Lecture 19: Effective Potential, and Gravity

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Pierce Perkins
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1 Lectue 19: Effective Potential, and Gavity The expession fo the enegy of cental-foce motion was: 1 ( ) l E = µ + U + µ We can teat this as a one-dimensional poblem if we define an effective potential: ( ) ( ) l V = U + µ Effective means that V acts like a potential enegy, even though it isn t one thee is no foce coesponding to the gadient of the second tem With this definition, the enegy becomes: 1 E = µ + V ( ) = T ( ) + V T is the adial kinetic enegy

2 Example: -k/ potential Most of the emaining discussion of cental-foce motion will focus on motion unde gavity How much can we lean about the motion simply fom the effective potential? Stat with a sketch: E U V

3 We see that the motion in fo a given enegy can be quite diffeent fo diffeent values of l if l = 0, the system can each = 0 If thee is a non-zeo angula momentum, thee will be an excluded egion at small. The sepaation between the two masses will neve be smalle than this value The minimum of the potential coesponds to the location of a cicula obit i.e., this is an equilibium point fo adial motion fo 1/ potentials the equilibium is stable (convenient fo those of us who like to maintain a elatively constant distance fom the sun!) The position is given by: dv k l = = 0 3 d µ = l µ k Inceases apidly as l inceases

4 Minimum Enegy We also see that fo a given l thee is a minimum enegy the paticle can have This is the enegy of a cicula obit E k = + min min µ min l k l µ k µ k µ k = + = + = l l l l l µ µ k µ k

5 Gavitation Now that we ve leaned some of the basic ules of centalfoce motion, we ll see how they apply to the only foce descibed by Newton: gavity Now we know thee ae thee othe foces: 1. Electomagnetism (which you ll study in anothe couse, using math simila to what we use fo gavity). The weak foce 3. The stong foce The last two can only eally be undestood in the context of quantum field theoy In one looks at the inteaction between fundamental paticles, gavity is by fa the weakest of the foces Howeve, it is the only foce that both acts ove long distances, and is always attactive, making it the most significant influence on astonomical objects

6 The Basic Rule The foce of gavity between two point masses is given by: M e F m F GMm = e whee G is a univesal constant: 6.67 x Nm kg - The foce is linea: if thee wee seveal point masses in the poblem, the total foce on m would be the vecto sum of all the individual foces If thee is a continuous distibution of matte, one needs to integate to find the total foce on m: V ( ) ρ F = Gm dv

7 The Gavitational Field Ou definition of the gavitational foce equies it to act ove lage distances Not as easy to undestand as the foce between two objects that touch each othe One conceptual way to think about it is that the pesence of a mass altes the space aound it, in a way that othe masses can feel In fact, such a pictue is at the coe of geneal elativity, the moden theoy of gavity We can epesent this modification of space as adding a gavitational field to evey point in space The field due to a mass M is: g GM = e This is also the acceleation that any mass in the field will have

8 The Gavitational Potential The lines of the gavitational field look like: They don t fom closed loops Moe pecisely, it s easy to show that g = 0 This means that we can define a potential coesponding to the field Note that this is not a potential enegy, which coesponds to a foce! The equied fom is GM Φ = + C Φ Φ = e = GM e

9 As with all potentials, the constant C is completely abitay Howeve, it s usually convenient to teat the potential at infinite distances as zeo, which coesponds to C = 0 The elationship between gavitational potential and potential enegy is the same as that between the gavitational field and gavitational foce: Fo a paticle of mass m, the potential enegy is U = mφ

10 Gavitational Self-Enegy It s often inteesting to know how much gavitational enegy is stoed in a given object (o collection of objects) In othe wods, how much enegy would we gain by pulling the object apat? If the value is negative (as it always will be fo gavity!) that means we need to supply enegy to pull the object apat. To find the answe, we calculate the enegy needed to assemble the object piece by piece Simple example: thee point masses How much gavitational enegy is stoed in the following configuation? m 3 m 1 d 13 d 1 d 3 m

11 We stat with all thee masses infinitely apat. Then we find the change in potential enegy when m 1 is moved to its final position This is 0, since the othe two masses ae still infinitely fa away Now m is bought in to its final position. It now must move though the field geneated by m 1, so the change in potential enegy is: Gm1m U = mφ = d1 Similaly, when m 3 is bought in, we have: Gm m Gm m U3 = d13 d3 So the total self-enegy is: Gm1m Gm m Gm m U = d d d Note that the answe doesn t depend on the ode in which the masses wee assembled

12 In one of the homewok poblems, you ll show that the self-enegy of a unifom sphee of adius R and mass M is U = If the sphee s adius wee to become smalle, the gavitational self-enegy would decease Thus, to conseve enegy oveall, the object would need to elease enegy (somehow ) In the 1800 s, this was the accepted explanation fo why the sun shines 3 5 GM R i.e., it s a ball of contacting gas, that convets gavitational enegy to heat and light

13 Sounds geat, but U The cuent gavitational self-enegy of the sun is ( Nm kg )( 10 kg) 3 = = 8 5 ( 7 10 m) Note that this epesents the opposite of the total of all the enegy eleased in the past The sun s luminosity was know to be about 4 x 10 6 W But that implies that the sun could have been shining fo at most 6 x s, o less than 100 million yeas And geologists had good evidence that the Eath was olde than that! J The geologists wee ight it s the stong nuclea foce, not gavity, that powes the sun!

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects