A. Use the idea that df = grad f · dr to work out the gradient of f in both spherical polar and in cylindrical coordinates

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A1. a) for point P on the Earth with longitude f from the Greenwich meridian, and latitude angle l from the equator, write down the x,y, and z components of a vector from the center of the Earth to point P.

b) Repeat for point P', with longitude f' and latitude l'.

c) Determine the angle G between the vectors of parts a) and b) in terms of f, f', l and l'.

d)Calculate (not just look up) the great circle distance from New York (40.72N, 74.00W) to Dublin, Ireland (53.20 N, 6.15W). [Note that a net search would give you this information, and an engine on the net will let you check your answer. I used HotBot for my search.]
 
 

B. (Begin in Lab) Use the dot product for the following calculations.
 
 

a) Find the angle W between the Earth's north direction, and the line from Earth to Sun. The Earth's axis is tilted to the plane of the Earth orbit by an angle a = 23.5°. You will need another angle (call it b) which runs from 0 to 2p as the Earth travels through its orbit during one year.

B1. For a vector A and some arbitrary direction whose unit vector is e, A can be written as the sum of two vectors, one of which is parallel to e, and one of which is perpendicular to e :

A = Aparallel + Aperpendicular .

a) Draw a sketch showing A, e, Aparallel and Aperpendicular .

b) Use the dot product and write Aparallel in terms of A and e .

c) Use the cross product and write Aperpendicular in terms of A and e.
 
 

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C. Work out the Lagrange equation of motion for an atwood machine with masses m1 and m2, and a massless, frictionless pulley of radius R. Start with the lagrangian function and work through the prescription for the equation of motion

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D. A 250 kg mass is to be dropped from 0.4 m onto a (massless) platform sitting on a spring whose force constant is 17000 N/m. You are to install a dashpot (damping force proportional to velocity) so that the platform will come to rest without overshoot in the shortest possible time. Determine the damping constant for the dashpot.

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E. A 10-g mass slides without friction along a circle of radius 0.5 m. There is a small (10^8 kg) black hole located 0.02 m from the perimeter of the circle. Use Lagrange's equations to determine the frequency of small oscillations of the 10-g mass on the circle due to the attraction of the small black hole. (Keep only the most important terms for small angles of oscillation about the line between the center of the circle and the black hole.)

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F. It has been stated that the vibrating Earth has a resonance period of 54 min and a Q of about 400. After a large earthquake the Earth 'rings' (continues to vibrate) for two months.

a) By what factor has the amplitude of vibrations decreased in that time?

b) By what factor has the energy of vibration decreased?

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G. An osage orange hangs in a tree, 60 ft above the ground. A strong wind blows it off the tree and 35 ft across a road where it strikes the side of a passing car. Use numerical methods to determine the wind speed for this to happen. Assume a turbulent drag force
 
 

F(drag) = 1/2 D A v^2,
 
 

where D is the density of the air, A is the cross-sectional area of the sphere, and v is the sphere's speed. [ An osage orange is about the size of a softball.]

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H. A proposed automobile design has the bumper effectively attached to the body of an 1100-kg car by a spring of constant 150 N/m. The body structure is capable of withstanding a 5.5 g's without suffering damage due to crumpling. The car must pass a 'crash test' at 5 mph with a stationary obstacle.

a) Is this a satisfactory design? State the reasons for your conclusion.

b) If the design is not satisfactory, improve the design to meet the crash test specs without serious objections to your modified design.
 
 

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I. When a ball of mass m is dropped just above a ball of mass M, the lighter ball (m<M) rebounds to a greater height than its release point if elastic collisions occur.

a) Discuss qualitatively what goes on in this situation, causing the lighter ball to rebound to a great height.

b) Consider the following situation: m = 1/3 kg and let M= 1 kg . To the lower side of each mass is attached a spring of constant k = 80000 N/m, and an unstretched length of 1/50 m. The masses are dropped with a separation greater than 1/50 m from a height of 1/2 m. M comes in contact with the ground first via its spring, and then some time later it comes in contact with mass m via the spring of mass m. These collisions are elastic, but not instantaneous. Discuss qualitatively how one might arrange for mass m to rebound to a great height, much greater than 1/2 m.

c) Adapt the 'contact.ms' program to model the collisions. [Bragging rights on this go to the person who achieves the highest rebound height of m, with m released from rest at 1/2 m.]

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J. Show that for an uniform oblate ellipsoid (slightly flattened at the poles; shaped slightly like a doorknob) the gravitational potential at distances far from the ellipsoid contains an additional term, when the potential is expanded to lowest order in the radius over the observation distance. Work out the exact form of this term (it is not proportional to 1/distance).

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K. A particle of mas m slides on a frictionless table top with a small hole in the center. The mass m is connected by a light cord through the hole to a mass M hanging directly beneath the hole. Let the mass m= 1/10 kg and M equal 2 kg. At t=0, let r= 1/2 m, and dr/dt =1/4 m/s and the angular velocity be 3/2 radian/second.

a) Write down the lagrangian of this system, and obtain the equations of motion.

b) Determine the constants of the motion.

c) Determine (before going to the full Maple simulation) what the maximum and minimum values are for the distance of m from the center of the table.

d) Carry out the simulation, and turn in a plot for motions over at least two orbits of m around the center.

e) Compare the max and min values of r in part c) to those in part d).

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M. The eccentricity of the Earth's orbit around the Sun is given as 0.0167. The slightly elliptical orbit means while that the Earth is sweeping out equal areas in equal times, it does not advance by the same angle each day around the sun. (Its angular velocity is a faster closer to the sun, and slower farther away). If the Earth's orbit were perfectly circular, local apparent noon (LAN), the time the sun is directly overhead would be the same every day of the year. Since the orbit is slightly elliptical, the time of LAN (take it to be halfway between sunrise and sunset) varies during the year. Work out the variation of LAN during the year (to first order in the Earth's eccentricity), showing the number of minutes of variation there is (the latest LAN minus the earliest LAN).

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N. An auto has brakes on all four wheels. Select reasonable values for tire static friction with the ground, and geometrical factors for the auto.

a) Draw a neat sketch of the forces exerted during braking, both horizontal and vertical.

b) determine the maximum deceleration during braking. (Torque with respect to auto CM comes into play.).

c) the braking force exerted by the front wheels on the ground

d) the braking force exerted by the rear wheels on the ground.

e) comment on the practice of putting disc brakes on the front and drum brakes on the rear wheels of many autos.

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P. Consider a uniform rectangular plate of sides a and b, which is forced to rotate about an axis through a diagonal of the plate with an angular velocity W.

a) Show that the principal moments of inertia of this plate are (M/12) a^2, (M/12)b^2, and (M/12)(a^2+b^2).

b) Identify the direction of torque needed to rotate the plate.

c) Find the magnitude of the torque, in terms of M, a, b, and W.

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Q. A symmetrical rigid body is powered in free space by two jet engines which supply a constant torque N about the z-axis (the symmetry axis) of the body. Find the general solution for the angular velocity as a function of time, and describe how the angular velocity vector moves with respect to the body axes.

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R. A football has principal moments of inertia in the ratio of 1.7:1. This ball has been thrown poorly, so it acts wobbly, with the nose of the ball making an angle to the angular momentum direction of 25 degrees. If the nose of the ball makes one revolution about the direction of angular momentum every 0.8 seconds, determine the rate at which the ball rotates about its own symmetry axis (psi-dot).

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S. The fractional difference in the Earth's moments of inertia is about 1/300, the Earth being slightly flattened at the poles. Use this information to calculate the rate at which the Earth's angular velocity vector precesses (wobbles) about the direction of the Earth's rotational angular momentum (you could say this is the rotational North pole)

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V. An object has principal moments of inertia I1, I2, and I3 about its principal axes. This object is constrained to rotate about a single direction such that the direction cosines of the three principal axes to this direction are g1, g2, and g3. The center of mass is located a distance D below the axis of rotation. Find the frequency of small oscillations in terms of gravity, the mass M of the body, and the parameters given above.

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Y. Consider a particle constrained to move on a cylindrical surface, the defining equation of which is
 
 

x^2 + y^2 = R^2.
 
 

The particle of mass M is subject to a force proportional to the distance from the origin.
 
 

a) Show that the potential energy function is U = k (R^2 + z^2).

b) Write out the Lagrangian for this system in cylindrical coordinates.

c) Write out the Hamiltonian (it may not equal the total energy) and obtain Hamilton's equations of motion

d) Show that Lagrange's equations and Hamilton's equations are identical for this system. (Do not solve the equations)

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Z. A uniform solid sphere of mass M and radius R rolls without slipping on a turntable which is rotating at a uniform angular frequency W. Show that the path of the sphere CM is a circle, and that the angular velocity of the sphere CM in this path is 2W/7. [Hard problem, can be tackled with use of vectors, cross products, derivatives, etc.]

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AA. a) Do an analysis to find the frequency of oscillation of a mass M at the bottom of a bowl of radius R. b) Sanity-check your answer to part a) by comparing to the frequency of a well-known physical system which is equivalent to the mass in the bowl.
 
 

AB. Find the frequency of oscillation of a sphere of mass M and radius r which rolls without slipping at the bottom of a bowl of radius R.
 
 

AC.Consider the frequencies of oscillation of a 1-kg mass in the following 1-D potential
 
 

U(x) = 3*sin(x) + 2/(x+0.2) + x, in joules.
 
 

a) Find the potential minima accurately (start with a plot to get your bearings.

b) Find the frequency of small oscillations around each minimum between x=0 and x=5.
 
 

AD. A proton (m = 1.67 x 10^-27 kg, q = +1.6 x 10^-19 C) with velocity 2.0 x 10^7 m/s heads directly for a stationary proton which is free to move. Determine the distance of closest approach between the protons.
 
 

AE. Determine the time for the two protons in the previous problem to go from a separation of 1.0 m to the closest approach distance.
 
 

AF. Model a water molecule having a mass 16 u at the origin, a mass u a distance L along the x-axis, and a mass u at a distance L from the origin at an angle of 110° fromthe x-axis.

a) Calculate the CM of this system.

b) Calculate the inertia tensor for the water molecule in a coordinate system centered on the molecule's CM, with axes parallel to the x and y axes.

c) Perform a similarity transformation on the inertia tensor by going to a coordinate system rotated by 55°. This should result in a diagonal tensor, giving the principal moments of the water molecule. (The principal axes are supposed to lie along the symmetry directions of the body.)