TIFR 2014 Problem 2 Solution has been written for TIFR entrance preparation series. A problem on continuous bounded function from Real analysis.
TIFR 2014 Problem 2 Solution has been written for TIFR entrance preparation series. A problem on continuous bounded function from Real analysis.
TIFR 2014 Problem 1 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects. The image is a front […]
An ice skater spins at (4\pi )rad/s with her arms extended. If her moment of inertia with arms folded is (80\%) of that with her arms extended, what is her angular velocity when she folded her arms? Discussion: Conservation of angular momentum gives $$ I_1\omega_1=I_2 \omega_2$$ $$ (I_1) (4\pi)=0.8 I_1 \omega_2 $$ Hence, (\omega_2=5\pi)
Let's discuss a problem on angular momentum and prove that the tension in the thread is inversely proportional to the cube of the distance from the hole. The Problem: A small mass m tied to a piece of thread moves over a smooth horizontal plane/ The other end of the thread is drawn through a […]
TIFR 2013 Problem 40 Solution has been written for TIFR entrance preparation series. A problem on alternatig series from mathematical analysis.
A ring of radius (r) is located in the (x-y) plane is given a total charge (Q=2\pi R\lambda). Show that (E) is maximum when the distance (z=r/\sqrt{2}). Discussion: The elctric field at distance z from the centre of the ring on the axis of the ring with charge (Q=2\pi R\lambda) is given by $$ E=\frac{\lambda […]
A total charge (q) is spread uniformly over the inner surface of a non-conducting hemispherical cup of inner radius (a). Calculate the electric field. Discussion: Consider a circular strip symmetric about (z-axis) of radius (r) and width (ad\theta) The charge on the strip is $$ dq=q\frac{2\pi r ad\theta}{2\pi a^2}=\frac{qr d\theta}{a}=q sin\theta d\theta$$ (a) At the […]
TIFR 2013 Problem 39 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects. The image is a front […]
If the electric potential is given by (\chi=cxy), calculate the electric field. Discussion: $$ E_x=-\frac{\partial\chi}{\partial x}=-cy$$ $$ E_y=\frac{\partial \chi}{\partial y}=-cx$$ Hence electric field $$ \vec{E}=-c(y\hat{i}+x\hat{j})$$
TIFR 2013 Problem 38 Solution has been written for TIFR entrace preparation series. A problem using the idea of eigenvector from Linear Algebra.