Problem 1
Marines of bodyweight \(70[\mathrm{~kg}]\) reach an altitude of \(2923[\mathrm{~m}]\) in \(7.75\) hours. Calculate the total potential energy gained. What is the resulting power? What is more sensible, to calculate the power based on \(7.75\) hours, or based on 24 hours?
Problem 10
A cylindrical hot water tank (height \(1[\mathrm{~m}]\), radius \(22[\mathrm{~cm}]\) ) is heated during the day from \(20\left[{ }^{\circ} \mathrm{C}\right]\) to \(80\left[{ }^{\circ} \mathrm{C}\right]\). (a) Check that the heat capacity would be enough for two baths of \(120[\mathrm{~L}]\) at \(50\left[{ }^{\circ} \mathrm{C}\right]\) or seven showers \((40[\mathrm{~L}])\) at the same temperature, assuming the temperature of tap water is \(20\left[{ }^{\circ} \mathrm{C}\right]\). (b) The tank is insulated on all sides with \(d=10\) \([\mathrm{cm}]\) urethane foam. The air temperature is \(10\left[{ }^{\circ} \mathrm{C}\right]\). Estimate the loss by conduction for the first hour, assuming all insulated surfaces are flat (a very good approximation) and ignore the temperature decrease with time. Look at your results and argue why this is an acceptable assumption. Use \(h=10\left[\mathrm{Wm}^{-2} \mathrm{~K}^{-1}\right]\) for convection, ignore radiation.
Problem 24
Consider three \(\mathrm{AC}\) power lines with phases differing by \(2 \pi / 3\) in the following way \(V_{0} \cos \omega t, V_{0} \cos (\omega t+2 \pi / 3), V_{0} \cos (\omega t+4 \pi / 3)\). Each of the lines has its return current to ground; the loads are organized such that these currents also are in phase \(I_{0} \cos \omega t, I_{0} \cos (\omega t+2 \pi / 3), I_{0} \cos (\omega t+4 \pi / 3)\). (a) Show that the return currents cancel in the ground. (b) Estimate \(I_{0}\) such that the power delivered is the same as in the DC case. (c) Calculate the power loss in the three lines and compare with the DC case.
Problem 25
A HVDC line, made of copper with resistivity (or specific resistance) \(\rho=1.7 \times 10^{-8}\) \([\Omega \mathrm{m}]\) has a length \(l=800[\mathrm{~km}]\) and a diameter of \(2 r=22[\mathrm{~cm}]\). It is delivering 1000 [MW] of electricity at a grid point with a voltage of \(1000[\mathrm{kV}]\). Calculate (a) the electric resistance of the line; (b) the electrical current; (c) the heat loss in the cable, both in absolute value and in percentage of the transmitted power; (d) the voltage difference \(\Delta V\) between the beginning and the end of the line. Note: in practice, the resistance of the cable is higher and losses are in the order of 3 to \(2.5 \% / 1000[\mathrm{~km}]\).
Problem 30
A student gets a mortgage of \(£ 100000\) (or \(\$ 100000\) ). This has to be paid back in 25 years with an interest rate of \(10 \%\). Calculate the annual payment on basis of annuity
