Mathematical Tripos Dynamics and Relativity 2019

Listed below are my answers to the MT Dynamics and Relativity questions in Paper 4 of Part 1A held in June 2019. Answers to MT D&R questions will be added each October to correspond to the start of Michaelmas Term.

1. Question 3A. uln2
2. Question 4A. t = √ (m/(ɣg) ) e^-2ɣh/m tanh^-1( √(1-e^-2ɣh/m) )
3. Question 9A. A=-1/4, B=3/4
4. Question 10A. (a) x = 1-r+rcos(⍵t), y = rsin(⍵t), z = vt, where r = mv/( qB √ (1-2v²/c² ) ) and ⍵=v/r.
5. Question 10A. (b) (i) (0,0), three saddle points also exist where the equilibrium is stable in one direction (ii) stable along x and y axes. At (0,0) ∂V/∂x=0, ∂V/∂y=0, ∂²V/∂x²>0, ∂V/∂y=0,∂²V/∂y²>0 (iii) A=3^5/4/(2^3/2√𝜋). At saddle point ⍵=1.389q/√(m𝜀) taking a=1

IB Physics Series and Parallel Resistors

A tutorial sheet on series and parallel resistors is given below.

1. A 2 Ω and a 4 Ω are place in series with a battery. Which resistor has the greater power?
2. A 2 Ω and a 4 Ω are placed in parallel across a battery. Which resistor has the greater power?
3. A 2 Ω and a 4 Ω are connected in parallel. In series with this parallel group is a 1 Ω. When a battery of zero internal resistance is connected across this combination, which resistor has (a) the greatest power? and (b) the least power? [1 Ω,4 Ω]
4. Two 5 Ω resistors are placed in parallel. In series with this combination is a 10 Ω. The power of the 10 Ω is 20 W when the entire combination is connected across a battery of zero internal resistance. What is the power of the 10 Ω when one of the 5 Ω resistors is removed? [13.9 W]