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d Consider a potential expressed in spherical coordinates. What is the force arising from this potential Be sure to give your. answer in spherical coordinates,mg cos r mg sin,mg cos r sin. I should note that the direction vector is simply a spherical coordinate. description of k,2 15 points Consider the vector field. and a scalar field,a Compute the divergence of the vector field v. In Cartesian coordinates,b Compute the gradient of the scalar field f. 2xy i x2 j, c Now compute the curl of the result from the previous part f.

We proved this in general since the curl of a gradient is always zero. but to be sure,x uy z uz y 0 0 0,y uz x ux z 0 0 0. z ux y uy x 2x 2x 0, 3 15 points Consider a piece of wood in roughly the shape of an oar It. has a length L which is which has a density per unit Length of. a What is the mass of the rod,As a reminder the 1 d version of the integral is. b What is the center of mass,xCM x x dx, c What is the moment of inertia if rotated around the left end. Please give your answer in terms of M and L,Same basic idea.

4 15 points Consider the central force, which acts like a bungee cord pulling a planet in to the center of a solar. At some instant the planet is a distance R from the center and moving. with a purely tangential velocity v0, a In terms of k m and R what angular velocity 0 is required to. maintain the planet in a circular orbit,Hint A circular orbit requires r to be constant. As a reminder the radial acceleration is given by, b What is the angular momentum of the planet in terms of k m. The speed of the orbit is R so,mR2 0 R2 km, c The planet is then given a kick pointed purely in the radial direction.

maintaining the same angular momentum in case you missed it. Express as a function of r and, Hint There is definitely a conservation law at work. It s simply, d E C 3 points Following this write but do not solve a differential. equation describing the evolution of r incorporating only r and. Very importantly,is conserved,But our radial equation yields. Plugging in our result from the previous part, 5 20 points Consider two blocks of equal mass m connected by a spring. of constant k and confined to move in one dimension The entire system. moves without friction At equilibrium the spring has a length. a Write down the Lagrangian of this system as a function of x1 and x2. and their time derivatives Assume x2 x1, The Lagrangian is simply the sum of two free Lagrangians plus a.

coupling term,1 2 2 1 2 2 1,L m x 1 m x 2 k x2 x1 2. b Write the Euler Lagrange equations for this system. mx 1 k x2 x1,and for x2,mx 2 k x2 x1, Note that the sum of these two expressions guarantees overall con. servation of linear momentum,c Make the change of variables. Write the Euler Lagrange equations in these new variables. First note that,which simply gets rid of the So,x 21 x 22 2X 2. L mX 2 m k 2,and the Euler Lagrange equations, d What are the frequencies of oscillation if the two masses are out of.

phase with one another,Reading off from the previous equation for. Note that this is the exact solution that you d get by computing the. frequency for the reduced mass m 2, 6 15 points Okay This is the toughest problem even though it looks. innocuous Be careful and make sure you show your work I know you. I throw a marble of mass m downward through a fluid with a linear. viscous coefficient b very very fast, You don t need to compute the terminal velocity vT explicitly but I. would suggest doing so for your own sake I initially throw the marble at. a Write but do not solve a differential equation describing the velocity. v of the marble, As a suggestion you should define v as positive if it is going down. There is a simple force,which is canceled if, But writing the first version as a differential equation.

or cleaning up a little bit, b I care about you all deeply so I will tell you that the solution to the. motion has the form,v t A Bert, Solve explicitly for A B and r and determine the motion of the. marble at all times, Hint As a strong suggestion you should make careful note of terms. which vary over times and which ones don t,First note that. So plugging into the cleaned up version above, Since the term on the left has no time dependance we can say.

and similarly,Finally we can look at the initial conditions t 0. Just to summarize,v v0 vT e gt vT vT, c Sketch the velocity of the marble How long will it take before the. marble is moving at only v 2vT,First for convenience define. so the exponent is just,We re solving for,2vT 7vT e t vT. t ln 7 ln 7 1 95,I don t need exact labels just a trendline.

PHYSICS 311 Classical Mechanics Midterm Exam Solution Key 2019 1 20 points Short Answers 5 points each a Consider the complex number A 4 3i Please compute AA Sol First note A 4 3i so AA 16 9 25 b As best you are able in a sentence or a few words and an equation please describe a conservative force Sol A conservative force has a number of important

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