A Vibrating MoleculeA good approximation to the potential energy between two atoms is given bythe so-called Leonard Jones potential. In dimensionless units, the potentialis given byU(r) = Aσr6−σr12, (1)where A is the strength of the interaction, σ is a sort-of length scale for theinteraction, and r is the distance between atoms. The potential is a familiarone from chemistry as shown in Figure 1. What is the separation distanceFigure 1: The Leonard Jones Potentialthat minimizes the energy of the system (this is where the system is mostlikely to be found)? What is the frequency of oscillation ab out this minimum?Faster Route to China?Suppose man drilled a hole through the Earth from the US to China as amethod of transcontinental travel. Follow the steps below to show that themotion on a trip through a hole drilled through the earth is simple harmonicmotion. Find the period for this trip and determine how fast you would need1to travel to circumnavigate the earth in the same amount of time. Figure 2should prove useful. y r d R Figure 2: Diagram with Relevant VariablesNote that it can be shown (c.f. Gauss) that a particle at location r experi-ences a gravitational force due only to the mass within the circle of radiusr.i. Using the universal gravitation law, write down the force felt by theobject at point r assuming the mass within the sphere or radius r isgiven by some function M(r) (which we find later).ii. Since the motion is periodic in y, write r in terms of y.iii. Since the object is confined to the tunnel, we only want the y componentof force. Using trig, find the y-comp onent of this force.iv. We want to learn M(y). Assuming the density of Earth is constantand equal to ρ, write down M(y) (remember ρ = m/V where V isvolume...another function of r).2v. Replace ρ with MEarth/VEarthand write VEarthas a function of the radiusof Earth, R.vi. Put it all together and find the y-component of force as a function of y.vii. Write F = ma = m¨y. Does this look familiar?viii. Deduce ω.ix. From ω determine the period, T , and determine how long the round triptakes.x. IF v = s/T and s is the distance around the world, determine v as afunction of d. Pick some reasonable values of d and see what you get!A Glancing BlowSuppose NASA puts a s atellite in orbit 3R, where R is the radius of theEarth, above the surface of the earth. Suppose further, it is a spy satellitethat Generic Rogue Terrorist Group wants to knock out of orbit. Due tomiscalculations, only a glancing blow is received and the deviation from theoriginal orbit, δ is small compared to R . Show that for δ/R  1, the motionabout the original orbit is simple harmonic motion. Determine the period ofthis motion. Compare it with the period of rotation about the earth if theorbit was / is very nearly circular. Remember the universal gravitation law:F (r) =GMmr2= ma = m¨r. (2)3Peter Griffin Annexes Joe’s PoolIn E Peterus Unum, an episode of Family Guy, Peter learns his property is notpart of the US and takes the opportunity to declare his own sovereign state,Petoria. Wanting a pool, he decides to invade his next-door neighbor, Joe,and seize his pool. This aggravates the USA who responds with the full mightof its military. While floating in the pool, (Peter floats as fat is less densethan water) the US plans a military strategy. First, assuming Peter is 300 lbsand about 6 ft × 2 ft × 3 ft, calculate his density. Determine what percentof Peter is below the water and what percent of him is above. Suppose, themilitary fires a warning shot over Peter’s belly and that this causes a smallpiece of debris of 20 lbs to land on him. Find his new equilibrium position(model Peter as a solid rectangular box for sake of ease). Annoyed by itspresence, Peter flicks the debris at Meg and off of himself. Show Peter undergoes approximate simple harmonic motion as 20/300  1. Determine theperiod of Peter’s