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.

IB Physics Two Masses and a String

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A tutorial sheet with problems involving two masses connected by a string is given below.

A 2.0 kg block and a 3.0 kg block are tied to the ends of a light inextensible string. Find the tension in the string in each case.

  1. The blocks are released from rest with the 3.0 kg hanging below the 2.0 kg.
  2. The blocks are released from rest with the 2.0 kg hanging below the 3.0 kg.
  3. The blocks are pulled along a smooth horizontal surface by a force of 1.0 N applied parallel to the surface acting on the 2.0 kg block.
  4. The blocks are at rest on a rough horizontal surface of coefficient of static friction 0.4. A force of 1.0 N parallel to the surface is applied to the 2.0 kg block.
  5. The blocks are at rest on a rough horizontal surface of coefficient of static friction 0.4. A force of 3.0 N parallel to the surface is applied to the 2.0 kg block.
  6. A smooth pulley is at rest in the laboratory. The string is passed over the pulley and the blocks are released from rest.
  7. In question 6 the axle of the pulley accelerates upwards uniformly at 2.0 m s-2.
  8. In question 7 what must be the acceleration of the pulley if the blocks stay at rest relative to each other?