News

Entertainment

Science & Technology

Life

Culture & Art

Hobbies

News

Entertainment

Science & Technology

Culture & Art

Hobbies

1. Isaac Newton Just Discovered a New Force in Nature In a new paper published in Nature, researchers from the University of Cambridge have demonstrated a newly discovered force in nature, which they have named the Newtonian force. This force is the result of a combination of the gravitational force and the force of inertia, and was predicted by Sir Isaac Newton in the 17th century. The researchers used a combination of theoretical calculations and experiments to prove the existence of the force and its properties. The research could lead to new insights into the workings of the universe, and could also help to explain why some objects appear to move against the expected gravitational force. 2. Isaac Newton's Revolutionary Discoveries Sir Isaac Newton is one of the most influential scientists in history. He made groundbreaking discoveries in the fields of mathematics, physics, astronomy, and optics. His most famous work is the three laws of motion, which laid the foundation for modern physics. He also discovered the law of universal gravitation, which states that all objects in the universe attract each other with a force proportional to their mass. His discoveries revolutionized science and paved the way for the development of many of the technologies we use today. 3. 5 Amazing Facts About Sir

Fig. 1 Shown in Fig. 1 is a semicircle centered at C $latex (\frac{1}{2}, 0)$ with radius = $latex \frac{1}{2}$. Its equation is $latex (x-\frac{1}{2})^2+y^2=(\frac{1}{2})^2, 0 \le x \le 1, y \ge 0.$ Simplifying and solving for $latex y$ gives $latex y = x^{\frac{1}{2}}\cdot(1-x)^{\frac{1}{2}}.\quad\quad\quad(1)$ We see that Area (sector OAC) = Area (sector OAB) +…

The Binomial Theorem (see "Double Feature on Christmas Day", "Prelude to Taylor's theorem") states: $latex (x+y)^n = \sum\limits_{i=0}^{n} \binom{n}{i}x^{n-i}y^i, \quad n \in \mathbb{N}.$ Provide $latex x$ and $latex y$ are suitably restricted, there is an Extended Binomial Theorem. Namely, $latex (x+y)^r = \sum\limits_{i=0}^{\infty} \binom{r}{i}x^{r-i}y^i, \underline{|\frac{x}{y}|<1}, r \in \mathbb{R}$ where $latex \binom{r}{i} \overset{\Delta}{=} \frac{r(r-1)(r-2)...(r-i+1)}{i!}.$ Although Issac Newton…

In 1665, following an outbreak of the bubonic plague in England, Cambridge University closed its doors, forcing Issac Newton, then a college student in his 20s, to go home. Away from university life, and unbounded by curriculum constraints and tests, Newton thrived. The year-plus he spent in isolation was later referred to as his annus…

In 1665, following an outbreak of the bubonic plague in England, Cambridge University closed its doors, forcing Issac Newton, then a college student in his 20s, to go home. Away from university life, and unbounded by curriculum constraints and tests, Newton thrived. The year-plus he spent in isolation was later referred to as his annus…

We know from "arcsin" : $latex \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}}.$ Integrate from $latex 0$ to $latex x\; (0<x<1):$ $latex \int\limits_{0}^{x}\frac{d}{dx}\arcsin(x)\;dx = \int\limits_{0}^{x}\frac{1}{\sqrt{1-x^2}}\;dx$ gives $latex \arcsin(x)\bigg|_{0}^{x} = \int\limits_{0}^{x}\frac{1}{\sqrt{1-x^2}}\;dx.$ i.e., $latex \arcsin(x) = \int\limits_{0}^{x}\frac{1}{\sqrt{1-x^2}}\;dx.\quad\quad\quad(\star)$ Rewrite the integrand $latex \frac{1}{\sqrt{1-x^2}}$ as $latex (1-x^2)^{-\frac{1}{2}}= (1+(-x^2))^{-\frac{1}{2}}$ so that by the extended binomial theorem (see "A Gem from Issac Newton"), $latex…