## Archive of posts in category “physics”

• ### Even solutions to bound states in an odd number of $\delta$ potential wells

We try solving the even function solutions to the time-independent Schrödinger equation for the potential $V=-\alpha\sum_{j=-n}^n\delta(x-ja)$ such that $E<0$ (bound states).

• ### Defend our earth against aliens’ bullets!

The aliens intiated their attack to the earth! They shoot bullets with mass $m$ and speed $v$ from a far-awar planet. To defend, humans built a field $U=\alpha/r$ that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid.

• ### How to construct mechanics in higher dimensions?

We can derive the equation of motion for mechanical systems in a Galileo universe with $\iota$ time dimensions and $\chi$ space dimensions by generalizing the principle of relativity and Hamilton’s principle.

• ### When a supersonic airplane flies over your head

Suppose a supersonic airplane has Mach number $M$. It flies horizontally. At some time, it flies past over your head at height $h$. Then, the distance between you and it when you have just heard it is $Mh$.

• ### Finding curvature radius physically

Sometimes the curvature radius of a curve can be found by using physical methods although it seems that you must use calculus to find it. In this article, the curvature radius of the curve $x^2=2py$ at the point where the curvature radius is smallest is found by using physical methods without using calculus (with only high school knowledge). The answer is that the smallest curvature is exactly $p$, and the point with smallest curvature is the vertex.

• ### A whirling point charge

In the vacuum, inside a fixed ring of radius $R$ with fixed charge $Q$ uniformly distributed, there is a point charge with charge $q$ and mass $m$ moving in the plane of the ring due tue the electrostatic force. It moves in the small region around the center of the ring, and the motion is periodic along a closed curve. The area of the region enclosed by the curve is $S$. Denote the distance from the center to the point charge as $r$, and $r\ll R$. Find the magnetic induction $B$ at the center of the ring.

• ### An example of non-uniform elements: heavy elastic rope

To illustrate the concept about non-uniform elements, we study a simple problem: suppose a uniform heavy elastic rope has mass $m$, original length $L_0$, and stiffness $k$, and find the mass distribution and length of it when hung vertically. We can use the element method to solve this problem, but the elements are non-uniform in terms of length. The elements add up to get the total length $L=\frac{mg}{2k}+L_0$.

• ### Kinetic energy, momentum, and angular momentum of rigid bodies

In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy $T$ or the angular momentum $\mathbf M$ of a rigid body. Then, we introduce the concept of principal inertia $\mathbf J_{\mathrm{pri}}$. We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.

• ### The image of a circular object through a thin lens

The image of a circle with radius $r$ and centered at $C\left(-2f,0\right)$ through a thin lens at $x=0$ with focal length $f$ and centered at $O\left(0,0\right)$ is a conic section with the focus being $\left(2f,0\right)$, the directrix being line $x=f$, and the eccentricity being $\frac rf$.

• ### Mapping from Kepler problem to free particle on 3-sphere

There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for $E<0$ is $\mathbf u:=\frac{p^2-p_0^2}{p^2+p_0^2}\hat{\mathbf n}+\frac{2p_0}{p^2+p_0^2}\mathbf p,$ where $\mathbf u$ is the position of the particle on 3-sphere (a 4-dimensional vector), $\mathbf p$ is the momentum of the original particle in Kepler problem, $\hat{\mathbf n}$ is a vector perpendicular to the 3-dimensional hyperplane where $\mathbf p$ lies, and $p_0:=\sqrt{-2mE}$.