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.