A finite potential energy function U(x) allows $\psi(x),$ the solution of the time-independent Schrödinger equation, to penetrate the classically forbidden region. Without assuming any particular function for U(x), show that $\psi(x)$ must have an inflection point at any value of x where it enters a classically forbidden region.

A pendulum has length of 250 mm. What is the system’s natural frequency in hertz?

Consider a flat-plate solar collector placed horizontally on the flat roof of a house. The collector is 1.5 m wide and 4.5 m long, and the average temperature of the exposed surface of the collector is $42^\circ C.$ Determine the rate of heat loss from the collector by natural convection during a calm day when the ambient air temperature is $8^\circ C.$ Also, determine the heat loss by radiation by taking the emissivity of the collector surface to be 0.85 and the effective sky temperature to be $15^\circ C.$

In turbulent flow, one can estimate the Nusselt number using the analogy between heat and momentum transfer (Colburn analogy). This analogy relates the Nusselt number to the coefficient of friction, $C_f,$ as(a) $\mathrm{Nu}=0.5 C_{f} \mathrm{Re} \mathrm{Pr}^{1 / 3}$ (b) $\mathrm{Nu}=0.5 \mathrm{C}_{f} \mathrm{Re} \mathrm{Pr}^{2 / 3}$ (c) $\mathrm{Nu}=C_{f} \text { Re } \mathrm{Pr}^{1 / 3}$ (d)$\mathrm{Nu}=C_{f} \operatorname{Re} \mathrm{Pr}^{2 / 3}$

A heating element in the shape of a cylinder is maintained at 2000$^{\circ} F$ and placed at the center of a half-cylindrical reflector as shown. The rod diameter is 2 in. and that of the reflector is 18 in. The emissivity of the heater surface is 0.8, and the entire assembly is placed in a room maintained at 70$^{\circ} F$. What is the radiant energy loss from the heater per foot of length? How does this compare to the loss from the heater without the reflector present?

In natural-convection problems, the variation of density due to the temperature difference $\Delta \mathrm{T}$ creates an important buoyancy term in the momentum equation. If a warm gas at THE moves through a gas at temperature T0 and if the density change is only due to temperature changes, the equation of motion becomes $\rho \frac{D \mathbf{v}}{D t}=-\nabla P+\mu \nabla^{2} \mathbf{v}+\rho \mathbf{g}\left(\frac{T_{\boldsymbol{H}}}{T_{0}}-1\right)$. Show that the ratio of gravity (buoyancy) to inertial forces acting on a fluid element is $\frac{L_{\mathrm{g}}}{V_{0}^{2}}\left(\frac{T_{H}}{T_{0}}-1\right)$ where L and V0 are reference lengths and velocity, respectively.

A flat plate is subject to air flow parallel to its surface. The average friction coefficient over the plate is given as $C_{f}=1.33\left(\mathrm{Re}_{L}\right)^{-1 / 2} \quad \text { for } \mathrm{Re}_{L}<5 \times 10^{5}$ (laminar flow) $C_{f}=0.074\left(\mathrm{Re}_{L}\right)^{-1 / 5} \text { for } 5 \times 10^{5} \leq \mathrm{Re}_{L} \leq 10^{7}$ (turbulent flow) The plate length parallel to the air flow is 1 m. Using EES (or other) software, determine the effect of air velocity on the average convection heat transfer coefficient for the plate. By varying the air velocity for $0<V \leq 20 \mathrm{m} / \mathrm{s}$ plot the average convection heat transfer coefficient as a function of air velocity. Evaluate the air properties at $20^\circ C$ and 1 atm.

It is well known that wind makes the cold air feel much colder as a result of the wind chill effect that is due to the increase in the convection heat transfer coefficient with increasing air velocity. The wind chill effect is usually expressed in terms of the wind chill temperature (WCT) , which is the apparent temperature felt by exposed skin. For outdoor air temperature of $0^\circ C,$ for example, the wind chill temperature is $−5^\circ C$ at 20 km / h winds and $− 9^\circ C$ at 60km/h winds. That is, a person exposed to $0^\circ C$ windy air at 20km/h will feel as cold as a person exposed to $−5^\circ C$ calm air (air motion under 5km/h) .For heat transfer purposes, a standing man can be modeled as a 30-cm-diameter, 170-cm-long vertical cylinder with both the top and bottom surfaces insulated and with the side surface at an average temperature of $34^\circ C.$ For a convection heat transfer coefficient of $15 W/m^2 \cdot K,$determine the rateof heat loss from this man by convection in still air at $20^\circ C.$ What would your answer be if the convection heat transfer coefficient is increased to $30W/m^2 \cdot K$ as a result of winds? What is the wind chill temperature in this case?

A total of 10 rectangular aluminum fins $(k = 203 W/m \cdot K)$ are placed on the outside flat surface of an electronic device. Each fin is 100 mm wide, 20 mm high and 4 mm thick. The fins are located parallel to each other at a center-to-center distance of 8 mm. The temperature at the outside surface of the electronic device is $60^\circ C.$ The air is at $20^\circ C,$ and the heat transfer coefficient is $100 W/m^2 \cdot K.$ Determine (a) the rate of heat loss from the electronic device to the surrounding air and (b) the fin effectiveness.

A 4-m-high and 6-m-wide wall consists of a long 18-cm × 30-cm cross section of horizontal bricks $(k = 0.72 W/m \cdot K)$ separated by 3-cm-thick plaster layers $(k = 0.22 W/m \cdot K).$ There are also 2-cm-thick plaster layers on each side of the wall, and a 2-cm-thick rigid foam $(k = 0.026 W/m \cdot K)$ on the inner side of the wall. The indoor and the outdoor temperatures are $22^\circ C$ and $−4^\circ C,$ and the convection heat transfer coefficients on the inner and the outer sides are $h_1 = 10 W/m^2 \cdot K$ and $h_2 = 20 W/m^2 \cdot K,$ respectively. Assuming one-dimensional heat transfer and disregarding radiation, determine the rate of heat transfer through the wall.

Describe the two proofreading functions of RNA polymerase in prokaryotes.

Circular fins of uniform cross section, with diameter of 10 mm and length of 50 mm, are attached to a wall with surface temperature of $350^\circ C.$ The fins are made of material with thermal conductivity of $240 W/m \cdot K,$ and they are exposed to an ambient air condition of $25^\circ C$ and the convection heat transfer coefficient is $250 W/m^2 \cdot K.$ Determine the heat transfer rate and plot the temperature Variation of a Single fin for the following boundary conditions: (a) Infinitely long fin (b) Adiabatic fin tip (c) Fin with tip temperature of $250^\circ C.$ (d) Convection from the fin tip

At a windspeed of 30 mph and air temperature of $35^{\circ} \mathrm{F}$, estimate the rate of change of the windchill temperature with respect to air temperature if the windspeed is held constant.

True/False. $D_{-\mathbf{k}} f(x, y, z)=\frac{\partial f}{\partial z}$