In the analysis of the bending of a uniformly loaded circular plate, the equation $w(r)$ of the deflection curve of the plate can be shown to satisfy the third-order differential equation

$$\frac { d ^ { 3 } w } { d r ^ { 3 } } + \frac { 1 } { r } \frac { d ^ { 2 } w } { d r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { d w } { d r } = \frac { q } { 2 D } r,$$

where $q$ and $D$ are constants. Here $r$ is the radial distance from a point on the circular plate to its center. (a) Find the general solution of the equation.

A superball is dropped from a height $h$ onto a hard steel plate (fixed to the Earth), from which it rebounds at very nearly its original speed. If we consider the ball and the Earth as our system, during what parts of the process is momentum conserved for a piece of putty that falls and sticks to the steel plate.

A vertical flat plate is maintained at a constant temperature of $120^{\circ} \mathrm{F}$ and exposed to atmospheric air at $70^{\circ} \mathrm{F}$. At a distance of $14\ \mathrm{in}$. from the leading edge of the plate the boundary layer thickness is $1.0\ \mathrm{in}$. Find the thickness of the boundary layer at a distance of $24\ \mathrm{in}$. from the leading edge.

The coefficient of restitution for steel onto steel is measured by dropping a steel ball onto a steel plate. If the ball is dropped from a height $H$ and rebounds to a height $h$, then the coefficient of restitution is $\sqrt{\frac{h}{H}}$. Suppose that a steel ball is dropped from a height of $1\ \mathrm{m}$ onto a steel plate. Each time the ball hits the plate, it rebounds to $r$ times it previous height $(0<r<1)$. If the ball travels a total distance of $2\ \mathrm{m}$, find the coefficient of restitution for steel on steel.

To produce a polarized laser beam, a plate of transparent material is placed in the laser cavity and oriented so the light strikes it at the polarizing angle. Such a plate is called a Brewster window. Prove that if $\theta_{\mathrm{P} 1}$ is the polarizing angle for the $n_1$ to $n_2$ interface, then $\theta_{\mathrm{P} 2}$ is the polarizing angle for the $n_2$ to $n_1$ interface.

An avalanche may occur when the temperature keeps snow crystals from sticking together. The room temperature in an avalanche research lab is -30°C. Scientists study changes in snow crystals by melting snow on a hot plate that heats to only -1°C. In a regular lab, the room temperature is about 18°C and a hot plate heats to about 300°C. How many degrees warmer is the hot plate than the room temperature in each lab? Which difference is greater? How much greater?

Castor oil flows over the surface of the flat plate at a free-stream speed of $2 \mathrm{~m} / \mathrm{s}$. The plate is $0.5 \mathrm{~m}$ wide and $1 \mathrm{~m}$ long. Plot the boundary layer and the shear stress versus $x$. Give values for every $0.5 \mathrm{~m}$. Also find the friction drag on the plate. Take $\rho_{c o}=960 \mathrm{~kg} / \mathrm{m}^3$ and $\mu_{c o}=985\left(10^{-3}\right) \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^2$

A square $2014$-$\text{T}4$ aluminum alloy plate $400 \mathrm{~mm}$ on a side has a $75 -\text{mm}$-diameter circular hole at its center. The plate is heated from $20^{\circ} \mathrm{C}$ to $45^{\circ} \mathrm{C}$. Determine the final diameter of the hole.

Solve the given problems by setting up and solving appropriate inequalities. Graph each solution.

The mass $m$ (in g) of silver plate on a dish is increased by electroplating. The mass of silver on the plate is given by $m = 125 + 15.0t,$ where $t$ is the time (in h) of electroplating. For what values of $t$ is $m$ between $131$ g and $164$ g?

A machinist bores a hole of diameter 1.35 cm in a steel plate that is at $25.0^{\circ} \mathrm{C}$. What is the cross-sectional area of the hole (a) at $25.0^{\circ} \mathrm{C}$ and (b) when the temperature of the plate is increased to $175^{\circ} C$? Assume that the coefficient of linear expansion remains constant over this temperature range.

Solve the preceding problem if the plate is made of aluminum with $E=72\ \mathrm{GPa}$ and Poisson's ratio $\nu=0.33$. The plate is loaded in biaxial stress with normal stress $\sigma_x=79\ \mathrm{MPa}_{\text {, angle } \phi}=18^{\circ}$, and the strain measured by the gage is $\varepsilon=925 \times 10^{-6}$.

The heat transfer coefficient for a gas flowing over a thin flat plate $3 \mathrm{~m}$ long and $0.3 \mathrm{~m}$ wide varies with distance from the leading edge according to

$$h_C(x)=10 x^{-1 / 4} \frac{\mathrm{W}}{\mathrm{m}^2 \mathrm{~K}}$$

If the plate temperature is $170^{\circ} \mathrm{C}$ and the gas temperature is $30^{\circ} \mathrm{C}$, calculate (a) the average heat transfer coefficient, (b) the rate of heat transfer between the plate and the gas, and (c) the local heat flux $2 \mathrm{~m}$ from the leading edge.

Engine oil at $40^\circ C$ is flowing over a long flat plate with a velocity of 6 m/s. Determine the distance $x_{\mathrm{cr}}$ from the leading edge of the plate where the flow becomes turbulent, and calculate and plot the thickness of the boundary layer over a length of $2 x_{\mathrm{cr}}.$

A vertical rectangular plate with a width of $16 \mathrm{~m}$ and a height of $12 \mathrm{~m}$ is located $4 \mathrm{~m}$ below a water surface. The resultant hydrostatic force on this plate is (a) $2555 \mathrm{kN}$ (b) $3770 \mathrm{kN}$ $(c)$ $11,300 \mathrm{kN}$ (d) $15,070 \mathrm{kN}$ (e) $18,835 \mathrm{kN}$