Recap. Centripetal acceleration: v r. a = m/s 2 (towards center of curvature)

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1 a = c v 2 Recap Centipetal acceleation: m/s 2 (towads cente of cuvatue) A centipetal foce F c is equied to keep a body in cicula motion: This foce poduces centipetal acceleation that continuously changes the body s velocity vecto. F = m a m v c = (Newtons) Thus fo a given mass the needed foce: inceases with velocity 2 c inceases as adius educes. 2

2 Example: The centipetal foce needed fo a ca to ound a bend is povided by fiction. If total (static) fictional foce is geate than equied centipetal foce, ca will successfully ound the bend. The highe the velocity and the shape the bend, the moe fiction needed! F s > mv2 As F s = µ s N - the fiction depends on suface type (µ s ). F f F f Eg. If you hit ice, µ becomes small and you fail to go aound the bend. Note: If you stat to skid (locked bakes) µ s changes to its kinetic value (which is lowe) and the skid gets wose! Moal: Don t speed aound tight bends! (especially in winte)

3 Motion on a Banked Cuve The nomal foce N depends on weight of the ca W and angle of the bank θ. Thee is a hoizontal component (N h ) acting towads cente of cuvatue. This exta centipetal foce can significantly educe amount of fiction needed v 2 If tan θ = g then the hoizontal N h povides all the centipetal foce needed! N N h θ N v = mg W=mg In this case no fiction is necessay and you can safely ound even an icy bend at speed

4 Ice Skating and Bike Racing Ice skates can t tilt ice so they lean ove to get a helping component of eaction foce to ound shap bends. F n F c

5 Feis Wheel Vetical Cicula Motion Ball on Sting N > W N W=mg Feel pulled in and upwad Bottom of cicle: Centipetal acceleation is diected upwads. Net foce is thus diected upwads: F net = N - W = ma c N=appaent weight (like in elevato) Thus: N = W + ma c T W T > W i.e. feel heavie/lage tension

6 Feel thown out and down N < W N > W N Top of the Cicle W W N W=mg T T W Component of W povides tension T > W Weight is only foce fo centipetal acceleation downwad N = W m a c i.e. lighte / less tension Special condition if W = m a c feel weightless (T=0) then a c = g = v 2 o v = g (lage, highe v)

7 Questions: Newton s Law of Univesal Gavitation What ole does centipetal acceleation play in the motions of heavenly bodies? What foces ae acting to cause thei motion? We know the planets ae moving in cuved paths (obits) aound the Sun. What foce is eve pesent to cause the necessay centipetal acceleation? Answe: It must be gavity but how? Newton s eath shatteing beakthough!

8 Newton ealized that the motion of a pojectile launched nea the Eath s suface and the moon s obit aound the Eath ae simila! He ealized that the moon is also unde the influence of gavity and is actually continuously falling towads Eath. Famous sketch fom Newton s Pincipia : Imagine a pojectile launched hoizontally fom an incedibly high mountain. (Olympus Mons) The lage the initial velocity, the futhe it will tavel. At vey high velocities, the cuvatue of the Eath becomes impotant in detemining the ange v

9 In fact, if velocity is high enough it will neve land It will keep falling (fee-fall), but the Eath s suface (cuvatue) keeps dopping away at the same ate! Range inceases as v inceases Cicula obit aound the Eath---Wow! obit So the same foce that contols the motion of objects nea Eath s suface (as descibed by d = ½ a.t 2, and v = a t) also acts to keep the moon in obit! Question: What is the natue of this foce?

10 Natue of Univesal Gavitational Law Newton s 2 nd law applied to fee-falling object: F = m g (weight foce) Thus: mass is key to the geneal desciption of gavity intuitive. But how does gavitational foce vay with distance? Expect foce to decease in stength as distance inceases intuitive. 4A point mass m A Actual: Foce 1 Many foces in natue exhibit a Aea of foce field inceases by 2 elationship

11 Newton s Gavitational Law The gavitational foce between two objects is popotional to thei masses and invesely popotional to the squae of the distance between thei centes. F = G m 1 m 2 2 (Newtons) F is an attactive foce vecto acting along line joining the two centes of masses. G = Univesal Gavitational Constant G = 6.67 x N.m 2 /kg 2 m 1 F 1 F 2 (vey small) Note: G was not measued until > 100 yeas afte Newton! - by Heny Cavendish (18 th cen.) m 2 (F 1 = -F 2 )

12 Newton poved this 1/ 2 dependence using Keple s laws (next lectue) and he applied his knowledge of centipetal acceleation and his ideas on gavity to the moon Centipetal acceleation of moon fo cicula motion: Moon s obit Eath v a = 60.3 Eath adii, o 3.84 x 10 8 c = v2 m (i.e. 0.4 million km) Moon v = 1.02 km/s a c = m/s 2 = 2.7 x 10-3 m/s 2 Newton agued that Eath s gavitational acceleation (i.e. foce) deceases with 1/ 2 If so, then the acceleation due to gavity at the moon s obital distance (g ) is: g 9.81 = = 2.7 x 10-3 m/s 2 Moon s centipetal acceleation is povided by Eath s gavitational acceleation at luna obit.

13 Gavitational Attactive Foce As G is vey small the gavitational attaction between the two evey-day objects is extemely small. Example: 2 people of mass 150 kg and 200 kg sepaated by 0.1 m G m 1 m x 10 F = -11 x = 150 x x 0.1 = 2 x 10-4 N (i.e N) Newtons Howeve, as masses of planets and in paticula stas and even galaxies ae HUGE, then the gavitational attaction can also be enomous! Example: Foce of attaction between Eath and Moon. mass of Eath = 5.98 x kg mass of Moon = 7.35 x kg F 2 x N! = 384 x 10 3 km (i.e. 200,000,000,000,000,000,000 N)

14 m e e How is Weight Related to Gavitation? F F e = adius of Eath = 6370 km m = mass of an object m e = mass of Eath = 5.98 x kg Gavitational foce of attaction: G m 1 m F = 2 (Newtons) if m = 150 kg, F = 1472 N (o ~ 330 lbs wt) But this foce ceates the object s weight: By Newton s 2 nd law (F=ma) we can also calculate weight: W = m g = 9.81 x 150 = 1472 N By equating these expessions fo gavitational foce: m g = G m e m o at suface: g = G m e e 2 2 e 2 Result: g is independent of mass of object!!

15 Acceleation due to gavity g is: 1. Constant fo a given planet and depends on planets mass and adius. 2. Independent of the mass of the acceleating object! (Galileo s discovey). Howeve, the gavitational foce F is dependent on object mass. In geneal, the gavitational acceleation (g) of a planet of mass (M) and adius (R) is: g = G M 2 R Planet Mecuy Venus Eath Moon Mas This equation also shows that Jupite g will decease with altitude: e.g. At 100 km height g = 9.53 m/s 2 Satun Uanus At moon s obit g = 2.7 x 10-3 m/s 2 Neptune Pluto 2 g m/s no solid suface

16 Newton s 3 F F d law: Each body feels same foce m 1 m 2 2 Gm 1 m F= 2 acting on it (but in opposite 2 diections) Thus each body expeiences an acceleation! Example: Boy 40 kg jumps off a box: Foce on boy: F = m g = 40 x 9.81 = 392 N Foce on Eath: F = m e a = 392 N o a = = x x m/s 2 ie. almost zeo! Example: 3 billion people jumping off boxes all at same time (mass 100 kg each) 3 x 10 9 x 100 x a = x = 5 x m/s 2 Conclusion: The Eath is so massive, we have essentially no effect on its motion!

Chap 5. Circular Motion: Gravitation

Chap 5. Circular Motion: Gravitation Chap 5. Cicula Motion: Gavitation Sec. 5.1 - Unifom Cicula Motion A body moves in unifom cicula motion, if the magnitude of the velocity vecto is constant and the diection changes at evey point and is