D. C. Pandey Solutions for Chapter: Solved Paper 2015 JEE Main Joint Entrance Examination Physics, Exercise 1: Objective Problems

From a solid sphere of mass $M$ and radius $R\text{,}$ a spherical portion of radius $\left(\frac{R}{2}\right)$ is removed as shown in the figure. Taking gravitational potential $V=0$ at $r=\infty ,$ the potential at the centre of the cavity thus formed is ($G=$gravitational constant)

A pendulum made of a uniform wire of cross-sectional area $A$ has time period $T\text{.}$ When an additional mass $M$ is added to its bob, the time period changes $T_M\text{.}$ If Young's modulus of the material of the wire is $Y\text{,}$ then $\frac{1}{Y}$ is equal to ($g=$gravitational acceleration)

Consider a spherical shell of radius $R$ at temperature $T\text{.}$ The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u=\frac{U}{V}\propto T^4$ and pressure $p=\frac{1}{3}\left(\frac{U}{V}\right)\text{.}$ If the shell now undergoes an adiabatic expansion, the relation between $T$ and $R$ is

A solid body of constant heat capacity $1 \mathrm{J} {}^{\circ }\mathrm{C}^{-1}$ is being heated by keeping it in contact with reservoirs in two ways
(i) Sequentially keeping in contact with $2$ reservoirs such that each reservoir supplies the same amount of heat.
(ii) Sequentially keeping in contact with $8$ reservoirs such that each reservoir supplies the same amount of heat.

In both the cases, the body is brought from initial temperature $100 °\mathrm{C}$ to final temperature $200 °\mathrm{C}$. Entropy change of the body in the two cases respectively, is

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as $V^q\text{,}$ where, $V$ is the volume of the gas. The value of $q$ is $\left(\gamma =\frac{C_p}{C_V}\right)$

For a simple pendulum, a graph is plotted between its Kinetic energy $\left(\mathrm{K}.\mathrm{E}.\right)$ and Potential energy $\left(\mathrm{P}.\mathrm{E}.\right)$ against its displacement $d$.  Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

A train is moving on a straight track with speed $20 \mathrm{m} {\mathrm{s}}^{-1}$. It is blowing its whistle at the frequency of $1000 \mathrm{Hz}$. The percentage change in the frequency heard by a person standing near the track as the train passes him is close to (speed of sound $=320 \mathrm{m} {\mathrm{s}}^{-1}$)

A ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{M}{8}$ at rest at $O.$ These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant, the angular speed of the system is $\frac{8}{9}\omega$ and one of the masses is at a distance of $\frac{3}{5}R$ from $O.$ At this instant, the distance of the other mass from $O$ is,