Other Problem Solutions: Expert Help and Step-by-Step Explanation

Consider a parallel-plate capacitor with a K-dielectric that is only half filled. Although it occupies a little portion of the capacitor's surface area, the dielectric is the same height as the distance between its plates. When the dielectric is all gone, the capacitor has a capacitance of ${\displaystyle C_0}$. How does this capacitor's capacitance C(f) change with f?
In terms of ${\displaystyle C_0}$ and K, express C(f).

In Figure, two identical springs,each with a relaxed length of 50 cm and a spring constant of 450 N/m, are connected by a short cordof length 10 cm. The upper spring is attached to the ceiling. A box that weighs 97 N hangs from the lower spring. Two additional cords, each 85 cm long, are also tied to theassembly, and they are limp.

a) If the short cord is cut, so that the box then hangs from the springs and the two longer cords, does the box move up or down from its initial location?
b) How far does the box move?
c) How much total work do the two spring forces do on the box during that move?

Determine the current (magnitude and direction) in the 8.0 and 2.0-? resistors in the drawing.

Find the current (magnitude and direction) in the 8.0 and 2.0-? resistors in the drawing.

In Figure P1, assume that in each case the velocity vector shown is replaced with a wire carrying a current in the direction of the velocity vector. For each case, find the direction of the magnetic force acting on the wire.

Given that $f\left(t\right)=4e^{-3(t-4)}$
a) Find $L\left[\frac{df\left(t\right)}{dt}\right]$ by differentiating f(t) and then using the Laplace transform tables in lecture notes.
b) Find $L\left[\frac{df\left(t\right)}{dt}\right]$ using the theorem for differentiation
c) Repeat a) and b) for the case that $f\left(t\right)=4e^{-3(t-4)}u(t-4)$

To simplify:
The given redical:
An expression: ${\displaystyle \sqrt[4]{{(x^2-4)}^4}.}$
Use absolute value bars when necessary.

Prove that ${\displaystyle \sum _{i=1}^n\left(\begin{array}{c}i\\ 2\end{array}\right)=\left(\begin{array}{c}n+1\\ 3\end{array}\right)}$
$\mathrm{\forall }n\ge 2$

Prove that ${\displaystyle T(x,y)=(3x+y,2y,x-y)}$ defines a linear transformation ${\displaystyle T:{\mathbb{R}}^2\to {\mathbb{R}}^3}$. Give the full and correct answer.