Module 5 - Physics Notes Advanced Mechanics

MODULE 5: ADVANCED MECHANICS GALILEO’S ANALYSIS OF PROJECTILES  Horizontal and vertical components of projectile motion are independent of each other  Horizontal motion of a moving object is not subject to gravitational forces, and therefore doesn’t experience acceleration  Vertical motion of an object near the surface of the Earth is affected by downwards force of gravity which gives it an acceleration 9.8ms-1 Equations used in straight line motion v = u + at v2 = u2 + 2as s = ut + 1/2at2 Horizontal component of motion Ux = u cosθ Vx = ux (ax = 0) v x2 = ux2 Δx = uxt Vertical component of motion Uy = u sinθ v y = u y + a yt vy2 = uy2 + 2ay Δy Δy = uyt = 1/2ayt2 NEWTON’S LAW OF GRAVITY 1. There is a force of attraction between any pair of point masses in the universe 2. This force is directional proportional to masses of objects 3. This force is inversely proportional to the square of distance between them F= GMm 2 r *Superposition principle: adding a third mass does not affect the force between mass 1 and 2, however net force on any one mass is no the superposition (vector sum) of forces towards the other two masses. MASS, WEIGHT, & ACCELERATION DUE TO GRAVITY Gravitational field g of a massive object at any point P is defined as the gravitational force felt per unit mass of another object placed at that point. g= F g GM = 2 m r Mass: amount of matter in an object Weight: object placed in gravitational field feels a downwards gravitational force W F=F=mg= GMm =acceleration due¿ gravity r2 *average surface value of the gravitational field on Earth’s surface is 9.8ms-1. Possible reasons for variation include:  Earth’s crust varies in structure, thickness and density  The Earth’s globe is flattened; distance of surface to centre of Earth is less  Spin of Earth; spin effect is greatest at equator, as you travel from equator to poles, the spin effect on g shrinks to zero  Altitude above surface of planet decreases g UNIFORM CIRULAR MOTION An object in a circular path with constant speed is undergoing circular motion. v= 2 πr T Tangential to the path followed Change in direction of velocity over time is acceleration, a centripetal acceleration towards centre a= v2 r By Newton’s second law, there must be a net force directed to the centre to cause this acceleration. ∑ F=m v2 r e.g. – gravity is the net centripetal force keeping a satellite in orbit around the Earth - Friction between tyres and road is the centripetal force keeping a car in a round-about. - Tension in the rope keeps a lasso spun above the head. The object’s velocity always acts perpendicular, the centripetal force neither adds or takes away from its speed. i.e. it is in constant uniform circular motion. Centripetal force just changes the direction and without it, velocity would leave the circle at a tangent with constant speed by Newton’s 1st law of inertia. CENTRIFUGAL FORCE A fictitious ‘centrifugal force directed radially outwards and of equal magnitude to the centripetal force (such that net force is zero). e.g. astronaut remains at rest on the floor even if the rocket ship is subject to an unbalanced centripetal force. ANGULAR VELOCITY ω= dθ dt - Rate of change of angular displacement with time: - Angular frequency, multiplying the frequency (number of revs per unit time) by the angle covered in a revolution: ω=2 πf - Angular velocity can be written as ω= v r THE CROSS PRODUCT Means to multiply the perpendicular (non-corresponding) components of the vectors to produce a new vector defined as: a ×b=absinθ *if the vectors are perpendicular ( θ = 90) then their cross product is max *if the vectors are parallel ( θ = 0 or 180) it is zero ANGULAR MOMENTUM L≡ r × p Angular momentum L of particle about the origin is the cross produce of r with the particle’s momentum p P = mv L is perpendicular to both r and p and therefore normal to the plane containing them. TORQUE The rotational equivalent of force (just as a force can be thought of a push or a pull which causes a change in linear momentum. Thought as a twist which causes a change in angular momentum, (it is conserved if there is no external torque) τ ≡r × F=rFsinθ No torque is produced by a force applied at the axis, or parallel to the position from the axis. Right Hand Rule to find direction of rotation: the thumb points in the direction of torque, and fingers grip in the direction of rotation. KEPLER’S LAW OF PERIODS Kepler’s third law of planetary motion states: T 2 ∝ r 3 Thus, the orbits of two planets, A and B, about the Sun are related by: r3A T 2 A = r3B T 2B Newtown’s laws allow us to accurately describe planetary motion since planets are very close to being circular. Kepler’s third law is derived by equating the force of gravity to the net centripetal force: GM =v 2 Then, If the orbit is circular, then speed is given by: v= 2 πr  T GM 2 πr =( ) r T 2 3 r GM = 2 T2 4 π ORBITAL VELOCITY *orbital velocity is independent of the mass of the satellite – only depends on the mass of central body being orbited and radius - the rate of change of velocity (acceleration) due to gravity is independent of the satellite’s mass, so they will share the same motion at every point and follow the same orbital trajectory *both the spacecraft and astronauts on board share the same motion, the spacecraft floor exerts no normal force to constrain the astronaut’s motion, so astronaut experiences apparent weightlessness. √ v= GM r ORBITAL TRANSFERS The spacecraft is accelerated out of Earth orbit in the direction of the Earth’s orbital velocity “launch window”. This is when the orbital velocity relative to the sun corresponds to the desired heading such that it is put towards the required orbital transfer velocity. THE SLINGSHOT EFFECT A manoeuvre used by space probes to change speed and/or direction relative to the sun by using the motion of a planet. To gain speed, space probe approaches planet head on To lose speed, space probe approaches a planet from behind and slingshots around it. ORBITAL DECAY Satellites is Low Earth Orbit (LEO) are still within the Earth’s upper atmosphere - En c oun t e rs ma l la mo u n to fa t mo s p h e r i cd r a gd u et oc o l l i s i o n swi t hmo l e c u l e s - Dr a gs l o wl yc a u s e st h e mt og r a d u a l l ys p i r a lt o wa r d se a r t h * t h el o we ri t sa l t i t u dedr o p s ,t h ef a s t e rt h ed e c a yd u et ot h ei n c r e a s ei na i rd e n s i t y I fl e f tunc he c k e d, t hes a t e l l i t ewi l lb u r nu pd u et ot h eh e a tg e n e r a t e db yf r i c t i o nwi t ht h ei n c r e a s i n g l y d e ns ea t mos p he r e . Ge os t at i onar yOr bi t( GEO)me a n st h a ts a t e l l i t ei ss t a t i o n a r yr e l a t i v et oEa r t h ’ ss u r f a c e ESCAPEVELOCI TY Mi ni mum v e l oc i t yr e q u i r e df o rap r o j e c t i l et oj u s te s c a p ef r o mEa r t h ’ sg r a v i t a t i o n a lfie l d . √ v esc = 2GM r MULTI PLYI NGVECTORS Dotpr o duc t :mu l t i pl y i ngc o r r e s p o n d i n gc o mp o n e n t so fv e c t o r st o g e t h e r . a ∙ b=abcosθ ENERGY:ACONSERVEDQUANTI TY s ∫ F ∙ ds+ E= 21 m v 2 o WORK W =F ∙ ∆ s POTENTI ALENERGY Wo r kd oneb yac o ns e r v a t i v ef o r c ei si n d e p e n d e n to ft h ep a t ht a k e na n dd e p e n d so n l yo nt h ep o s i t i o n o ft hee n dp oi nt soft h ep a t h . ( g r a v i t y ) Ne a rt heEar t h’ ss ur f a c e :g r a v i t a t i o n a lp o t e n t i a le n e r g yi sme a s u r e dr e l a t i v et ot h eg r o u n d U=mgh Farf r om t heEar t h:gr a v i t a t i o n a lp o t e n t i a le n e r gyi sme a s u r e dr e l a t i v et o ∞ U= −GMm r I ft hef or c ei sn otc ons e r v a t i v e ,t a ki n gd i ffe r e n tp a t h sl e a d st oc o n fli c t i n gp o t e n t i a le n e r g i e sf o rt h e s a mepo i nt . e . g . f r i c t i o n:s i g nc h a n g e sd e p e n d i n go nwh i c hwa yt h ep a r t i c l ei smo v i n g . PRI NCI PLEOFCONSERVATI ONOFMECHANI CALENERGY Sol o nga sonl yc o n s e r v a t i v ef o r c e sa c t ,t h es u mo ft h eki n e t i ce n e r gya n dp o t e n t i a le n e r g yi sa c ons t a ntE. T +U =E THEWORKENERGYTHEROM 1 . Thewor kdoneb ya( ne t )f o r c ei se qua lt ot hec ha ng ei nki ne t i ce ne r g yo ft hepa r t i c l e ∆ T =W 2 . Forac ons e r v a t i v ef o r c e , t hewo r kdo nei smi nust hec ha ng ei npo t e nt i a le ne r g y . −∆ U =W c NONCONSERVATI VEFORCES Th ec h a n g ei napa r t i c l e ’ st o t a lme c h a n i c a le n e r g yEa si tt r a v e l sf r o mo n ep o i n tt oa n o t h e ri st h e r e f o r e e qua lt ot het ot a lwo r kd o n eb yn o n c o n s e r v a t i v ef o r c e s W NC=∆ T + ∆ U =∆ E THELAW OFCONSERVATI ONOFENERGY I nas y s t e mi s ol a t e df r omi t ss u r r o u n d i n g s , t h e r ei sas c a l a rq u a n t i t yc a l l e de n e r gywh i c ha tt h ee n do f a n ypr o c e s si st h es a mea swh e nt h a tp r o c e s sb e g a n , t h a ti s ,i tr e ma i n sc o n s t a n t . ELASTI CVSI NELASTI CCOLLI SI ONS El a s t i c :ki ne t i ce ne r gyi sc o n s e r v e d I ne l a s t i c :ki ne t i ce ne r gyi sl o s ta sh e a t