Sujit Prasad Kushwaha

Conservation of Linear Momentum The total quantity of motion contained in a body is called linear momentum . It is denoted by $\vec{P}$ and defined as the product of mass and velocity of the body. For a body of mass $m$ and velocity $v$, its linear momentum $\vec{P}$ in vector form is given by: $$\vec{P} = m\vec{v}$$ In terms of magnitude only, $$P = mv$$ SI unit of linear momentum is $\text{kg m s}^{-1}$ or Ns, and dimensional formula is $[MLT^{-1}]$. Law of Conservation of Linear Momentum It states that, "the total linear momentum of a system of particles remains constant, provided that no external force acts on the system" . Proof: From Newton's second law of motion , $$F = \frac{dP}{dt}$$ In absence of force, $$\frac{dP}{dt} = 0 \;\Rightarrow\; P = \text{constant}$$ Proof (for a system of particles): Let us consider a system of $n$ particles of total mass $M$. Then, $$M = m_1 + m_2 + \cdots + m_n$$ So, $$P = P_1 + P_2 + \cdots + P_n$$ Differe...

Gravity and Gravitation Central Force A central force between two particles is one which is directed along the line joining the two particles and whose magnitude is a function of the distance between them. If one particle is fixed in its position, the central force acting on the other is $\vec{F}$, then the torque acting on it is: $$ \vec{\tau} = \vec{r} \times \vec{F} = 0 $$ Since, $$ \vec{\tau} = \frac{d\vec{J}}{dt} $$ $$ 0 = \frac{d\vec{J}}{dt} $$ $$ \Rightarrow \vec{J} = \text{constant} $$ Again, we know, $$ \vec{J} = \vec{r} \times \vec{p} $$ $$ \vec{r} \cdot \vec{J} = \vec{r} \cdot (\vec{r} \times \vec{p}) = (\vec{r} \cdot \vec{r})\, \vec{p} = 0 $$ This shows that $\vec{r}$ and $\vec{J}$ are perpendicular to each other, and the motion of the particle is confined to a plane. Central force can be represented as: $$ \vec{F} = \frac{C}{r^2}\,\hat{r} $$ For gravitational force, $$ C = Gm_1 m_2, \qquad \text{then} \qquad \vec{F} = \frac{Gm_1 m_2}{r^2}\,\hat{r} $$ For...

Situated just 7 kilometers north of Prithvichowk in the scenic Pokhara Valley, Mahendra Cave is a captivating natural wonder that has intrigued both travelers and locals. Located in the Batulechaur-16 area of northern Pokhara, this limestone cave offers more than just breathtaking geological formations; it's also rich in history, folklore, and spirituality. Whether you're an adventurer, a history enthusiast, or someone seeking spiritual peace, Mahendra Cave offers an unforgettable experience. The Historical Background: The story of Mahendra Cave's discovery is as fascinating as the cave itself. Local villagers, troubled by the loss of their domestic animals to wild tigers, tracked the predators back to what is now known as Mahendra Cave. This discovery, believed to have occurred in 2012 B.S., quickly made the cave a point of interest. Its significance grew even more when the late King Mahendra Bir Bikram Shah Dev visited the site in 2016 B.S. The cave was subsequently named...

This is the mathematics formulae book for class 12 students. Here, we have covered all the formulae needed to solve the questions of Mathematics. Permutation The total arrangement of total objects taken some or all at a time is called permutation. The permutation of 'n' objects taken 'r' at a time is denoted by $^np_{r}$ and is defined by; 1. $^np_{r}=\frac{n!}{(n-r)!}$ 2. Permutation of an object, all different;$$=\frac{n!}{p!\,q!\,r!}$$ where,  n=total objects p=repetition of first kind  q=repetition of second kind r=repetition of third kind 3. Circular permutation= $(n-1)!$ 4. If clockwise and anticlockwise arrangement are not distinguishable [necklace of beads and beads into bracelet]; Total number of permutation=$\frac{(n-1)!}{2}$ Combination The selection of total objects taken some or all at a time is called combination. The combination of 'n objects taken 'r' at a time is denoted by $^nc_{r} \, or \, c(n,r)$ and defined by; $$^nc_{r}=\frac{n!}{(n-r)!...

Fear is a universal emotion, deeply ingrained in the human experience. It is an instinctive response to potential threats, but in the modern world, fear often transcends its primal origins. Whether it's the fear of failure, rejection, or the unknown, it can significantly impact various aspects of life, including studies, relationships, business, and more. However, fear doesn't have to be an insurmountable obstacle. By understanding and confronting fear, people can transform it into a powerful force that drives success and personal growth. Students often grapple with the fear of failure, which can manifest as anxiety during exams or hesitation in participating in class. This fear can stem from high expectations, both self-imposed and from others. It can lead to procrastination, avoidance, and a lack of confidence in one's abilities. However, those who learn to manage this fear can turn it into a motivator. By setting realistic goals, seeking help when needed, and embracing...