Gravitation
• Attractive force between two masses (m1,m2)
• r = distance between their centers.
1. What is the Gravitational Force on an object at the Surface of the Earth?
•Object has mass (m)
•Radius of the Earth: RE=6.4*106 m
•Mass of the Earth: ME=6.0*1024 kg
•Big G: G = 6.67*10-11 Nm2/kg2
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.
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)
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.
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?
For a Rigid Rotating Object
Linear velocity is proportional to rotational velocity
v ~ r ω
r = distance from axis of rotation.
“~” means “proportional to”
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?
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?
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.
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?
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 Nasa– Wiley Animation and Discussion– Wiley discussion2– Effects on Airplane Flight– Coriolis Effect on Wind Direction
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.
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?
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
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: τnet= 0
– If torque produces counterclockwise rotation it is (+)
– If torque produces clockwise rotation it is (+)
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?
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 ~ mr2
• For a small mass, a distance r from a pivot: I = mr2 Ex: Pendulum: Longer r is slower.
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 =(mr2) ×(v/r) = r × m v = r × p
• Applying an external Torque to an object or system:– Increases ω and increases L but…..
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.