# Qual Study

##### by Brian Rayburn 06/22/2011

# Qual Study

by Brian Rayburn# 1. Thermo/StatMech

## 1.1. Black Body Radiation

### 1.1.1. Winter 2008 Prob 1

1.1.1.1. 1. The concept of a black body radiator is central to many areas of physics including solid state physics, atmospheric radiation and astrophysics. a) [3 pts] Explain what is meant by the term "black body" and why it is important. b) [4 pts] Give an expression for the energy density of a black body as a function of wavelength or frequency (your choice). c) [3 pts] Explain how the expression given in part (b) is relevant to the color and absolute (range-compensated) brightness of a star.

### 1.1.2. Winter 2007 Prob 2

1.1.2.1. Prove the total power radiated from a black-body is: Powerlunit area = a p w h e r e a is a constant. (Possible useful data: Boltzmann's constant kb = 1.38x10-l6 ergI0K; n4 e x - 1 15 (10 point) ->

### 1.1.3. Winter 2007 Prob 3

1.1.3.1. The purpose of this problem is to calculate the temperature of the earth. The following simplifying assumptions will be made: (1) The only source of heat for the earth is the sun, which radiates as a black-body having a temperature of Ts 6000°K;(2) The earth has the same temperature everywhere, and also behaves as a black-body. (Possible useful km, 1 data: Radius of the sun Rs = 7x10~ Distance from the sun to earth D = 1 . 5 ~o8km, .3 Boltzmann'sconstantkb=1.38x10~16ergPK.) f?=dik5 o C Ea~-11-. G . Y r t a k* (a) Calculate the total power radiated by the sun. (4 point) (b) Calculate the temperature of the earth. (4 point) (c) List three possible complicating factors that may affect the validity of the assumption used above. (2 point)

### 1.1.4. Fa04 Pr4

1.1.4.1. The power radiated per unit area of a blackbody in the frequency range between v and v + d v can be expressed as RT(v)dv, where the spectral radiance is given by RT(v)= ~~~V'E(V)IC', ~ ( v is the average energy of an electromagnetic where ) standing wave of frequency v. (8). (a). Write down the expression for the average energy ~ ( v ) . [You may just state the result, but you can derive it if necessary.] (b). Show that the total power radiated per unit area of the blackbody is given by RT = f14, give an expression for the Stefan-Boltzmann constant o in terms and of fundamental constants and the value of a dimensionless definite integral. (c). Show that the spectral radiance R,.(v) is a function of the form R,.(v) = g ( T )f (V IT). Use this result to show that v,,, a T , where v,, is the

## 1.2. Ensembles

### 1.2.1. Winter 2008 Prob 2

1.2.1.1. 2. a) [5 pts] Briefly describe the three ensembles used for most statistical mechanics problems, the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble. Pay particular attention to the differences. b) [5 pts] Does analysis of an ideal gas using the three ensembles lead to the same ideal gas law? Explain why or why not.

### 1.2.2. Winter 2007 Prob 1

1.2.2.1. N asymmetric atoms are arranged along a straight line to form a chain molecule. Each atom is assumed to be capable of being either in an A state or a B state. In an A state the length contributed to the chain is La and the energy is E,. In a B state the length contributed to the chain is Lb and the energy is Eb. (a) Derive the relation between the length L of the chain molecule and the tension T applied between both ends of the molecule. Use the canonical ensemble at constant tension. (5 point) (b) Can this system exhibit a negative temperature, and if so for T small and negative, which state is preferred? (5 point)

## 1.3. Other

### 1.3.1. Winter 2008 Prob 3

1.3.1.1. 3. Consider a closed rectangular box held at temperature T somewhere near room temperature. The box has a base with sides of length a and height of length L. The box is filled with helium and a gravitational force mg is acting on each atom. The gravitational potential energy for an atom is mgz where z is the vertical height of the atom measured from the bottom of the box. You may neglect quantum effects. a) [I pt] Write down the classical Hamiltonian for a single atom of the gas in this box. b) [ l pt] Write down and evaluate the phase-space integral for the classical partition function for a single atom in the gas. c) [l pt] Write down an expression for the probability of finding the atom between z and z + dz in the box. You do not need to evaluate the expression. d) [l pt] Now write down the N-particle classical partition function for the gas in the box, and briefly state the relevant assumptions made. e) [3 pts] Find expressions for the average energy and specific heat of the gas and examine the limits for a very short box and a very tall box. f) [3 pts] Discuss the applicability of your result to the air in this room. Be succinct.

### 1.3.2. F04 Pr2

1.3.2.1. At room temperature T, a gas of N molecules, each with mass M, is confined within a cylinder of radius b and length L. If the cylinder is rotated at an angular velocity w about its axis of symmetry, find the density of the gas as a function of the radial distance from the axis. Neglect gravitational and transient effects. (6). (10 points)

## 1.4. Ideal Gas

### 1.4.1. Fall 2007 Prob 1

1.4.1.1. Problem ( ) 4. A planet with mass m rotates around our sun. The distance of closest approach of the planet to the sun is measured to be I; and the speed of the planet at this point is measured to be v, . (a). What two physical quantities are conserved (are constants) as the planet rotates around the sun? [2 points] (b). Find expressions for the conserved quantities in part (a) in terms of the known quantities [2 points] my r,, and v,. (c). Find expressions for the maximurn distance r, of the planet from the sun and the speed v, of the planet at this point in terms of the quantities in (a). [4 points] (d). Find an expression for the period of the planet in terms of the quantities in (a). [2 points]

### 1.4.2. F05 Pr3

1.4.2.1. A monatomic ideal gas at temperature Ti is initially contained in the left-hand side of a well-ins~~lated while at the same time the right-hand side of the box is evacuated. At box, a later time the internal partition is rapidly removed, and the system is allowed to come to equilibrium. To a first approximation, there is no heat exchanged with the walls. (6), Present your reaso.ning as you answer each of the following questions. (a). When equilibrium is reached, what fraction of the gas atoms will b e in each half of the box? What is the final temperature of the gas? What is the final pressure of the gas? (4 points) (b). What is the relationship between the initial and final entropy of the system? (6 points)

### 1.4.3. F04 Pr1

1.4.3.1. A mole of an ideal gas is expanded from volume V, to volume 2V, through two processes: (1)reversible isothermal expansion, and (2) free expansion. (5). Find the entropy changes of the gas, the heat bath, and the universe in both cases. (10 points)

## 1.5. Equilibrium

### 1.5.1. Fall 2007 Prob 2

1.5.1.1. Two equal blocks of copper (mass m) at temperatures TI and T2 are brought into thermal contact while isolated from the rest of the universe. a. Determine the final temperature of the system after it has come to thermal equilibrium (show how to get it!). b. Find the entropy change of the system and show that AS > 0 for ally finite temperatures T I # T2.

### 1.5.2. W05 Pr1 (equipartition theory)

1.5.2.1. Consider a classical system in contact wit11 n thermal ~.ese~.voj~. has ollly n that linear degree of freedom, The reservoir is held at the fixed Ice1vin temperat~ture T. For example, let the energy be E = cq, where c is a constailt ~ n c q is a pos.itive l generalized cool.dinate, (7), Use the eq~ipartition tl~eore~n calculate the average energy for tllis system, to (10 points)

## 1.6. Partition Functions

### 1.6.1. Fall 2007 Prob 3

1.6.1.1. One can use basic statistical mechanics to model the absorption of oxygen in the air by hemoglobin, and the effect of having a small amount of carbon monoxide in the air. One can do this by looking at the possible states of a single hemoglobin molecule: no gas molecules bonded to it, an oxygen molecule bonded to it, or a carbon monoxide molecule bonded to it. The energies of the system for these possibilities are 0, -E, and -E' respectively. a. Determine the grand partition function for the system if no carbon monoxide is present. b. The chemical potential of oxygen in the atmosphere is approximately -0.6 eV. Explain why this value can be used as the chemical potential in the oxygen-hemoglobin system in the lungs. Use this value, and the binding energies E = 0.7 eV and E' = 0.9 eV to determine the numeric probability of an oxygen molecule bonded to a hemoglobin molecule in the absence of carbon monoxide in the air at room temperature. c. If carbon monoxide is present in a ratio of 1:100 compared to oxygen then its chemical potential in the air is -0.7 eV (due to its low concentration). Find the new probability of an oxygen molecule bonded to a hemoglobin molecule when carbon monoxide is also present at room temperature.

### 1.6.2. Fa04 Pr3

1.6.2.1. Statistical Mechanics The velocity of molecules in a gas is described in spherical coordinates. Write down the Maxwell-Boltzmann distribution p(v) for the molecular speed v of the (7a). molecules. Note that (7b). Sketch p(v) vs. v. What is the average molecular speed? (7c). What is the most probable molecular speed? (7d). Use the theorem of equipartition of energy to find the variance <vZ> and the RMS (root mean square) molecular speed. (7e). J p.(v) dv = 1 0 What is the probability that a molecule has a speed greater than some value vo? Note: Leave your answer in the form of an integral. (70. n t : 7 - I = exp(-m21m = Jn a dl -- = r2exp(-a')&. and da J - (10 points)

## 1.7. Engine Cycles

### 1.7.1. Fall 2006 Prob 1

1.7.1.1. A uniform ladder leans against a smooth vertical wall. If the floor is also smooth, and the initial angle between the floor and the ladder is 0 , show that the ladder, in sliding down, will lose contact with the wall when the angle between the floor and the ladder is . 3. 0 sm -1[2 sm ()] •

### 1.7.2. Fall 2006 Prob 2

1.7.2.1. Almost every refrigerator uses some version of a Joule-Thomson process to lower the pressure and cool the working material during the cycle. This adiabatic process can be modeled by pushing the working material through a porous plug that allows a high pressure on one side of the plug and a low pressure on the other side of the plug, with the material leaking through the plug. a. Determine the change in energy of the gas from it being on the left side of the diagram to it being on the right side of the diagram. (4 points) b. Show that the enthalpy is constant during this process. (3 points) c. Show that the working material cannot be an ideal gas for this process to cool the working material. (2 points) d. Describe why a non-ideal gas can be the working material cooled in this process. (1 points)

## 1.8. Chemical Potential

### 1.8.1. Fall 2006 Prob 3

1.8.1.1. Almost every refrigerator uses some version of a Joule-Thomson process to lower the pressure and cool the working material during the cycle. This adiabatic process can be modeled by pushing the working material through a porous plug that allows a high pressure on one side of the plug and a low pressure on the other side of the plug, with the material leaking through the plug. a. Determine the change in energy of the gas from it being on the left side of the diagram to it being on the right side of the diagram. (4 points) b. Show that the enthalpy is constant during this process. (3 points) c. Show that the working material cannot be an ideal gas for this process to cool the working material. (2 points) d. Describe why a non-ideal gas can be the working material cooled in this process. (1 points)

## 1.9. Quantum Statistics

### 1.9.1. F05 Pr1

1.9.1.1. Consider a quantum system whose lowest energy states have the following eigenenergies Enand degeneracies g,,: (4). Write down the partition function. [I point] (a). Assuming that the series converges rapidly enough so that it is possible to neglect those terms that pertain to states n 2 3, at what temperature would you feel justified in neglecting states with n 2 3 1 Justify your answer. [3 points] (b). Find the average energy of the system for temperatures below the "critical" temperature T, found in part (b), If the "system" happened to be an atom, estimate a numerical value for and for the critical temperature. [3 points] (c). Find and plot the specific heat of the system for T s 2 T,, i.e., extend the plot above T . , What feature of the plot is caused by the truncation of the partition function in (b)? [3 points] (d). (10 points)

### 1.9.2. F05 Pr2

1.9.2.1. Consider a system that has two identical particles in it and that has three single-particle quantum states, each with the same energy. (5). (a). List all of tlie possible states of the system if the particles are treated as (al) classical particles, (a2) fermions, (a3) bosons. [2 points each] (b). What is tlie entropy of tlie system in each of the cases in (a)? [4 points]. (10 points)

### 1.9.3. W05 Pr2

1.9.3.1. If w e take the energy of the ground state of the hydrogen atom t o be zero, then the first excited state has an energy of 10.2 eV. We are curious as t o what percentage of hydrogen atoms are in the ground state or first excited state at various ternperat~nfes. a first approximation, lets consider-just one atom.Recall that the As quailt~tum n~nrribers the hydrogen-atom are n , 1, and 112. for Note: 16 = 8 . 6 2 ~ 1 0 - ~ e ~ I 1 ~ . (8), How many independent states of equal energy can a hydrogen atom in its first excited state have? (a). (b). %That .is the entropy of the hydrogen atom in its first excited state? (c). Calculate the I-Ielrnholtz free Calculate the I-Iel~nholtz energy of the first: excited state level for any free t.emperature T, (cl). of the gsouncl state for any temperature T does to (e). At MIhat te1nlxsilt111.e~ the atom p~.efel- be in the ground state, wd at what tein]~eratui.es does the atom prcfe~. be .in the excited state? Explaill your answer, to on and cotunrne~it whether 01. not the I-I-atoms on tile surface of the sun are liltely t.o be iu the gsound state or tile first excited state.

# 2. Classical

## 2.1. Rotation

### 2.1.1. W2008 Prob 2

2.1.1.1. A small ring is free to move along a smooth wire of y=f(x), where y=f(x) is the shape of the wire) and x-axis is horizontal. The wire is rotating about the y-axis with a constant Find a function of y=f(x) so that the ring may stay in any position. angular velocity of o. What is the normal force in this case?

### 2.1.2. W2008 Prob 1

2.1.2.1. Particle A of mass m is in motion on a smooth horizontal plane. Particle B of mass m is under the plane connected to particle A by a light rope through a smooth pinhole located at point 0 of the plane. Assuming the distance between particle A and the pinhole 0 is b which at time t=O and particle A has initial velocity ~ = ( 9 ~ b / 2 ) ' ~ is perpendicular to OA. Find the equation of motion for particle A and show that the motion of particle A will be restricted in the range of b to 3b from pinhole 0.

### 2.1.3. F05 Pr3

2.1.3.1. Three identical point particles, each having a mass m, are connected together by massless rods so that the positions of the three particles in a laboratory frame are given by I; = az , r, = & a? , and y3 = aS,+ 2a? , where a is a constant and i a unit vector + is pointing along the positive x-axis, etc, (3), - A - (a), Compute the inertia tensor for this system using the laboratory frame of reference. (b), If the system were rotated with angular speed w, along an axis in the laboratory frame given by A,, laboratory frame. =- [ 4 1 c g ] , compute the angular momentum of the system in the & (c). There are three axes in the laboratory frame about which you can rotate this object and find the angular momentum and angular velocity to be co-linear. Calculate the direction of any one of them. (10 points)

## 2.2. Osciallators

### 2.2.1. W2008 Prob 3

2.2.1.1. A compound pendulum is made by a thin rod of mass m and length 2b. The end of the rod is constrained to slide along a smooth horizontal x-axis. Find the motion of the pendulum and the time period of the pendulum in small oscillation.

### 2.2.2. F2007 Prob 2

2.2.2.1. A circular, uniform disk of radius b is frictionlessly pivoted through a point P on its circumference so that it hangs vertically in the Earth's gravitational field. Choose point P to be the origin of a Cartesian coordinate system with the y-axis pointing vertically and the x-axis pointing horizontally. When displaced from its equilibrium position, the disk moves in the x-y plane. (a). Compute the moment of inertia of the disk about an axis parallel to the z-axis passing through the disk's center of mass. [I point] (b). Find the moment of inertia of the disk about the z-axis axis (passing through P). [I point] In the following questions, assume that the disk is displaced a small angle 8 from its , equilibrium position and released from rest at time t = 0. (c). Draw a carefully labeled diagram of all of the forces that act on the disk, and write down the equations that result from applying Newton's laws to this problem. [4 points] (d). Find an expression for the period of oscillation of the disk. (e). Find expression for the vertical V and horizontal H forces that the pivot exerts on the disk as a function of tim [2 points]

### 2.2.3. F2007 Prob 3

2.2.3.1. A pendulum of mass ml and length b is ; attached to block m! that is constrained to move along a horizontal frictionless track as shown in the diagram on the right. The pendulum moves in a vertical plane containing the traclc. When both masses are at rest at their respective eq~~ilibriurn positions, x = xo. . - - - - . (a). Find the equations of motion of mi and m2. -. - . [5 points] In answering the following question, assume that the pendulum undergoes small oscillations about its equilibrium position. (b). Suppose that at time t = 0 the pendulum and the supporting block m2 are at their respective equilibrium positions with m2 at rest but with mlmoving towards the right with a speed of Yo. Find the functions x(t) and B(t) for any time t 0. [5 points]. 2

### 2.2.4. W2007 Prob 2

2.2.4.1. Problem 2. The Inass in the sketch to the right is supportecl by a massless spring of stiffness I;. The mass is also subject to a viscous retarcling force that is a linear function of the velocity si~ch might be caused by air drag at low as speeds. a) Write a differential equation that describes the motion of the mass about its equilibriu~~~ position (2 pts). ~l b) Find the general solutioll of ~ n o t i o for the Inass for arbitrary values of the spring stiffness ancl air resistance (2 pts). c) Draw a graph of the Inass displace~ne~lt versus time for the three possible situations of overdal~zpilzg, urzderclnnzpi~zg, and critical clanzl~irzg briefly describe the motion you have and drawn (2 pts). d) For the case of overdarizping, write the solution for the position of the mass as a function of time, describe how to obtain the integration constants in your solution, and cornlnent on the decay co~lstant contants in the solutio~l pts). or (4 +

### 2.2.5. F2006 Prob 1

2.2.5.1. A spring of stiffness k supports a box of mass M in which is placed a block of mass m. The system is pulled downward a distance d from the equilibrium position and then released. Neglect any air resistance. (a). (6 points) Find the force of reaction between the block and the bottom of the box as a function of time. (b). (4 points) For what value of d will the block just begin to leave the bottom of the box at the top of the vertical oscillations?

### 2.2.6. F2006 Prob 2 (different! mag fields)

2.2.6.1. An electron moves in a force field due to a uniform electric field E and a uniform magnetic field B which is at right angles to E. Let E = Y and B = E The initial position of the electron is at the origin with an initial velocity vo = xVo in the x-direction. Show that the resulting path of the electron is a cycloid whose parametric equations with respect to the elapsed time t have the form ofx(t) = A sine rot) + bt, yet) = A [l-cos( rot)], and z(t) = O. Find expressions for A and b in terms of the given quantities. zB.

### 2.2.7. W2006 Prob 1

2.2.7.1. For a damped, driven harmonic oscillator: a) (8 points) Find the steady state solution in the form x(t) A(w) expi(Ult-lfl), giving expressions for the amplitude A(w) 0 is real) and the phase angle <p.? b) (2 points) Find the resonant frequency, i.e. the frequency for which the amplitude A(ro) is a maximum. Under what condition is there no resonance? k.CC·\ =

### 2.2.8. W05 Pr1

2.2.8.1. A mass r 2moves along the top,frictionless su~face n of another mass In, as shown in the diagram on the right, The mass m,;is connected to one end of a spring with . spring.cdnsti~~t The other end of the spring is lc. anchored to mass m,that is free to move horizontally on africtionless s~u-face is coi~lcident that with the x-axis of a Cartesian coordinate system. The distance of the left end of m, from the origin of the coo~.d-inate system is x,, (g), .-..-. -. . .. .- . .__ , (a). Find the equations of motion involving x, and x2. (b). Find an equation for x2that does not involve x,. (c). With what frequency does the system oscillate? (10 points)

## 2.3. Central Force/Two Body

### 2.3.1. Fall 2007 Prob 1

2.3.1.1. Problem ( ) 4. A planet with mass m rotates around our sun. The distance of closest approach of the planet to the sun is measured to be I; and the speed of the planet at this point is measured to be v, . (a). What two physical quantities are conserved (are constants) as the planet rotates around the sun? [2 points] (b). Find expressions for the conserved quantities in part (a) in terms of the known quantities [2 points] my r,, and v,. (c). Find expressions for the maximurn distance r, of the planet from the sun and the speed v, of the planet at this point in terms of the quantities in (a). [4 points] (d). Find an expression for the period of the planet in terms of the quantities in (a). [2 points]

### 2.3.2. Winter 2007 Prob 1

2.3.2.1. Problem 1. Collsider the interaction between two neutral hydrogen atoms as they are brought together from infinite separation. When the atoins are far fro111each other they are slightly attracted to each other clue to dipole-dipole interactions (van cler Waals interaction). As they are brought even closer to each other the attraction increases. If they are brought very close together, the iliteraction becomes very strongly repulsive to prevent overlap of the atoms. In between these two extremes exists a minimum energy situation at an internuclear separation that corresponcls to the boncl length for the hydrogen molecule. a) Draw the potential as a function of the separation distance and label the important points on the graph (2pts). b) Write a general expression for the potential energy as a function of the separation distance using the parameters labeled in your graph (2 pts). c) We would like to study small amplitude oscillations (vibrations) about the equilibrium small displacelnents from the equilibrium position and separation distance R,. By consideri~lg using the binomial expansion, use your potential energy function to derive an expression for the restoring force under these srnall displacements and find an expression for the force constant (6 pts).

### 2.3.3. F05 Pr2

2.3.3.1. Two particles having masses m, and m, move so that their relative ve'locity is v and the velocity of their center of mass is V,. If ME m, -I-rn2is the total mass of the system and p n rn,m2/(m,tm2) its reduced mass, prove that the total kinetic energy T is given by is (2). T = (1/2)MV,-V, -t (1/2) pv-v. (10 points)

## 2.4. Vibration

### 2.4.1. Winter 2007 Prob 3

2.4.1.1. Problem 3. Carbon dioxide (CO,) is a linear tri-atomic molecule. In 01-derto relate some of the obsesvecl the is infrared (IR) spectroscopy of CO, to its struct~~re, molec~~le commonly ~llocleleclas three lnasses connected by two identical springs. Consicler only the notion in one dimension along the axis of the ~nolecule this entire problem. for of a) Describe the norinal ~nocles vibration for this system of Inasses and springs (2 pts). b) Write the Lagrailgian for this system and use Lagrange's equations to find the equations of motion for the three lnasses (4 pts). c) Solve the equatiolls of lnotion for the nortnal (eigen) frequencies and comment on which of the frequencies is associated with each of the normal modes you described in part (a) (4 pts').

## 2.5. Kinematics

### 2.5.1. Fall 2006 Prob 3

2.5.1.1. A uniform ladder leans against a smooth vertical wall. If the floor is also smooth, and the initial angle between the floor and the ladder is 0 , show that the ladder, in sliding down, will lose contact with the wall when the angle between the floor and the ladder is . 3. 0 sm -1[2 sm ()] •

### 2.5.2. W06 Pr2

2.5.2.1. A mass m is released from rest at the top of a building. The air exerts a drag force F d = - cy2 where Y=dz/dt and ceconstant. Define z as the distance fallen, i.e. take z=O at the top of the building. ~~t\<. 4 ~~I~cft~nts) Write down the~~~~n;;~o~n an~::;~ri~~~mllnal ct? ~ ~ b) (4 points) Transform the equation of motion into a differential equation for the change ~ in the kinetic energy as a function of z and find the solution. K£ = ~ m-Y"';:: t..,.,Icl~:z.=7 d\(G =- l-mJ1J7 . tJ£} &:t:- 2 d"b r!JIJ . c) (3 points) Verify that your solution reduces to the correct result in the limit C-7 O.

### 2.5.3. W06 Pr3

2.5.3.1. A rod with mass M and length R. rotates with angular velocity OJ = de/dt in ahorizorttaLplaneabout a fixed axis through point P. (The axis of rotation is perpendicular to the plane ofthe paper.) A bead ofmass m - - - - - p " - - - - - . : . . . - . . . . ! - - - slides freely along the rod; its distance along .• ,y the rod from the point P is ret). The moment t- 1J) ' , inz ab ' Y ,v/":1;q0lui) 0 f inertia ~ f a rod rotatm~ a, out one en d IS '/l'{o~l4 l ro(/ = MR. /3. Neglect fricticn. I l= i:Iw1- a) (4 point~~ Fin.d.the equations or,motion~~r~(t)~d 8(t) from the Lagrangian. -t"r ~(~1 I y-rO"'I'Gt)=Q . 'l, e '0"- I 'l-. :\ iM.<; 1" ~w ( \(l..,. 'l'~>-; \.-.-. ~ - _.._.-.__._ ... .. --....__.. - } ~ -t",,). 'i1'l&,ldI + Y'f\) ,t-J -+- ~(2 +() CLbB- ~ 0 - Q&ei l+eJt\ ~) (4 points) Show that ~he e~uation of ~otion fo~ SCt) is a statemen~ of the conservation ~(bf angular momentum for this system. Fmd~n terms of ret), OJ.o]-nd roo wher~"~~ = !OJ(t=O) and r o = r(t=O) are the initial values. Show that 00::::: 000 = constant when M » m:. \ 'F~~==-" v ~~)~~C~ c) (2 points) For this part, take OJ = OJo:;:: constant. Solve the equation of motion for ret) -flo subject to the initial conditions r(t . '-, - : ; ; get to the end of the rod (r= R.)? ~=~ t.. /. ~ 0) ~ r, r (t = 0) =O.! At what time t* does the bead dt oV" . I 1\ J

### 2.5.4. W05 Pr2

2.5.4.1. Two bloclcs of equal mass ITI are connected by a uniform, flexible cord of Inass 1-n, ancl length b. One block is placed 011 a horizonta1,j-ictiorzless table and the other block hangs over thefiictiolaless edge of the table as show11 i11 the cli agram 011 the right, Fincl the acceleration of the bloclcs ancl cord

## 2.6. Other

### 2.6.1. F05 Pr1

2.6.1.1. A right cylinder of mass m, length L, and cross-sectional area A Elo ats in a fluid of density p so that the axis of the cylinder is vertical. (1). (a) Using the level of the fluid as a reference level, what is the height of the top of the cylinder at equilibrium? (b). The cylinder is lifted a snzall anzownt A while still submerged and then released from rest, What i.s the expltsssion for the height of the top of the cylinder as a function of time after release? (Assume that the fluid is non-viscous, i.e., there is no friction exerted by the fluid on the cylinder.) (c). Now assume that the fluid is modified so that it becomes viscous and it exerts a frictional force linearly proportional to the cylinder's velocity, i.e., FviEcous V. =- What is the smallest value of that will ensure that the cylinder, when released the same as in part (b), never has its top drop below the level computed in part (a)? (10 points)

### 2.6.2. F04 Pr1 (Rotational Reference Frame/Coreolus Effect)

2.6.2.1. The Leaning Tower of Pisa (see figure on the right) is at 43.7ON.The Tower was recently reopened after a several-year effort to stop it from falling over. A UrvlBC student drops a ball from the top of the 56m high tower, which leans at an angle of 4' to the south as shown. The Coriolis acceleration of the Enrth is 2 & f , where fi is the angular velocity of the Earth about its axis of rotation and V is the instantaneous velocity of the ball. (a). Derive an expression for the displacement from the vertical where the ball lands. In what direction is h s displacement? (b). Compute the value of this displacement from your expression in (a).

### 2.6.3. F04 Pr2 (Satellite Orbit)

2.6.3.1. (10). A television commwjcations satellite is in geostationary orbit at 9O0W longitude over the United States. Another European communications..satellite fails, and the company needs to move the U.S. satellite to a position at 0' longitude (the Greenwich meridian). The company wants to rapidly change the orbit of the satellite so that it will reach 0' longitude in 2 weeks. (Assume that tbe satellite's transit time to its new orbit is very small with respect to two weeks.) How much does the radius of the orbit of the satellite need to be changed to malte this change? Is this change an increase or decrease? (10 points)