Gravitation. AP Physics C

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1 Gavitation AP Physics C

2 Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What the popotionality above is saying is that fo thee to be a FORCE DUE TO GRAVITY on something thee must be at least masses involved, whee one is lage than the othe.

3 N.L.o.G. As you move AWAY fom the eath, you DISTANCE inceases and you FORCE DUE TO GRAVITY decease. This is a special INVERSE elationship called an Invese- Squae. F g α 1 The stands fo SEPARATION DISTANCE and is the distance between the CENTERS OF MASS of the objects. We us the symbol as it symbolizes the adius. Gavitation is closely elated to cicula motion as you will discove late.

4 N.L.o.G Putting it all togethe m1m Fg α G = constant of popotionality G = Univesal Gavitational Constant G = 6.67x10 F g m m = G 1 7 Nm kg F F g g = mg Use this when you ae on the eath = G m m 1 Use this when you ae LEAVING the eath

5 Ty this! Let s set the equations equal to each othe since they BOTH epesent you weight o foce due to gavity F F g g = mg Use this when you ae on the eath = G m m 1 Use this when you ae LEAVING the eath Mm mg = G M g = G M = Mass of the Eath = adius of the Eath = 5.97x10 = 6.37x kg m SOLVE FOR g! 7 4 (6.67x10 )(5.97x10 ) g = = 9.81m / s 6 (6.37x10 )

6 How did Newton figue this out? Newton knew that the foce on a falling apple (due to Eath) is in diect popotion to the acceleation of that apple. He also knew that the foce on the moon is in diect popotion to the acceleation of the moon, ALSO due to Eath Newton also sumised that that SAME foce was invesely popotional to the distance fom the cente of Eath. The poblem was that he wasn t exactly sue what the exponent was.

7 How did Newton figue this out? Since both the acceleation and distance wee set up as popotionalities with the foce, he decided to set up a atio. Newton knew that the acceleation of the apple was 9.8 and that the appoximate distance was 4000 miles to the cente of Eath. Newton also knew the distance and acceleation of the Moon as it obits Eath centipetally. It was the outcome of this atio that led him to the exponent of. Theefoe ceating an invese squae elationship.

8 Newton s Law of Gavitation (in moe detail) To make the expession moe mathematically acceptable we also look at this fomula this way: The NEW "" that you see is simply a unit vecto like I,j, & k-hat. A unit vecto, emembe, tells you the diection the foce is going. In this case it means that it is between the two bodies is RADIAL in natue. The NEGATIVE SIGN is meant to denote that a foce poduces "bound" obits. It is only used when you ae sue you need it elative to whateve efeence fame you ae using...so BE CAREFUL! It may be wise to use this expession to find magnitudes only.

9 Example What is the gavitational foce between the eath and a 100 kg man standing on the eath's suface? M F = Mass of = adius of g = G m man the Eath = 5.97x10 the Eath = 6.37x10 M Eath = 6.67x kg m 11 (100)(5.97 x10 6 (6.37x10 ) 4 ) = 9.81 x 10 N Because the foce nea the suface of Eath is constant, we can define this foce easie by ealizing that this foce of gavitation is in diect popotional to the man s mass. A constant of popotionality must dive this elationship. Fg α mman Fg = mmang We see that this constant is in fact the gavitational 9.81x10 = 100g acceleation located nea g = 9.8 m / s / s the Eath s suface.

10 Example How fa fom the eath's suface must an astonaut in space be if she is to feel a gavitational acceleation that is half what she would feel on the eath's suface? Mm mg = G M g = G M = Mass of = adius of g = = G ( M + eath (6.67x10 the Eath = 5.97x10 the Eath = 6.37x ) 4 kg m = )(5.97x GM g 4 ) Eath Eath 6.37x10 6 = This value is fou tenths the adius of Eath..64x10 6 m

11 A couple of things to conside about Eath You can teat the eath as a point mass with its mass being at the cente if an object is on its suface The eath is actually not unifom The eath is not a sphee The eath is otating Let's assume the eath is a unifom sphee. What would happen to a mass (man) that is dopped down a hole that goes completely though the eath? Digging a hole at the Fobidden City in Beijing will cause you to end up somewhee in Agentina. But don t be supised if you dig somewhee else and wate stats to pou in!

12 Digging a hole When you jump down and ae at a adius fom the cente, the potion of Eath that lies OUTSIDE a sphee a adius does NOT poduce a NET gavitational foce on you! The potion that lies INSIDE the sphee does. This implies that as you fall the sphee changes in volume, mass, and density ( due to diffeent types of ocks) M = Vsphee = 4 3 ρ, π M V 3 Mm G4πmρ Fg = G Fg = 3 F = k g inside 3 4π = ρ 3 G4πmρ k = 3 This tells us that you weight actually DECREASES as you appoach the cente of Eath fom within the INSIDE of the sphee and that it behaves like Hook s Law. YOU WILL OSCILLATE.

13 Enegy Consideations Wok is the integal of a Foce function with espect to displacement. Putting in the basic expession fo gavitational foce Pulling out the constants and binging the denominato to the numeato. The negative sign should not supise you as we aleady knew that Wok was equal to the negative change in U o mgh.

14 Escape Speed Conside a ocket leaving the eath. It usually goes up, slows down, and then etuns to eath. Thee exists an initial minimum speed that when eached the ockets will continue on foeve. Let's use consevation of enegy to analyze this situation! We know that ENERGY will neve change. As the ocket leaves the eath it's kinetic is lage and its potential is small. As it ascends, thee is a tansfe of enegy such that the diffeence between the kinetic and potential will always equal to ZERO.

15 Escape Speed This expession is called the escape speed! Due to the otation of the eath, we can take advantage of the fact that we ae otating at a speed of 1500 km/h at the Cape! NOTE: THIS IS ONLY FOR A SYSTEM WHERE YOU ARE TRYING TO GET THE OBJECT IN ORBIT!!!!!

16 Keple'sLaws Thee ae thee laws that Johannes Keple fomulated when he was studying the heavens THE LAW OF ORBITS - "All planets move in elliptical obits, with the Sun at one focus. THE LAW OF AREAS - "A line that connects a planet to the sun sweeps out equal aeas in the plane of the planet's obit in equal times, that is, the ate da/dt at which it sweeps out aea A is constant. THE LAW OF PERIODS - "The squae of the peiod of any planet is popotional to the cube of the semi majo axis of its obit."

17 Keple s 1 st law The Law of Obits "All planets move in elliptical obits, with the Sun at one focus.

18 Keple s nd Law The Law of Aeas "A line that connects a planet to the sun sweeps out equal aeas in the plane of the planet's obit in equal times, that is, the ate da/dt at which it sweeps out aea A is constant.

19 Keple s nd Law How do we know that the ate at which the aea is swept is constant? Angula momentum is conseved and thus constant! We see that both ae popotional to the same two vaiables, thus Keple's second law holds tue to fom.

20 Keple s 3 d Law The Law of Peiods "The squae of the peiod of any planet is popotional to the cube of the semi majo axis of its obit." Gavitational foces ae centipetal, thus we can set them equal to each othe! Since we ae moving in a cicle we can substitute the appopiate velocity fomula! The expession in the RED cicle deived by setting the centipetal foce equal to the gavitational foce is called ORBITAL SPEED. Using algeba, you can see that eveything in the paenthesis is CONSTANT. Thus the popotionality holds tue!

21 Kinetic Enegy in Obit Using ou ORBITAL SPEED deived fom K.T.L and the fomula fo kinetic enegy we can define the kinetic enegy of an object in a bit moe detail when it is in obit aound a body. The question is WHY? Why do we need a new equation fo kinetic enegy? Well, the answe is that geatly simplifies the math. If we use egula kinetic enegy along with potential, we will need both the obital velocity AND the obital adius. In this case, we need only the obital adius.

22 Total Enegy of an obiting body Notice the lack of velocities in this expession as mentioned in the last slide. So by inspection we see that the kinetic enegy function is always positive, the potential is negative and the total enegy function is negative. In fact the total enegy equation is the negative invese of the kinetic. The negative is symbolic because it means that the mass m is BOUND to the mass of M and can neve escape fom it. It is called a BINDING ENERGY.

23 Enegy fom a gaphical pespective As the adius of motion gets lage. The obiting body s kinetic enegy must decease ( slows down) and its potential enegy must incease ( become less negative). By saying become less negative means that we have defined ou ZERO position fo ou potential enegy at INFINITY.

24 Please make you selection... How do you move into a highe velocity obit? Question: If we have an obiting Eath satellite and we want to put it in a highe velocity obit, how can we use the satellite s thustes to make the adjustment? a) Fie Backwads b) Fie Fowads 50% 50% Backwads = speed up Fowads = slow down 1 Fie Backwads 3 Fie Fowads

25 Fastest Respondes (in seconds) Paticipant 1 Paticipant Paticipant 3 Paticipant 4 Paticipant 5

26 How do you move into a highe velocity obit? 1) If you fie backwads thinking you will speed up the satellite you put it into a lage obital adius which ultimately SLOWS DOWN the satellite as the KE deceases. ) By thusting backwads you ae ADDING enegy to the system moving the total enegy close to ZERO, this esults in a lage adius which also causes the KE to decease. 3) Fie fowads gently so that you do NEGATIVE WORK. This will cause the satellite to fall into a smalle obit inceasing the KE and inceasing the speed. It also makes the potential enegy incease negatively because you ae moving fathe fom infinity. As the potential incease the KE again deceases.

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