Find the center of mass of a plate that is shaped like the region between $y = x^2$ and $y = 2x$, where the density varies as $1 + x + y.$

It takes 100,000 years for a plate to move about 2 kilometers. What is the rate of motion in centimeters per year?

A thin square plate is uniformly deformed as shown .Determine the shear strain $\gamma_{x y}$ after deformations (a) at corner $P$, and (b) at corner $Q$.

Find the centroid of a thin, flat plate covering the region enclosed by the x-axis, the lines x = 2 and x = -2, and the parabola $y=x^{2}.$

Two triangular plates are congruent. The area of one of the plates is 60 square inches.What is the area of the other plate? Explain.

At $t=0$, the $0.6$-slug thin plate has angular velocity $\omega=101+10 \mathrm{~J}(\mathrm{rad} / \mathrm{m})$ and is subjected to the force $\mathbf{F}=$ $-10 \mathrm{k}$ (Ib) acting at the point $(0,6,0)$ in. No other forces or couples act on the plate. What are the components of its angular acceleration at that instant?

A parallel-plate capacitor has circular plates of 8.20 cm radius and 1.30 mm separation. (a) Calculate the capacitance.

How does the energy stored in a capacitor change if the charge on each plate is doubled as the capacitor remains connected to a battery?

The assembly shown in Figure P5.45 consists of a steel $\left[E_1=30,000 \mathrm{ksi} ; A_1=1.25 \mathrm{in}^2\right]$ rod (1), a rigid bearing plate $B$ that is securely fastened to rod (1), and a bronze $\left[E_2=15,000 \mathrm{ksi}\right.$; $A_2=3.75 \mathrm{in}^2{ }^2$ ] post (2). The yield strengths of the steel and bronze are $62 \mathrm{ksi}$ and $75 \mathrm{ksi}$, respectively. A clearance of $0.125 \mathrm{in}$. exists between the bearing plate $B$ and bronze post (2) before the assembly is loaded. After a load of $P=65$ kips is applied to the bearing plate, determine (a) the normal stresses in bars (1) and (2). (b) the factors of safety with respect to yield for each of the members. (c) the vertical displacement of bearing plate $B$.

Determine the mass of inertia about the y-axis for the parabolic plate of the previous problem. State the radius of gyration about the y-axis.

Determine the cm of a thin, uniform, semicircular plate, Fig. 9-61. [Hint: Integrate; see Problem 63.

How much charge is on each plate of a $4.00-\mu \mathrm{F}$ capacitor when it is connected to a $12.0-\mathrm{V}$ battery?

Preganglionic sympathetic neurons develop from (a) neural crest, (b) neural tube, (c) alar plate, (d) endoderm

A $0.40~\mu \mathrm{F}$ capacitor is connected to a $9.0~ \mathrm{V}$ battery. How much charge is on each plate of the capacitor?