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Circular Motion - PowerPoint PPT Presentation


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Circular Motion. Definitions. Circular motion: when an object moves in a two-dimensional circular path Spin: object rotates about an axis that pass through the object itself. Definitions. Orbital motion: object circles an axis that does not pass through the object itself.

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slide2

Definitions

  • Circular motion: when an object moves in a two-dimensional circular path
  • Spin: object rotates about an axis that pass through the object itself
slide3

Definitions

  • Orbital motion: object circles an axis that does not pass through the object itself
slide4

Circle Terminology

  • Radius
  • Diameter
  • Chord
  • Tangent
  • Arc
slide5

Establishing Position

  • The simplest coordinate system to use for circular motion puts the tails of position vectors at the center of the circular motion.
slide6

Polar Coordinates

(r, θ)

  • magnitude of r = radius of circular path
  • θ = angle of rotation
  • θ is measured in radians
slide7

Radian Measure

  • Definition of a radian:

One radian is equal to the central angle of a circle that subtends an arc of the circle’s circumference whose length is equal to the length of the radius of the circle.

slide8

Radian Measure

  • There are exactly 2π radians in one complete circle.
  • Unit analysis:
    • 180° = π radians
slide9

Establishing Position

  • In circular motion, change of position is measured in angular units.
  • θ can have a positive or negative value.
slide10

Δθ

ω =

Δt

Speed and Velocity

  • ω represents the time-rate of change of angular position; this is also called the angular speed.
  • By definition:
slide11

Δθ

ω =

Δt

Speed and Velocity

  • ω is a scalar quantity.
  • It is commonly expressed as number of rotations or revolutions per unit of time.
    • Ex. “rpm”
slide12

Δθ

ω =

Δt

Speed and Velocity

  • If angular speed is constant, then the rotating object experiences uniform circular motion.
slide13

rad

or

s-1

s

Speed and Velocity

  • In the SI, the units are radians per second.
  • Written as:
slide14

Speed and Velocity

  • The velocity vector of a particle in circular motion is tangent to the circular path.
  • This velocity is called tangential velocity.
slide15

Speed and Velocity

  • The magnitude of the tangential velocity is called the tangential speed, vt.

vt = |vt|

slide16

l

vt =

Δt

Speed and Velocity

  • Another formula for tangential speed is:
  • arclengthl = r × Δθ
slide17

rΔθ

vt =

Δt

Speed and Velocity

  • average tangential speed:
slide18

Δv

a =

Δt

vt2

a =

r

Acceleration

  • Linear motion:
  • Circular motion:
slide19

Acceleration

  • The instantaneous acceleration vector always points toward the center of the circular path.
  • This is called centripetal acceleration.
slide20

vt2

ac =

m/s²

r

Acceleration

  • The magnitude of centripetal acceleration is:
  • For all circular motion at constant radius and speed
slide21

Acceleration

  • Another formula for centripetal acceleration:

ac = -rω2

slide22

Angular Velocity

  • Uniform angular velocity (ω) implies that the rate and direction of angular speed are constant.
slide23

Angular Velocity

  • Right-hand rule of circular motion:
slide24

Angular Velocity

  • Nonuniform circular motion is common in the real world.
  • Its properties are similar to uniform circular motion, but the mathematics are more challenging.
slide25

ω2 – ω1

Δω

α =

=

Δt

Δt

Angular Acceleration

  • change in angular velocity
  • notation: α
  • average angular acceleration:
slide26

ω2 – ω1

Δω

α =

=

Δt

Δt

Angular Acceleration

  • units are rad/s², or s-2
  • direction is parallel to the rotational axis
slide27

Tangential Acceleration

  • defined as the time-rate of change of the magnitude of tangential velocity
slide28

Δvt

at =

=αr

Δt

Tangential Acceleration

  • average tangential acceleration:
slide29

Tangential Acceleration

  • instantaneous tangential acceleration:

at =αr

Don’t be too concerned about the calculus involved here...

slide30

Tangential Acceleration

  • Instantaneous tangential acceleration is tangent to the circular path at the object’s position.
slide31

Tangential Acceleration

  • If tangential speed is increasing, then tangential acceleration is in the same direction as rotation.
slide32

Tangential Acceleration

  • If tangential speed is decreasing, then tangential acceleration points in the opposite direction of rotation.
slide33

Equations of Circular Motion

  • note the substitutions here:
slide35

Centripetal Force

  • in circular motion, the unbalanced force sum that produces centripetal acceleration
  • abbreviated Fc
slide36

mvt²

Fc =

r

Centripetal Force

  • to calculate the magnitude of Fc:
slide37

Centripetal Force

  • Centipetal force can be exerted through:
    • tension
    • gravity
slide38

Torque

  • the product of a force and the force’s position vector
  • abbreviated: τ
  • magnitude calculated by the formula τ = rF sin θ
slide39

Torque

τ = rF sin θ

  • r = magnitude of position vector from center to where force is applied
  • F = magnitude of applied force
slide40

Torque

τ = rF sin θ

  • θ = smallest angle between vectors r and F when they are positioned tail-to-tail
  • r sin θ is called the moment arm (l) of a torque
slide41

Torque

  • Maximum torque is obtained when the force is perpendicular to the position vector.
  • Angular acceleration is produced by unbalanced torques.
slide42

Torque

  • Zero net torques is called rotational equilibrium.
  • Στ= 0 N·m
slide43

F1

l2

=

F2

l1

Torque

  • Law of Moments: l1F1 = l2F2
  • Rearranged:
slide45

The Ideas

  • Geocentric: The earth is the center of the universe
  • Heliocentric: The sun is the center of the universe
  • Some observations did not conform to the geocentric view.
slide46

The Ideas

  • Ptolemy developed a theory that involved epicycles in deferent orbits.
  • For centuries, the geocentric view prevailed.
slide47

The Ideas

  • Copernicus concluded the geocentric theory was faulty.
  • His heliocentric theory was simpler.
slide48

The Ideas

  • Tycho Brahe disagreed with both Ptolemy and Copernicus.
  • He hired Johannes Kepler to interpret his observations.
slide49

Kepler’s Laws

  • Kepler’s 1st Law states that each planet’s orbit is an ellipse with the sun at one focus.
slide50

Kepler’s Laws

  • Kepler’s 2nd Law states that the position vector of a planet travels through equal areas in equal times if the vector is drawn from the sun.
slide51

Kepler’s Laws

Kepler’s 2nd Law

slide52

= K

Kepler’s Laws

  • Kepler’s 3rd Law relates the size of each planet’s orbit to the time it takes to complete one orbit.
slide53

= K

Kepler’s Laws

  • R = length of semi-major axis
  • T = time to complete one orbit (period)
slide54

= K

Kepler’s Laws

  • R is measured in ua (astronomical units), the mean distance from earth to the sun
slide55

= K

Kepler’s Laws

  • T is measured in years
slide56

Newton

  • determined that gravity controls the motions of heavenly bodies
  • determined that the gravitational force between two objects depends on distance and mass
slide57

Mm

Fg = G

Newton

  • derived the Law of Universal Gravitation:
  • G is called the universal gravitational constant
  • Newton did not calculate G.
slide58

Mm

Fg = G

Law of Universal Gravitation

  • It predicts the gravitational force, but does not explain how it exists or why it works.
slide59

Mm

Fg = G

Law of Universal Gravitation

  • It is valid only for “point-like masses.”
  • Gravity is always an attractive force.
slide60

Mm

Fg = G

Law of Universal Gravitation

  • Cavendish eventually determined the value of G through experimentation with a torsion balance.
slide61

Mm

Fg = G

Law of Universal Gravitation

  • G ~ 6.674 × 10-11 N·m²/kg²
  • Cavendish could then calculate the mass and density of planet Earth.