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.
A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to $$\theta(t)=\gamma t+\beta t^3$$ where (\gamma=0.4rad/s) and (\beta=0.0120 rad/s^3). What is the initial value of the angular velocity? Discussion: The angle through which the merry-go-round has turned varies with time according to $$\theta(t)=\gamma t+\beta t^3$$ where (\gamma=0.4rad/s) […]
TIFR 2013 Problem 24 Solution has been written for TIFR entrance preparation series. A problem on continuous image of a compact set from analysis.
Try this problem based on Angular Velocity and Acceleration, useful for Physics Olympiad. The Problem: A fan blade rotates with angular velocity given by $$ \omega=\gamma-\beta t^2$$ where (\gamma=5)rad/s and (\beta=0.800)rad/s. Calculate the angular acceleration as a function of time. Solution: The angular acceleration is given by $$\alpha=\frac{d\omega}{dt}=-2Bt=(-1.60)t $$ The unit of angular acceleration will […]
How many positive integers less than \(1000\) have the property that the sum of the digits of each such number is divisible by \(7\) and the number itself is divisible by \(3\) ? Suppose \(a,b\) are positive real numbers such that \(a\sqrt{a}+b\sqrt{b}=183\). \(a\sqrt{b}+b\sqrt{a}=182\). Find \(\frac{9}{5}(a+b)\). A contractor has two teams of workers: team A and […]
Try this problem, useful for Physics Olympiad, based on the propeller's angular velocity. The Problem: An airplane propeller is rotating at (1900)rpm (rev/min). (a)Compute the propeller's angular velocity in rad/s. (b) How many seconds does it take for the propeller to run through (35^\circ)? Solution: An airplane propeller is rotating at (1900)rpm (rev/min). (1)rpm = […]
Try this problem 23 from TIFR 2013 named - Complete not compact. Question: TIFR 2013 problem 23 True/False? Let \(X\) be complete metric space such that distance between any two points is less than 1. Then \(X\) is compact. Hint: What happens if you take discrete space? Discussion: Discrete metric space as we know it […]