Air at $20^\circ C$ with a convection heat transfer coefficient of $25 W/m^2 \cdot K$ blows over a horizontal steel hot plate $(k = 43 W/m \cdot K).$ The surface area of the plate is $0.38 m^2$ with a thickness of 2 cm. The plate surface is maintained at a constant temperature of $T_s = 250^\circ C$ and the plate loses 300 W from its surface by radiation. Calculate the inside plate temperature $(T_i).$

Set up the boundary-value problem for the steady-state temperature u(x, y). A thin rectangular plate coincides with the region in the xy-plane defined by $0 \leq x \leq 4,0 \leq y \leq 2$. The left end and the bottom of the plate are insulated. The top of the plate is held at temperature zero, and the right end of the plate is held at temperature f ( y).

Air at 20 degrees C with a convection heat transfer coefficient of $25 \mathrm{~W} / \mathrm{m}^2 \cdot \mathrm{K}$ blows over a horizontal steel hot plate $(k=43 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. The surface area of the plate is $0.38 \mathrm{~m}^2$ with a thickness of $2 \mathrm{~cm}$. The plate surface is maintained at a constant temperature of $T_s=250^{\circ} \mathrm{C}$, and the plate loses $300 \mathrm{~W}$ from its surface by radiation. Calculate the inside plate temperature $\left(T_i\right)$.

Air at $40^{\circ} \mathrm{C}$ flows over a long, 25-mm-diameter cylinder with an embedded electrical heater. In a series of tests, measurements were made of the power per unit length, $P^{\prime}$, required to maintain the cylinder surface temperature at $300^{\circ} \mathrm{C}$ for different free stream velocities V of the air. The results are as follows:

$$\begin{matrix} \text{Air velocity, V(m/s)} & \text{1} & \text{2} & \text{4} & \text{8} & \text{12}\\ \text{Power, }{P^{\prime}(\mathrm{W} / \mathrm{m})} & \text{450} & \text{658} & \text{983} & \text{1507} & \text{1963}\\ \end{matrix}$$

(a) Determine the convection coefficient for each velocity, and display your results graphically. (b) Assuming the dependence of the convection coefficient on the velocity to be of the form $h=C V^{n}$, determine the parameters C and n from the results of part (a).

The top surface of a metal plate $(k_{plate} = 237 W/m \cdot K)$ is being cooled by air $(k_{air} = 0.243 W/m \cdot K)$ while the bottom surface is exposed to a hot steam at $100^\circ C$ with a convection heat transfer coefficient of $30 W/m^2 \cdot K.$ If the bottom surface temperature of the plate is $80^\circ C,$ determine the temperature gradient in the air and the temperature gradient in the plate at the top surface of the plate.

Air at 20 degree C with a convection heat transfer coefficient of $20 W / m^2 \cdot K$ blows over a pond. The surface temperature of the pond is at 40 degree C. Determine the heat flux between the surface of the pond and the air.

A horizontal water jet of constant velocity V impinges normally on a vertical flat plate and splashes off the sides in the vertical plane. The plate is moving toward the oncoming water jet with velocity $rac{1}{2} V$. If a force F is required to maintain the plate stationary, how much force is required to move the plate toward the water jet?

A uniform rectangular plate has mass $m$ and edges of lengths $a$ and $b$.

$(a)$ Show by integration that the moment of inertia of the plate about an axis that is perpendicular to the plate and passes through one corner is $\frac{1}{3} m\left(a^2+b^2\right)$.

A rectangular plate is supported by three cables as shown. Knowing that the tension in cable $A D$ is $520 \mathrm{~N}$, determine the weight of the plate.

Consider a rectangular metal block $(k = 35 W/m \cdot K)$ of dimensions $100 cm \times 75 cm$ subjected to a sinusoidal temperature variation at its top surface while its bottom surface is insulated. The two sides of the metal block are exposed to a convective environment at $15^\circ C$ and having a heat transfer coefficient of $50 W/m^2 \cdot K.$ The sinusoidal temperature distribution at the top surface is given as $100 \sin (\pi x / L).$ Using a uniform mesh size of $\Delta x=\Delta y=25 \mathrm{cm}$ determine (a) finite difference equations and (b) the nodal temperatures.

A parallel-plate capacitor is charged until it carries charge +q on one plate and charge -q on the other plate. The capacitor is then disconnected from the power supply and isolated. What is the direction of the magnetic field that surrounds the charged capacitor?

Capacitors $C_1=6.00 \mu \mathrm{F}$ and $C_2=2.00 \mu \mathrm{F}$ are charged as a parallel combination across a $250-\mathrm{V}$ battery. The capacitors are disconnected from the battery and from each other. They are then connected positive plate to negative plate and negative plate to positive plate. Calculate the resulting charge on each capacitor.

Gabe Is spending two weeks on an archaeological dig. He finds a fragment of a circular plate that his leader thinks may bo valuable. The arc length of the fragment is about $\frac{1}{6}$ the circumference of the original complete plate and measures 1.65 Inches.
A similar plate found earlier has a diameter of 3,14 Inches. Could Gabe's fragment match this plate? a. Write an expression for the radius, r, of the earlier plate. b. What Is the measure. In radians, of a central angle, $\theta$, that Intercepts an arc that Is $\frac{1}{6}$ the length of the circumference of a circle? c. Write an expression for the arc length, S, intercepted by this central angle. d. How long would the arc length of a fragment be if it were $\frac{1}{6}$ the circumference of the plate? e. Could Gabe's plate be a matching plate?

An electron acquires $7.45 \times 10^{-16} \mathrm{J}$ of kinetic energy when it is accelerated by an electric field from plate A to plate B. What is the potential difference between the plates, and which plate is at the higher potential?