MATHS 3021 Capstone Project in Mathematical Sciences III
ASSIGNMENT 1
Due: 5pm Tuesday 19 August
Total marks = 12
1. [FB:1-4] Determine whether the equation
for the time rate of change of total energy E in a pendulum system is dimensionally correct. The pendulum system is a bob of mass m swinging on a string of length ℓ held fixed at the top, as shown below, where g denotes gravity acting vertically downwards. At time t the string is at angle θ(t) relative to vertical. (2 marks)
2. [FB:1-10] Consider the equilibrium heat conduction problem over 0 ≤ x ≤ 1, with conductivity k(u) > 0, depending on the temperature u(x):
where all quantities are dimensional in appropriate units. The next simplest choice for k(u) after the constant case is to assume that the conductivity depends linearly on temperature, i.e. k(u) = k0(1−bu) where b and k0 are positive constants. This corresponds to a substance where the conductivity decreases as the temperature increases.
(a) Given [k] = , what are the dimensions of k0 and b? (2 marks)
(b) Show, by solving the equilibrium 1-D heat equation, that
(Make certain discarding the positive sign solution of the quadratic equation is justified.) (4 marks)
(c) On a single graph sketch the equilibrium temperature for b = 0.05, 0.1. Also sketch the constant conductivity case on the same diagram. (2 marks)
(d) Discuss the physical significance of the condition b < 1/10. (2 marks)