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...

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)!...

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...

 

Welcome to www.sujitpk.com.np , your premier online hub for accessing top-notch calculus notes! Whether you're a student grappling with derivatives and integrals, a math enthusiast looking to expand your knowledge, or a professional in need of a calculus refresher, this website is the ultimate resource for you. Explore a vast collection of meticulously prepared calculus notes covering topics such as limits, differentiation, integration, and more. With a simple click, you can download these comprehensive notes and embark on a journey towards mastering calculus. Calculus Download Calculus Notes for BSC First Year in PDF format. Chapter 1: Tangents and Normals Tangents and Normals Chapter 2: Higher Order Derivatives and Mean Value Theorems Higher Order Derivatives Mean Value Theorems Chapter 3: Application of Derivatives Indeterminant Forms Asymptotes Tracing of Curves Curvature Chapter 4: Partial Differentiations and Maxima and Minim...