Archive of posts with tag “from zhihu”
A whirling point charge
In the vacuum, inside a fixed ring of radius with fixed charge uniformly distributed, there is a point charge with charge and mass 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 . Denote the distance from the center to the point charge as , and . Find the magnetic induction at the center of the ring.
- Categories: physics
- Tags: from zhihu
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 , original length , and stiffness , 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 .
- Categories: physics
- Tags: from zhihu
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 or the angular momentum of a rigid body. Then, we introduce the concept of principal inertia . 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.
- Categories: physics
- Tags: rigid body, linear algebra, classical mechanics, from zhihu
Hölder means inequality
The Hölder mean of with weights and a parameter is defined as , and the value at are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as increases. This property can be used to prove HM-GM-AM-QM inequalities.
- Categories: math
- Tags: calculus, from zhihu
A polynomial whose sum of coefficients is a factorial
The function is a polynomial of degree w.r.t. , and the sum of its coefficients is . This turns out to be properties of Eulerian numbers.
- Categories: math
- Tags: combinatorics, number sequence, from zhihu
The image of a circular object through a thin lens
The image of a circle with radius and centered at through a thin lens at with focal length and centered at is a conic section with the focus being , the directrix being line , and the eccentricity being .
- Categories: physics
- Tags: geometrical optics, from zhihu
The point on the circle farthest to two lines
Suppose is a point on the circle . When is the sum of distances from to two edges of extremal? It turns out to be related to angle bisectors (the intersections of and the bisector of or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of and edges of ) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting at lies, translate the region to make it center at , and see whether is inside the translated region.
- Categories: math
- Tags: elementary geometry, trigonometry, from zhihu
Solving linear homogeneous ODE with constant coefficients
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…
- Categories: math
- Tags: calculus, linear algebra, combinatorics, ode, long paper, from zhihu
Thoughts on a middle school thermal physics problem
To heat an object with hot water, if we divide the water into more parts and use each part to heat the object one after another, the final temperature will be higher. If the number of parts tends to infinity, then the final temperature will tend to the limit , where is the initial temperature of the object, is the temperature of the hot water, is the heat capacity of the object, and is the heat capacity of the hot water.
- Categories: physics
- Tags: calculus, number sequence, from zhihu