with no force? Can I get a torque with no net force? Can I apply a force to an object and get no torque?

15. Requirements for a Torque A Force A Lever Arm equals distance from the axis of rotation. The amount of torque (τ) we get depends on the Amount of force we apply (F ┴ ) Length of lever arm (r) τ = r * F ┴ = Torque about the pivot point

16. A Balance of Torques? Can we apply a number of torques and have no rotation? Can torques cancel out? A net torque causes rotation. Rotational Equilibrium: τ net = 0 –If torque produces counterclockwise rotation it is (+) –If torque produces clockwise rotation it is (+)

17. Example A meter stick is on a pivot at its center. –If a 1 kg mass is placed.8 meters to the left of the pivot, what is the torque produced about the pivot? –Can I place a.2 kg mass to the right of the pivot and balance the 1 kg mass? If so, where should the.2 kg mass be placed? –After placing the.2 kg mass, what is the force exerted by the pivot on the meter stick? What torque does this force produce?

18. Rotational Inertia (I) Resistance of an object to being rotated. It is more difficult to rotate an object about a point if more of its mass is further from that point. It is easier to rotate an object whose mass is closer in to the point of rotation. I ~ mr 2 For a small mass, a distance r from a pivot: I = mr 2 Ex: Pendulum: Longer r is slower.

19. Angular Momentum Measure of the resistance of an object to having its rotational motion changed. L = I × ω = I –L = angular momentum –I = rotational inertia –ω = rotational velocity (recall: v = rω) For a mass moving in a circle at speed v: L= I × ω =(mr 2 ) ×(v/r) = r × m v = r × p Applying an external Torque to an object or system: –Increases ω and increases L but…..

20. Conservation of Angular momentum If the net external torque on a system is zero, the angular momentum of the system is constant. –Example: L = mvr ; if r decreases with no net torque, then v increases. –Figure skaters spin faster when they pull in their arms. –Swimmers curl their bodies inward to turn faster after swimming a length. Angular momentum is a vector. If I reverse the direction of spinning, the direction of L reverses.