This question requires the use of the conduction charts given in the appendix at the end of this paper.

A wooden sphere of diameter $5$ cm is initially at a uniform temperature of $21\backslash$deg C$.$ Its surface temperature is suddenly raised to $83\backslash$deg C$.$ The thermal conductivity and thermal diffusivity of the wood are $0\mathrm{.}15$ W$/$m K and $0\mathrm{.}82$ x $10-7$ m$2/$s$,$ respectively.

a$)$ Calculate the temperature $2$ cm beneath the surface of the sphere $30$ min after the sudden increase in temperature.

b$)$ Calculate the energy gained by the sphere during this time.