Let u(x,t) describe the temperature of a thin metal ring with circumference 2pi. For convenience, let's orient the ring so that x spans the interval |-pi, pi|. Suppose that the ring has some internal heating that is angle-dependent, so that u(x, t) satisfies the inhomogeneous heat equation

u_t = ku_zz + f(x),

where k is the thermal diffusivity and f(x) describes the internal heating. Furthermore, assume that the temperature of the ring is initially zero.

a) Because the temperature u(x, t) is parameterized by the angle x, the temperature must be a 2pi-periodic function. This suggests that we should write the temperature as a complex Fourier expansion with time-dependent coefficients,

u(x, t) = SUM (A_n)e^(inz).

Substitue this expression into the heat equation, and obtain a differential equation for each (A_n)(t). Using the correct initial conditions, solve each ODE and write down the final solution.

b) Suppose f(x) = cos^2 (x). Find an explicit solution for u(x, t). (Hint: Expand cos^2(x) into a finite number of complex Fourier modes, and show that all but three of the A_n are always zero.)

... 61C we obtain the average heat capacity of the metal ingot of ... Here is a web link calculator for formula weights to use when calculating molar heat capacities ...

... For a metal the followings are the collision or scattering ... values of K is calculated using the equation K=( Cv ... K on T resulting entirely due to specific heat Cv ...

... rod are made up of the same metal aluminum, therefore ... or loosed) by a body is given by the following equation: Now, the amount of heat loosed by the hotter body ...

... where N is the total number of electrons in the metal. ... us the correct electronic contribution to the heat capacity ... Doing the same in equation (2) for Chi yields ...

...Heating for long periods like distillations, can result in ... 5. The equation is, AgNO3 + NaCl → AgCl + NaNO3 6 ... a clay triangle on a small metal ring clamped to ...

... mass of a solid b.) The change in volume of a liquid c.) The change in length of a metal rod d ...Heat given or removed by an object is given by the formula. ...

... to the O creating an intermediate metal alkoxide complex. ... If the aldol is heated in basic solution, the ... c. Fisher Projection formula of L-glucose is: Similarly ...

... personal motor vehicles out of the equation, we are ... The ultimate clean burn - produces CO2, water, heat. ... are very heavy, although storage in metal hydrides has ...

... X rays by firing electrons at a metal target is ... phase En n = 1,2,3 (heating) leads to ... 2m and remember p = ħk : Conservation equation (1D) -{iħ ...