Let's discuss the problem where we have to find the potential of metal sphere. The Problem: A metal sphere having a radius (r_1) charged to a potential (\phi_1) is enveloped by a thin-walled conducting spherical shell of radius (r_2). Determine the potential (\phi_2) acquired by the sphere after it has been connected for a short […]
A heavy load is suspended on a light spring with upper half rigidity (2k). The spring is slowly pulled down at the midpoint (a certain work (A) is done thereby) and then released. Determine the maximum kinetic energy (W_k) of the load in subsequent motion. Solution: If the middle of the spring is stretched out […]
TIFR 2013 Problem 27 Solution written for TIFR entrance preparation series. A problem on Bounded derivative and compactness from real analysis
The cosmonauts who landed at the pole found that the force of gravity there is (0.01) of that on the Earth, while the duration of the day on the planet is the same as that on Earth. It turned out that besides that the force of gravity on the equator is zero. Find the radius […]
Let's discuss a problem where we find out the time periods of revolution of two stars. The problem: Time Periods of Revolution of Two Stars The masses of two stars are (m_1) and (m_2) and their separation is (l). Determine the period (T) of their revolution in circular orbits about a common centre. Since the […]
A conveyor belt having a length (l) and carrying a block of mass (m) moves at a velocity (v). Determine the distance covered by the block before it stops. Discussion: The initial velocity of the block relative to the ground is determined from the conditions $$ v_0=at$$ $$ l=v_0t-at^2/2$$ Now, acceleration of the block due […]
Question: True/False? There exists a continuous surjective map from the complex plane onto the non-zero reals. Hint: Search for topological invariants. Discussion: Under a continuous function, connected set must go to connected set. The complex plane (\mathbb{C}) is connected. It's image must be connected. (\mathbb{R}-0) is not connected. So the statement is False.
Four small spheres, each of which you can regard as a point of mass (0.2)Kg are arranged in a square (0.4)m on a side and connected by extremely light rods. Find the moment of inertia of the system about an axis through the centre of the square. Discussion: The length of each side of a […]
Try this problem from TIFR 2013 Problem 25 based on Complete Metric on (0,1). Question: TIFR 2013 problem 25 True/False? There exists a complete metric on the open interval \((0,1)\) inducing the usual topology. Hint: Topologically, (0,1) can be made "equal" to \(\mathbb{R}\), which is a complete space with usual metric. Discussion: Suppose \(f:(0,1)\to \mathbb{R} […]
Try this problem based on Tangential and Radial Acceleration, useful for Physics Olympiad. First, try it yourself, then read the solution.