Try this beautiful problem, useful for Physics Olympiad based on Two Blocks on an Inclined Plane. The Problem: Two blocks of masses (4kg) and (8kg) are connected by a string and slide down a (30^\circ) inclined plane. The coefficient of kinetic friction between the (4Kg) block and the plane is (0.25); that between the (8Kg) […]
Try this problem useful for the Physics Olympiad based on the Position of a Particle. The problem: A particle of mass m is subject to a force $$ F(t)=me^{-bt}$$. The initial position and speed are zero. Find (x(t)). Solution: In the given problem $$ \ddot{x}=e^{-bt}$$ Integrating this with respect to time gives $$ v(t)=-\frac{e^{-bt}}{b}+A $$ […]
Try this problem from TIFR 2013 problem 5 based on Non-Cyclic Subgroup of \(\mathbb{R}\). Question: TIFR 2013 problem 5 True/False? All non-trivial proper subgroups of \((\mathbb{R},+)\) are cyclic. Hint: What subgroups comes to our mind immediately? Discussion: \((\mathbb{Q},+)\) is a subgroup of \((\mathbb{R},+)\). Is \((\mathbb{Q},+)\) a cyclic group? Suppose \((\mathbb{Q},+)\) is cyclic. Then there exists a generator say […]
Let's discuss a beautiful problem useful for Physics Olympiad based on the leaning ladder. The Problem: The Leaning Ladder A ladder leans against a frictionless wall. If the coefficient of friction with the ground is \(\mu\), what is the smallest angle that the ladder can make with the ground and not slip? Solution: Let the […]
Try this problem from TIFR 2013 problem 9 based on the existence of an element of order 51. You may use sequential hints if required.
TIFR 2013 Problem 10 Solutions 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 […]
Let's discuss a problem, useful for Physics Olympiad based on A Pulley System. Read the problem carefully, find it yourself and then read the solution.
Try this problem from TIFR 2013 problem 7 based on Eigenvectors of Similar Matrices. Question: TIFR 2013 problem 7 True/False? If \(A\) and \(B\) are similar matrices then every eigenvector of \(A\) is an eigenvector of \(B\). Hint: What does similar matrices mean? Discussion: There exists an invertible (change of basis) matrix \(P\) such that […]
Let's discuss a problem useful from Physics Olympiad based on Instantaneous Velocity and Acceleration. The Problem: Let (\vec{v}) and (\vec{a}) be instantaneous velocity and the acceleration respectively of a particle moving in a plane. The rate of change of speed (dv/dt) of the particle is: (a) (|a|) (b) ((v.a)/|v|) (c) the component of (\vec{a}) in the […]
TIFR 2013 Problem 2 Solutions 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 […]