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 […]
Try this problem, useful for the Physics Olympiad Problem based on total charge of a sphere. The Problem: Suppose a charge (Q) is distributed within a sphere of radius (R) in such a way that the charge density (\rho(r)) at a distance r from the centre of the sphere is $$ \rho(r)=K(R-r) \hspace{2mm }for\hspace{2mm} 0<r<R$$ […]
TIFR 2013 Math Solution Discussion Question: The sets \([0,1)\) and \((0,1)\) are homeomorphic. Hint: Check some topological invariant. TIFR 2013 Math Solution Also Visit: College Mathematics Program Discussion: In \([0,1)\), \(0\) seems to be a special point, as compared to \((0,1)\) where every point has equal importance (or non-importance). If \(f:X\to Y \) is a homeomorphism […]
Try this problem, useful for Physics Olympiad based on Total Charge of a Circular Wire. The Problem: A circular wire of radius (a) has linear charge density $$\lambda=\lambda_0cos^2\theta$$ where (\theta) is the angle with respect to a fixed radius. Calculate the total charge. Solution: Charge on an element (dl=ad\theta) is $$ \lambda_0cos^2\theta.ad\theta$$ Total charge $$ […]
Try this problem, useful for Physics Olympiad, based on a bouncing ball. The Problem: A ball is dropped from a height (h) above a horizontal concrete surface. The coefficient of restitution for the collision involved is (e). What is the time after which the ball stops bouncing? Discussion: The time required for the free fall […]