Phys 325 Homework 2A Name: ____________________________Due: Feb 5, 2015 by 1pm (lecture or 325 box)1. (10 points) For each of the following force functions, determine whether or not it represents aconservative force. If conservative, find the potential energy function . U(x,y,z)a) rF = x3ˆi + y−2ˆj + z4ˆkb) rF = xy3ˆi + x2ˆj + z4ˆkc) rF = exp{z}ˆi + 2yˆj + x exp{z}ˆkd) rF = 2x cos(αz)ˆi + z2αsin(αy)ˆj + (2zsin(αy) − x2αsin(αz))ˆk2. (10 points) A particle of mass m moves in 3-d in response to a force fieldF(x) = Az3 i + Ay3 j + 3Axz2 kIts speed is vo when it passes through the origin at (0,0,0).• Later on it is found to pass through position r = -i + 2j + 3k. Use conservation of energy tofind its speed there.3. (20 points) (Adapted from Taylor Pb 3.11) A rocket launches vertically from the earth withinitial total mass mo burning fuel at a constant rate k ( kg/sec) (so that m(t) = mo –kt) with exhaustvelocity u. It starts at speed vo = 0.The governing ODE is m(t)&v = ku − m(t)g• Solve this ODE for v(t) ( You can use separation of variables)• For the case mo = 2 x 106 kg and k = 104 kg/sec and u =3000 m/sec (roughlycorresponding to the space shuttle) , find its speed at t = 100 seconds. What was its initialacceleration? What was its acceleration at t = 100 seconds?• What is the value of your expression for the speed at t = 200 seconds ? What is wrongwith applying your analysis at t = 200?• Describe qualitatively what happens if (like the Saturn V) the burn rate k were smaller,such that the right sided of the ODE were (initially) negative) ? The above differential equationsays that the acceleration would be initially downwards! So where was the error?Homework 2B Name: ____________________________Due: Feb 5, 2015 by 1pm (lecture or 325 box)1. (25 points) cf Lecture Notes but with friction. Abead slides along a rough rigid rod with a coefficientof dry friction µs = µk > 0. The rod is rotating at aconstant rate dφ/dt = ω. Neglect gravity.• Find the differential equation that governs the radialmotion r(t).You will need the formula for acceleration in polarcoordinates: ra = (&&r(t) − r(t )&φ2)ˆr + (r(t )&&φ+ 2&r&φ)ˆφYou will also need to recognize that the total force on the bead is simply a normal force Nˆφplus a friction force −µkN sign(&r)ˆr that acts in the radial direction. You don't know Napriori but you do have an equation for it terms of the bead's motion.You should get a linear constant coefficient homogeneous ODE of the form A&&r + B&r + Cr = 0where A and B and C are given in terms of the system parameters m, µ and ω. and in which therearen't any non-analytic factors like absolute values or sgn functions. (you may wish to use theidentity sign(x) |x| = x. )• Find two basis solutions of this differential equation by trying r(t) = exp(λt) and determiningthe two values λ1 and λ2 that satisfy it.• Form a linear combination of those two basis solutionsr(t) = a exp(λ1t) + b exp(λ2 t)and then construct the values of a and b, and hence r(t) , for the case of initial conditions suchthat at t = 0, r = ro, and dr/dt = 0.2. (15 points) A particle of mass m slides (bothsideways and radially) on a smooth frictionlesshorizontal table. It is attached to a cord that is beingpulled downwards at a prescribed constant speed v bya force T. ( T may be varying.)• Use F=ma in polar coordinates to derive anexpression for the tension T of the cord pulling on theparticle (T will depend on the particle's coordinates rand φ and how they may be changing)• Show that the particle's polar coordinates{r(t), φ(t)} satisfy r2 dφ/dt = constant.Hint: the only horizontal force on the particle is T and it acts purely in the inward radialdirection. Also dr/dt is known, equal to
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