Homework Set 4 Physics 319 Classical Mechanics. m m k. x, x, x, x T U x x x x l 2. x x x x. x x x x

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Marianna Chambers
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1 Poble 78 a) The agangian i Hoewok Set 4 Phyic 319 Claical Mechanic k b) In te of the cente of a cooinate an x x1 x x1 x xc x x x x x1 xc x xc x x x x x1 xc x xc x, x, x, x T U x x x x l x k xc x l 4 xc 0 x / k x l c) The cente of a ove with unifo velocity The equation of otion fo x i k x x l x t l Aco t k Poble 73 If one efine x c X x, a the inie cat location in pace, the agangian i eithe o k X x T U xc x c k X x c c

2 Poble 79 k x T U x Aint x A in t kx In eithe cae the equation of otion i x 0 x Acot whee k Thee i no enegy integal fo thi poble 0 / It i clea that the point P i at the location x t Rx cot Ry in t P ˆ ˆ In te of the given angle the a i at the point x t x t l inxˆ l coyˆ P x t R int l co y t R cot l in U gy Rint l co The agangian i T U R R l t l g R t l The equation of otion i l R l in t R l co t gl in in in co co l R R R t g in l R co t g in Poble 734 Conie a unifo expanion ove the ping length On the fixe ie of the ping the coil oe not ove an the fee ie it ove xt If eaue the itance along the unextene an uncopee ping with =0 at the fixe en, then the coil at location ove by xcoil, t xt when the expanion in the ping i xt Siilaly, the velocity of the coil at location i vcoil xcoil t x t t,

3 The total kinetic enegy of the coil i v M x M Tcoil x, 6 Which nee to be ae to the kinetic enegy of the ca, x / The total agangian i M k x x 6 The uual calculation of the equation of otion lea to inuoial ocillation with (angula) fequency k / M / 3 Poble 735 In the fae that otate with the wie the poition i AB AB x 1 co ˆ in ˆ t x y But the unit vecto otate a xˆ t cotxˆ intyˆ An theefoe yˆ t intxˆ cotyˆ x t AB AB AB AB t t t t y t AB AB AB AB t t t t AB AB xt int in t AB AB y t cot co t AB AB AB T U co co co in in co co 1 co in in co in in AB AB co in 4 4 in in in Thi eult i obviouly coect a the centifugal foce fo the otation give an effective potential enegy of AB / AB 1co / 4 AB / /, an the kinetic enegy in the otating fae i AB / / The Eule-agange equation in thi 0

4 otating fae give in exactly, a they houl The (angula) fequency of all ocillation i Poble 81 a) The fit pat i taight fo the lectue whee thi aiu wa efine to be l G 1 b) Fo all eviation fo, the effective potential i, to econ oe in the eviation, Ueff Ueff Ein U G l l eff an the otion i a iple haonic otion with (angula) fequency k / l / Becaue fo all eviation a b, thi i a etateent of Keple econ law T ab/ l Poble 819 Fo the ellipe a 3300 k / 6400 k 8050 k Theefoe f 8050 k 6700 k 1350 k The eccenticity i 1350 / Becaue 1 a k, the height at 90 egee i 144 k Poble 89 Fo the lectue we know that on a cicula obit of aiu, the total enegy an Ge kinetic enegy an potential enegy ae, Ge, an l l G e, epectively (conitent with the viial theoe!) When the un a l i halve the angula oentu, an hence the kinetic enegy i coneve At thi Ge Ge ae intant, the potential enegy becoe an o the total l enegy vanihe Without futhe ao we know that the obit i a paabola out to with iniu aiu The paabola i tangent to the oiginal obit at the intant when the a change

5 Poble 830 The equation fo the obit i Becaue x co an Fo 1the obit i the paabola x, 1 co x y x x x x 1 x y y x x / Thi i Eq 860 ecalling i Taylo c Fo 1, by copleting the quae the obit i the hypebola o c / 1, one ay olve fo y y x 1 y x y 1 c / 1, an c / 1 x in the te efine by 861 Alo, 1 1 y x x Thi eult explain the iaga an equation fo the hypebola in ectue 15 Poble 833 Afte the atellite i place into the tanfe obit the elation c c R1 R3 1 1 c povie the eccenticity a apply Eliinating R3 R1 / R3 R1 To achieve a cicula obit at P, c3 R3ut be obtaine Theefoe c3 1 c 1 Thi i Taylo Eqn 873 R R R R R R R R R R R R 3 1 1

Test 2 phy a) How is the velocity of a particle defined? b) What is an inertial reference frame? c) Describe friction.

Test 2 phy a) How is the velocity of a particle defined? b) What is an inertial reference frame? c) Describe friction. Tet phy 40 1. a) How i the velocity of a paticle defined? b) What i an inetial efeence fae? c) Decibe fiction. phyic dealt otly with falling bodie. d) Copae the acceleation of a paticle in efeence fae