Phys 325 Homework 1A Name: __________________________Due: Thursday Jan 29, 2015 by 1pm (lecture or 325 box)1. (10 points) A force F(t) = 20 t3 acts on a particle of unit mass. At time t = 2 the particle isat x = 3 with velocity 4.• What is x(t) for later times t?2. (10 points) A particle of mass m is observed to be moving along the x-axis with avelocity that varies with the displacement x according to v(x) = Vo exp(βx), where β andVo are constants.• What are the physical dimensions of Vo and β ?• Find the force F(x) acting on the particle as a function of x and the given constants.• Check your expression for F(x) for dimensional consistency.3. (15 points) A particle of mass m moves in 1-d in a potential U(x) = A/x2 - B /xwith A and B positive.• Sketch a plot of U(x) for the case A=B =1.• For arbitrary positive A and B, construct the corresponding Force F(x) = - dU/dx.Sketch F(x) for the case A=B=1.• For arbitrary positive A and B, find the equilibrium point xeq and determine (byexamining U" there) whether or not motion near that point is stable.• The particle starts at position xo > 0 and speed vo. Find the inequality relatingA, B, m, xo and vo necessary and sufficient to assure that the particle's subsequent motionis bounded and periodic.4. (15 points) A particle of mass m moves in 1-d with a force fieldF(x) = -Ax exp(x2/a2) where a has units of length.• What is the associated U(x) ?•Show that x= 0 is an equilibrium point.•Find the period 2π/ω of oscillations assuming that deviations from equilibrium are small(|x| << a.) You may wish to use the methods of p 13-16 of the lecture notes.Phys 325 Homework 1B Name: ____________________________Due: Thursday Jan 29, 2015 by 1pm (lecture or 325 box)1. (15 points) A particle of mass m is under the influence of a force F that varies withtime and velocity likeF(t,v) =- k t2 vwhere k is a positive constant.• If the particle passes through the origin (x= 0) at time t = 0, with speed vo, what is thesubsequent velocity v(t) of the particle?• How much time does it take to stop?• Find an integral expression for the distance it goes ( you needn't evaluate the integral)2. (10 points) A particle of mass m is attracted to the origin under the influence of a forceF(x)F(x) = -κx-2where x is the distance from a fixed origin at x = 0 and κ is a positive constant.• It is released from rest with zero speed at x = d > 0. How much time does it take toreach the origin? (one of your integrals may respond to a clever substitution; try x = dsin2θ.)3. (15 points) A block with mass m slides on a horizontal surface. The block is initially(t=0) at x = 0 with (positive) velocity v0 . The drag force that the block experiences isproportional to the β th power of speed such thatF(v) = - k sgn(v) |v|βfor positive constant k and β ≥ 0 .The case β = 0 corresponds to a constant de-acceleration k/m, so the block comesto a stop at x = m vo2/ 2k at time vo m/k. (You may wish to confirm this for yourself)β = 1 corresponds to the case of linear viscous damping discussed in class, and tothe block going a distance vo m / k before stopping (and taking infinite time to do so).• If β =1.5, how much time does it take to stop and what will the sliding distance be?• Find the stopping time and the sliding distance if β = 2.4. (10 points) A particle of charge q and mass m is injected withvelocity rv = (vxoˆi + vyoˆj + vzoˆk) into a constant magnetic field rB = Bˆk.• What is the pitch of the resulting helical motion? ( Pitch is theamount of rise per turn of a helix.)Optional: you may also wish to determine the diameter of theresulting
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