Mathematical Tripos Dynamics and Relativity 2021

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Below are my answers to the 2021 MT Part 1A Dynamics and Relativity questions.

  1. 3C, t = m0( 1 - e-V0/U)/α
  2. 4C, Izz = M( a2 + b2 )/3
  3. 9C, Ve2 = 2λF0/m, escapes if Ve ≥V0, T = 8λ/( Ve√(1 - V02/Ve2) ) sin-1(V0/Ve)
  4. 10C, (c) (i) C=2 (ii) √( k/r1) √ ( Cr2/(r1 r2) - 1 ), √( k/r2) √ ( Cr1/(r1 r2) - 1 ) (iii) Use dA/dt = h/2, t = πab/h, t2 = π2a b2/(k(1-e2)) = π2a3/k = π2 (r1 + r2)3/(8k)
  5. 11C (ii) 𝜽 = 0, 𝜽 = cos-1(g/⍵2R), 𝜽 = 𝜋 provided g < ⍵2R. In this case 𝜽 = 0 and 𝜽 = 𝜋 are unstable and the period of small oscillations about 𝜽 = cos-1(g/⍵2R) is 2𝜋/⍵ ✕ 1/√(1 - g2/(⍵4R2)). If g > ⍵2R there is equilibrium when 𝜽 = 0 and 𝜽 = 𝜋. In this case 𝜽 = 0 is stable with a period of small oscillations 2𝜋/⍵ ✕ 1/√( g/(⍵2R) -1) )and 𝜽 = 𝜋 is unstable. If g = ⍵2R there is equilibrium when 𝜽 = 0 and 𝜽 = 𝜋. In this case the period of oscillations about 𝜽 = 0 is expressible in terms of a complete elliptic integral of the first kind and 𝜽 = 𝜋 is unstable. (iii)Force has perpendicular components N (towards the center of the circle) and Q (into the page). N = m(gcos𝜽 + ⍵2Rsin2𝜽 + R𝜽̇2), Q = 2mR⍵𝜽̇. F=√(N2 + Q2).
  6. 12C a = 1/(ɣm)( F - F·v v/c2 ), v = cqEt/√( c2m2 + q2E2t2 ). As t → ∞ , v → c.