3. Period Law: Simplified proof for circular orbits Equate gravitational and centripetal forces

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1 Physics 106 Lecture 10 Kepler s Laws and Planetary Motion-continued SJ 7 th ed.: Chap 1., 1.6 Kepler s laws of planetary otion Orbit Law Area Law Period Law Satellite and planetary orbits Orbits, potential, kinetic, total energy. Period Law: Siplified proof for circular orbits Equate gravitational and centripetal forces M v Fg = G = F cent = π recall: v = ω = where T period T ass M 4π cancels: G == T M F g v Kepler Third Law: 4π T = GM. M is ass of the central body. Holds also for eccentric orbits with = sei-ajor axis T Also: 4π = GM, constant for all satellites around the sae object with ass M. For the Solar Syste: M M sun = x 10 0 kg K T 4π S 4 = = = years eter GM 11 0 a sun / 1

2 Sae value for each planet in the solar syste! Sei-ajor Axis Period T /a =K sun Planet a (10 10 ) T (y) (10-4 y / ) Mercury Venus Earth Mars = Jupiter Saturn Uranus Neptune Pluto π GM sun The value of K depends on the object being orbited For exaple, for the Moon around the Earth, K Sun is replaced with K Earth Exaple, Find the ass of the Sun The distance between the Earth and the Sun is x The period of the Earth s orbit is.156 x 10 7 sec. Use Kepler s Third Law to find the ass of the Sun.

3 Exaple: Find the distance a fro Earth to the Moon Moon s period T = 7. days (about 1 onth) Earth s ass e = 5.98 x 10 4 kg G=6.67x10-11 N /kg Exaple: Find Jupiter s period in Earth-years Given: Jupiter s ean distance fro the Sun is 5.19 A. U. Define: 1 AU = 1 Astronoical Unit = Average radius of the Earth s orbit = 9,000,000 iles = 1.5 x eters

Orbital radius for Halley s Coet 10.. Since about 40 BC, Halley s coet has reappeared in the sky all over the Earth every 76 years on the average. The latest appearance was in 1986.

? A) 66 A. U. B).00 A. U. C).056 A. U. D) 4. A. U. E) 17.9 A. U. T a = 4π GM sun for any object orbiting the Sun Physics 106 Lecture 11 Oscillations I SJ 7 th Ed.: Chap 15.1 to, 15.

4 Orbital radius for Halley s Coet 10.. Since about 40 BC, Halley s coet has reappeared in the sky all over the Earth every 76 years on the average. The latest appearance was in The orbit is quite eccentric, but it still obeys Kepler s rd Law. What is the approxiate, sei-ajor axis of the orbit in A. U.? A) 66 A. U. B).00 A. U. C).056 A. U. D) 4. A. U. E) 17.9 A. U. T a = 4π GM sun for any object orbiting the Sun Physics 106 Lecture 11 Oscillations I SJ 7 th Ed.: Chap 15.1 to, 15.5 Oscillating systes Characterization Exaples: they are everywhere Key concepts Siple haronic otion: a paradig epresentation for displaceent, velocity, acceleration Oscillator equation SHM exaple: spring oscillator (Hooke s law) Energy relations, exaple SHM exaple: torsion pendulu SHM exaple: siple pendulu 4

5 Definition of an Oscillator : A syste executing periodic, repetitive behavior Syste state (t) = state(t+t) = = state(t+nt) T = period = tie to coplete one coplete cycle Exaple: Spring iclicker: If t=0 is chosen as below, which tie corresponds to period T? t=0 A B C D 5

Oscillating systes are everywhere in nature: MECHANICAL: pendula (swinging objects

g,string instruents) guitar, piano, drus, bells, horns, reeds otating object Siple

(t) a y (t) y θ x(t) v x (t) a x (t) unifor circular otion x Water Waves: Haronic

6 Oscillating systes are everywhere in nature: MECHANICAL: pendula (swinging objects on cables) auto suspensions usic/sound (e.g,string instruents) guitar, piano, drus, bells, horns, reeds otating object Siple systes spring x(t) v(t) a(t) vibrating string L pendulu θ θ(t) ω(t) α(t) y(t) v y (t) a y (t) y θ x(t) v x (t) a x (t) unifor circular otion x Water Waves: Haronic oscillations of the level at a point disturbance y1(t) = y10 cos( πft) v y (t) = y cos( πft + φ) 0 v The phase angle Φ sets the clock 6

Characterizing oscillations Aplitude x ax axiu displaceent away fro equilibriu (disturbance) displaceent Period T tie for otion to repeat Frequency f = 1/T

in the cycle the syste is, represented in angle (π corresponds to one full cycle) aplitude Frequency and aplitude of an oscillator 11

When object B is copared to object A, which of the following correctly describe the angular frequency and aplitude?

7 Characterizing oscillations Aplitude x ax axiu displaceent away fro equilibriu (disturbance) displaceent Period T tie for otion to repeat Frequency f = 1/T # of cycles per tie unit [Frequency] = Hertz (Hz) = t -1 Angular frequency: ω = π/τ (π corresponds to one full cycle) ω = πf (units: rad/s) tie Phase where in the cycle the syste is, represented in angle (π corresponds to one full cycle) aplitude Frequency and aplitude of an oscillator The figures show plots of the aplitude of two haronic oscillators versus tie. When object B is copared to object A, which of the following correctly describe the angular frequency and aplitude? A) B has larger frequency and larger aplitude than A B) B has larger frequency and saller aplitude than A C) B has larger period and larger aplitude than A D) B has saller frequency and saller aplitude than A E) B has larger period and larger aplitude than A ω = πf = π/t x(t) = x cos( ωt + ϕ) 7

8 iclicker: What is the phase difference between tie t1 and t? t1 t Α) π/4 Β) π/ C) π D) π/ E) π Matheatical representation of oscillation Haronic oscillation: oscillation that can be represented as sine or cosine function xt () = x cos( ωt+ φ) or xt () = x sin( ωt+ φ) ω T= π, since sine or cosine function repeats every π ω = π/t, which is angular frequency 8

9 Exaple 6s 5 0 s -5 epresent the above siple haronic oscillation in atheatical for, x(t). 9

m A 1 m mgd k m v ( C) AP Physics Multiple Choice Practice Oscillations

m A 1 m mgd k m v ( C) AP Physics Multiple Choice Practice Oscillations P Physics Multiple Choice Practice Oscillations. ass, attached to a horizontal assless spring with spring constant, is set into siple haronic otion. Its axiu displaceent fro its equilibriu position is.