Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUS10.3 The general equation of motion of a (non-relativistic) particle of mass m andcharge q when it is placed in a region where there is a magnetic field B and anelectric field E ism¨r = q(E + ṙ × B);here r is the position of the particle at time t and ṙ = dr/dt, etc. Write this asthree separate equations in terms of the Cartesian components of the vectorsinvolved.For the simple case of crossed uniform fields E = Ei, B = Bj, inwhichtheparticle starts from the origin at t =0withṙ = v 0 k, find the equations of motionand show the following:(a) if v 0 = E/B then the particle continues its initial motion;(b) if v 0 = 0 then the particle follows the space curve given in terms of theparameter ξ byx = mEmE(1 − cos ξ), y =0, z = (ξ − sin ξ).B 2 q B 2 qInterpret this curve geometrically and relate ξ to t. Show that the totaldistance travelled by the particle after time t is given by∫2E tBqt′B ∣sin 0 2m ∣ dt′ .10.4 Use vector methods to find the maximum angle to the horizontal at which a stonemay be thrown so as to ensure that it is always moving away from the thrower.10.5 If two systems of coordinates with a common origin O are rotating with respectto each other, the measured accelerations differ in the two systems. Denotingby r and r ′ position vectors in frames OXY Z and OX ′ Y ′ Z ′ , respectively, theconnection between the two is¨r ′ = ¨r + ˙ω × r +2ω × ṙ + ω × (ω × r),where ω is the angular velocity vector of the rotation of OXY Z with respect toOX ′ Y ′ Z ′ (taken as fixed). The third term on the RHS is known as the Coriolisacceleration, whilst the final term gives rise to a centrifugal force.Consider the application of this result to the firing of a shell of mass m froma stationary ship on the steadily rotating earth, working to the first order inω (= 7.3 × 10 −5 rad s −1 ). If the shell is fired with velocity v at time t =0andonlyreaches a height that is small compared with the radius of the earth, show thatits acceleration, as recorded on the ship, is given approximately by¨r = g − 2ω × (v + gt),where mg is the weight of the shell measured on the ship’s deck.The shell is fired at another stationary ship (a distance s away) and v is suchthat the shell would have hit its target had there been no Coriolis effect.(a) Show that without the Coriolis effect the time of flight of the shell wouldhave been τ = −2g · v/g 2 .(b) Show further that when the shell actually hits the sea it is off-target byapproximately2τg [(g × ω) · v](gτ + v) − (ω × 2 v)τ2 − 1 3 (ω × g)τ3 .(c) Estimate the order of magnitude ∆ of this miss for a shell for which theinitial speed v is 300 m s −1 , firing close to its maximum range (v makes anangle of π/4 with the vertical) in a northerly direction, whilst the ship isstationed at latitude 45 ◦ North.370