1 / 61

Circular Motion

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.

jenn
Download Presentation

Circular Motion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Circular Motion

  2. 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

  3. Definitions • Orbital motion: object circles an axis that does not pass through the object itself

  4. Circle Terminology • Radius • Diameter • Chord • Tangent • Arc

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

  6. Polar Coordinates (r, θ) • magnitude of r = radius of circular path • θ = angle of rotation • θ is measured in radians

  7. 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.

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

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

  10. 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:

  11. 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”

  12. Δθ ω = Δt Speed and Velocity • If angular speed is constant, then the rotating object experiences uniform circular motion.

  13. rad or s-1 s Speed and Velocity • In the SI, the units are radians per second. • Written as:

  14. Speed and Velocity • The velocity vector of a particle in circular motion is tangent to the circular path. • This velocity is called tangential velocity.

  15. Speed and Velocity • The magnitude of the tangential velocity is called the tangential speed, vt. vt = |vt|

  16. l vt = Δt Speed and Velocity • Another formula for tangential speed is: • arclengthl = r × Δθ

  17. rΔθ vt = Δt Speed and Velocity • average tangential speed:

  18. Δv a = Δt vt2 a = r Acceleration • Linear motion: • Circular motion:

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

  20. vt2 ac = m/s² r Acceleration • The magnitude of centripetal acceleration is: • For all circular motion at constant radius and speed

  21. Acceleration • Another formula for centripetal acceleration: ac = -rω2

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

  23. Angular Velocity • Right-hand rule of circular motion:

  24. 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.

  25. ω2 – ω1 Δω α = = Δt Δt Angular Acceleration • change in angular velocity • notation: α • average angular acceleration:

  26. ω2 – ω1 Δω α = = Δt Δt Angular Acceleration • units are rad/s², or s-2 • direction is parallel to the rotational axis

  27. Tangential Acceleration • defined as the time-rate of change of the magnitude of tangential velocity

  28. Δvt at = =αr Δt Tangential Acceleration • average tangential acceleration:

  29. Tangential Acceleration • instantaneous tangential acceleration: at =αr Don’t be too concerned about the calculus involved here...

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

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

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

  33. Equations of Circular Motion • note the substitutions here:

  34. Dynamics of Circular Motion

  35. Centripetal Force • in circular motion, the unbalanced force sum that produces centripetal acceleration • abbreviated Fc

  36. mvt² Fc = r Centripetal Force • to calculate the magnitude of Fc:

  37. Centripetal Force • Centipetal force can be exerted through: • tension • gravity

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

  39. Torque τ = rF sin θ • r = magnitude of position vector from center to where force is applied • F = magnitude of applied force

  40. 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

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

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

  43. F1 l2 = F2 l1 Torque • Law of Moments: l1F1 = l2F2 • Rearranged:

  44. Universal Gravitation

  45. 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.

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

  47. The Ideas • Copernicus concluded the geocentric theory was faulty. • His heliocentric theory was simpler.

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

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

  50. 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.

More Related