Embibe Experts Solutions for Chapter: Work, Energy and Power, Exercise 1: JEE Advanced Paper 1 - 2014

A thin and uniform rod of mass $\mathrm{M}$ and length $\mathrm{L}$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle ${60}^{\mathrm{o}}$ with vertical?
[ $\mathrm{g}$ is the acceleration due to gravity]

A small particle of mass $\mathrm{m}$ moving inside a heavy, hollow and straight tube along the tube axis, undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $\mathrm{L}={\mathrm{L}}_0$ the particle speed is $\mathrm{v}={\mathrm{v}}_0.$ The piston is moved inward at a very low speed $\mathrm{V}$ such that $\mathrm{V}\ll \frac{\mathrm{d}\mathrm{L}}{\mathrm{L}}{\mathrm{v}}_0,$ where $\mathrm{d}\mathrm{L}$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as $\frac{dK}{dt}=\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting the surface. The force on the ball during the collision is proportional to the length of compression of the ball. Which one of the following sketches describes the variation of its kinetic energy $\mathrm{K}$ with $\mathrm{t}$time most appropriately? The figures are only illustrative and not to the scale.

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is

A particle is moved along a path $\mathrm{A}\mathrm{B}-\mathrm{B}\mathrm{C}-\mathrm{C}\mathrm{D}-\mathrm{D}\mathrm{E}-\mathrm{E}\mathrm{F}-\mathrm{F}\mathrm{A},$ as shown in figure, in presence of a force $\overrightarrow{\mathrm{F}}=\left(\mathrm{\alpha }\mathrm{y}\widehat{\mathrm{i}}+2\mathrm{\alpha }\mathrm{x}\widehat{\mathrm{j}}\right)\mathrm{N},$ where $x$ and $\mathrm{y}$ are in meter and $\mathrm{\alpha }=-1\mathrm{N} {\mathrm{m}}^{-1}.$ The work done on the particle by this force $\overrightarrow{\mathrm{F}}$ will be ____ Joule.

A flat plate is moving normal to its plane through a gas under the action of a constant force $F$. The gas is kept at a very low pressure. The speed of the plate $v$ is much less than the average speed $u$ of the gas molecules. Which of the following options is/are true?

The work done on a particle of mass $m$ by a force $K\left[\frac{x}{{\left(x^2+y^2\right)}^{3/2}}\widehat{i}+\frac{y}{{\left(x^2+y^2\right)}^{3/2}}\widehat{j}\right]$ ($K$ being a constant of appropriate dimensions, when the particle is taken from the point $(a,0)$ to the point $(0,a)$ along a circular path of radius $a$ about the origin in the x-y plane is