Circular Motion & Gravity

Circular Motion Objects travel in a circle Rotate about an axis of rotation Tangential speed (vt) describes the rate at which the object moves around the circle Direction is tangential to the circular path

vt depends upon radius Given the object is rigid, e.g. a CD Object B must travel a greater distance to keep up with object A SB > SA But ΔtB = ΔtA Therefore, vB > vA

Comparison of Translational Motion & Uniform Circular Motion UCM = motion of an object traveling in a circle at a constant speed, vt Type of Motion Translational Uniform Circular Displacement Linear Δx Circumference 2πr Time Δt Period T Formula vavg = Δx/Δt vt = 2πr/T

Uniform Circular Motion Tangential speed vt is constant Because direction is changing, there is acceleration Centripetal acceleration

Centripetal Acceleration a = Δv/Δt When subtracting vectors, reverse the direction of vi Centripetal acceleration is, therefore, directed toward the center (axis of rotation) when θ is small

Centripetal Acceleration Centripetal means “center seeking” and is always directed toward the center Due to a change in direction of vt Phet simulation

Tangential Acceleration Tangential acceleration occurs when there is a change in tangential speed. For example, if a car is speeding up as it goes around a curve, It has tangential acceleration and Centripetal acceleration

Centripetal Force Because Fc acts at right angles to the object’s circular motion, it changes the direction of the objects velocity

Centripetal Force Is the cause of centripetal acceleration It is directed toward the axis of rotation It is the net force acting on an object in uniform circular motion, i.e. it is the cause of circular motion Centrifugal force is a misunderstanding of inertia

Centripetal Force & Newton’s 2nd Law

Centripetal Force Is just the name of any net force acting on an object in uniform circular motion Fc could take any form…. It could be frictional force, tension force, gravitational force, etc.

Motion of a Car Around a Curve On a horizontal turn, the centripetal force is friction

Circular Motion About a Banked Curve

Conical Pendulum

Vertical Circular Motion

Centifugal Force? If Fc is insufficient to maintain circular motion, the object will leave it’s circular path due to its own inertia, not because some force is pulling it away from the axis of rotation Thus, inertia is often mistaken for “centrifugal force”

Gravity

Gravitational Force Force of attraction between two masses Attractive only One of four fundamental forces Very weak (the weakest) When one object orbits another, gravitational force is a centripetal force

Newton’s Law of Universal Gravitation Gravitational force is… directly proportional to the product of the masses of the two bodies inversely proportional to the square of the distance between the centers of the two masses If the objects are large (e.g. planets, moons) then the radii would be included in r

Gravitational Force Exists Between Any Two Masses

Newton’s Cannon http://spaceplace.nasa.gov/en/kids/orbits1.shtml http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/newt/newtmtn.html

Importance of Gravitational Force Keeps you from floating away into space Gravitational force keeps the Moon and planets in orbit Keeps earth in orbit around sun Causes ocean tides

Black Holes: Extreme Gravity Extreme density Escape velocity > speed of light Detect by effects on surrounding matter

Gravitational Field Strength Increases as distance from mass center decreases Because gravitational field strength varies, weight varies with location

Gravitational Field Strength Describes the amount of gravitational force per unit mass at any given point Equals free-fall acceleration

Weight Changes with Location Because gravitational field strength varies, ag varies (acceleration of gravity). Since w = mag, weight must vary as ag varies Fg is an example of an inverse square law

7.3 Motion in Space Astronomer Planets orbit… Type of orbit Ptolomey Earth Epicycles Copernicus Sun Circular Kepler Elliptical

Kepler’s Laws of Planetary Motion 1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus. 2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times. 3. The Law of Periods: The square of the period of any planet is proportional to the cube of the average distance from the sun,

Kepler’s 1st and 2nd Laws Kepler's Law Simulation

Kepler’s 3rd Law Describes Orbital Period

Actual and Apparent Weight A bathroom scale records the normal force of scale acting on your body Step on the scale … the normal force equals your weight

Actual and Apparent Weight Now try this Step on the scale and have someone press down on your shoulders Predict and explain the result Step on the scale and have someone lift you slightly

Actual and Apparent Weight How does this relate to your experiences in an elevator? What would the scale read if, in an elevator, it descended with an acceleration of g?

Weight and Apparent Weightlessness

Torque a quantity that measures the ability of a force to rotate an object about an axis is not a force “rotating ability” the product of force and “lever arm” τ = F · d sinθ Lever arm (d) is distance perpendicular to direction of force to axis of rotation

Torque Sign (+) is counterclockwise (-) is clockwise Net Torque and when 2 or more forces act to rotate the same object, τnet = Στ τnet = τ1 + τ2 = F1d1 + F2d2

Torque Equilibrium Torque Equilibrium: Στ = 0

Torque Equilibrium The torque due to the boy is equal and opposite to that of the girl.

Net Torque

Center of Mass (COM) Point mass vs. extended object The point in a body at which all the mass can be considered to be concentrated when analyzing translational motion Unless an object rotates about a fixed point, (e.g. a hinge)… The point about which a mass or system of mass rotates during rotational motion

Center of Mass The extended object rotates about the CoM CoM follows the expected parabolic path

Center of Mass May not lie within the mass or system of masses

Simple Machines All machines are combinations of simple machines Purpose is to change magnitude or direction of an input force Mechanical Advantage describes the ratio of output and input forces

Ideal vs. Actual Mechanical Advantage Ideal MA MA if there were no friction Actual MA MA that takes friction into account

Machines and Work Machines do not change the amount of work Machines make work easier

Efficiency A measure of how well a machine works A less efficient machine produces less output per input A less efficient machine requires more input to get the same output