PHYS 321 Solutions to Practice Final (December 2002).

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1 PHYS Solutios to Practice Fial (December ) Two masses, m ad m are coected by a sprig of costat k, leadig to the potetial V( r) = k( r ) r a) What is the Lagragia for this system? (Assume -dimesioal motio) b) Fid the equatios of motio ad the costats of the motio c) Show that a circular orbit about the poit R = r + r is stable with respect to small perturbatios Solutios: a) The Lagragia is L= mv + mv k r r b) The equatios of motio are dv m + k( r r) = dt dv m + k( r r) = dt If we add the two equatios we fid that the total mometum mv+ mv = costat Hece the poit R= r+ r (ceter of mass) ca be take fixed if we desire Lettig the ew coordiates be r = r + R, r = r + R we fid the Lagragia may be rewritte dr dr L= m + m kr dt dt Trasformig to spherical polar coordiates we have L ( si ) eff = m r + r θ + r θϕ kr dr where we have dropped the term m because it is costat The costats of motio are therefore the total eergy, dt dr H = m( r + r θ + r si θϕ ) + m + kr, dt the total mometum, P mv mv m dr = + =, dt

2 ad the orbital agular mometum l = mr θ (We ca choose, if we wish, ϕ= ) c) I a circular orbit we have l Heff = mη + + k r +η 4m r +η r =, mrθ = kr we see that the effective potetial has a miimum at about that miimum we fid 9 l 7 mη+ k+ m k 4 η η+ η= mr so the motio is stable If we express the eergy as The differetial equatio describig a simple harmoic oscillator is r = 4 l ; expadig km x+ω x= a) By rescalig the time, t λ t, trasform the equatio to the form dx + x = dt dx Clearly, +ω x = ; with λω = we obtai the desired equatio λ dt b) What is λ? λ= ω dx c) If you were to plot the velocity, v = as a dt fuctio of x, what would the resultig x= Acos t +ϕ, curve look like? Sice v= Asi ( t +ϕ) Therefore as t icreases, the plot looks like a clockwise circle, as show to the right d) Suppose the equatio had a dampig term, dv v x dt + γ + = What would the graph of v vs x look like the? The solutio is the of the form γ t x= Ae cos( Ω t +φ) γ t γ t v= γae cos Ω t+φ ΩAe si Ω t+φ

3 where Ω= γ The graph is therefore a clockwise-rotatig iward spiral, as show to the right Recall how we derived a equatio of motio for the strig by cosiderig lumps of mass m = µ ξ separated by a distace ξ ad joied by massless strigs of tesio T, as show to the right a) Derive the coupled equatios of motio for the trasverse ψ t of the displacemets lumps ψ ψ Clearly, m + ψ ψ ψ = T T where we have writte ξ ξ the trasverse compoets of the restorig force i terms of the tesio ad the displacemets ψ ad the ve- b) Write the Lagragia ad the Hamiltoia i terms of ( t) locities ψ ( t) The Lagragia for the above system is T L= m ψ ψ+ ψ ξ ad the correspodig Hamiltoia is T H = m ψ + ψ+ ψ ξ c) Now cosider what happes whe the spacig ξ is allowed to become arbitrarily small Sums become itegrals, ξ L ξ L f dxf( x), = ξ where ξ x Thus fid a expressio for the Lagragia ad the Hamiltoia of the cotiuous strig, that is i the limit ψ ( t) ψ( ξ, t) =ψ ( xt, ) ψ I this limit, ψ+ ψ ξ so that x

4 ad (, t) (, t) ψ ξ ψ ξ L µ ξ T ξ t x ( xt, ) ψ ( xt, ) ψ l dx µ T t x ( xt, ) ψ ( xt, ) ψ H l dx µ + T t x 4 What are the momets of iertia of the followig objects? a) A circular hoop of radius R ad mass M, rotatig about ay diameter The MI of a circular hoop about a axis perpedicular to the plae ad through the ceter of the circle is I = MR By the perpedicular axis theorem, I = I + I = I Thus xx yy xx I I MR xx = = b) The pricipal momets of a uiform solid whose surface is give by the equatio x + y z + = a b Clearly I =ρ dx x + y Ixx =ρ dx z + y =ρ dxz + I If we chage variables to x + y = a η, z= bζ we fid η 4 I = πρab dηη dζ η 4 = πρab η d η η 8π 4 = ρab 5 Sice 4π M=ρ dx= πρab d η η = ρab we have I == Ma 5 Similarly,,

5 Ixx = I +ρ dxz = Ma + πρab 5 d η η = M( a + b ) 5 ( ) c) A pedulum cosistig of a thi uiform rod of mass m ad legth l, to oe ed of which is fixed a sphere of mass M ad radius R, about a pivot at the other ed (that is, the ed without the sphere), as show to the right The MI of the rod is 5 Ml, ad that of the sphere about its ceter is MR We ow exploit the parallel axis theorem to write the MI of the 5 MR + M l + R so the etire MI is their sum, l ( l ) 5 sphere about the pivot as I= M + MR + M + R 5 Two masses, m ad M, are coected by a strig of legth l that passes (frictiolessly) through a hole i a table That is, oe mass slides without frictio o the table, whereas the other hags vertically below the table Write the Lagragia ad fid the equatios of motio, costats of the motio, etc Assumig it is M that is daglig, the Lagragia is L= Mz + m( r + r θ ) Mgr + costat we igore the costat We have take the horizotal distace of m from the hole to be r Clearly, if m gets closer to the hole, M drops by the same amout, so z = r The coordiate θ is cyclic, so λ= mr θ is coserved The equatio of motio for r is λ ( M+ m) r = Mg mr We ca itegrate this oce to obtai the (coserved) eergy, λ ( M+ m) r + + Mgr = E mr

6 a) Uder what iitial coditios does the system exhibit stable oscillatios? λ As log as λ the effective potetial + Mgr has a sigle miimum, ad becomes ifiite as r Hece stable oscillatios about the mr miimum are possible b) Provide a qualitative physical explaatio for this pheomeo, i terms of forces, coservatio laws, ad so forth Basically, coservatio of agular mometum guaratees that as log as λ, the effective potetial becomes ifiitely repulsive as m approaches the hole The agular frequecy θ icreases as r so that the cetrifugal force icreases as r This meas the cetrifugal force ca always grow large eough to support the weight Mg that is, a stable orbit is always possible 6 The Great Pyramid of Egypt is about 48 feet high It is composed of roughly 8 millio limestoe blocks weighig about 5 lb a) What is the work eeded to assemble the pyramid, eglectig frictio? The work eeded to assemble the pyramid is basically that eeded to raise the blocks, ie mgz for a block raised to height z We may write h M= ρ dx= ρa z dz h W = ρ gzdx = ρgza z dz z A( z) = A h which gives M=ρAh ( ζ) dζ= ρah W = gρah ζ( ζ) dζ= gρ Ah = Mgh 4 Puttig i the umbers ad covertig to Joules, we get J b) Assumig a huma beig ca provide a steady mechaical output of 5 watts for a 8-hour day, how may ma-days will be required to assemble the pyramid? The mechaical output of a laborer is 6 44 J/day so that about 6 6 ma-days are required for the costructio

7 c) Commet o the statemet by the Greek historia Herodotus, that a work force of, me labored for years to build the pyramid If we take the time as years, oly some laborers would be required Assumig a equal umber worked at quarryig the stoe ad a equal umber at ferryig the stoes to the site, we still have fewer tha, workers total Allowig for the fact that these were probably ot slaves but peasats employed betwee harvest ad the ext platig, ie oly half a year, we would the eed twice as may workers at most, or, The umber give by Herodotus is too large by 5- This is ot so surprisig, sice he was oly repeatig what he was told by a Egyptia tour guide livig about years after the Great Pyramid was built!

Problem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to:

Problem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to: 2.003 Egieerig Dyamics Problem Set 9--Solutio Problem 1 Fid the equatio of motio for the system show with respect to: a) Zero sprig force positio. Draw the appropriate free body diagram. b) Static equilibrium