D. C. Pandey Solutions for Chapter: JEE Advanced 2016 (Solved), Exercise 1: Paper 2

A uniform string of length $20 \mathrm{m}$ is suspended from a rigid support. A short wave pulse is introduced at its lowest end. It starts moving up the string. The time taken to reach the support is (Take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A screw gauge with a pitch of $0.5 \mathrm{mm}$ and a circular scale with $50$ divisions are used to measure the thickness of a thin sheet of aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the $45\mathrm{th}$ division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet, if the main scale reading is $0.5 \mathrm{mm}$ and the $25 \mathrm{th}$ division coincides with the main scale line?

A pipe open at both ends has a fundamental frequency $f$ in air. The pipe is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now

A water cooler of storage capacity $120 \mathrm{litres}$ can cool water at a constant rate of $P \mathrm{watts}$. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly $3 \mathrm{kW}$ of heat (thermal load). The temperature of water fed into the device cannot exceed $30 °\mathrm{C}$ and the entire stored $120 \mathrm{litres}$ of water is initially cooled to $10 °\mathrm{C}$. The entire system is thermally insulated. The minimum value of $P$ (in $\mathrm{watts}$) for which the device can be operated for $3 \mathrm{hours}$ is

(specific heat of water is $4.2 \mathrm{kJ} {\mathrm{kg}}^{-1} {\mathrm{K}}^{-1}$ and the density of water is $1000 \mathrm{kg} {\mathrm{m}}^{-3}$).

A uniform wooden stick of mass $1.6 \mathrm{kg}$ of length $l$ rests in an inclined manner on a smooth, vertical wall of height $h\left(<l\right)$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $30°$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio $\frac{h}{l}$ and the frictional force $f$ at the bottom of the stick are $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $p_i={10}^5 \mathrm{Pa}$ and volume $V_i={10}^{-3} {\mathrm{m}}^3$ changes to a final state at $p_f=\left(\frac{1}{32}\right)\times {10}^5 \mathrm{Pa}$ and $V_f=8\times {10}^{-3} {\mathrm{m}}^3$ in an adiabatic quasi-static process, such that $p^3{V}^5=$constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $p_i$ followed by an isochoric (isovolumetric) process at volume $V_f$. The amount of heat supplied to the system in the two-step process is approximate,

There are two Vernier calipers both of which have $1 \mathrm{cm}$ divided into $10$ equal divisions on the main scale. The Vernier scale of one of the calipers $\left(C_1\right)$ has $10$ equal divisions that correspond to $9$ main scale divisions. The Vernier scale of the other caliper $\left(C_2\right)$ has $10$ equal divisions that correspond to $11$ main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in $\mathrm{cm}$) by calipers $C_1$ and $C_2$ respectively, are

The ends $Q$ and $R$ of two thin wires, $PQ$ and $RS,$ are soldered (joined) together. Initially, each of the wire has a length of $1 \mathrm{m}$ at $10 °\mathrm{C}$. Now, the end $P$ is maintained at $10 °\mathrm{C},$ while the end $S$ is heated and maintained at $400 °\mathrm{C}.$ The system is thermally insulated from its surroundings. If the thermal conductivity of wire $PQ$ is twice that of the wire $RS$ and the coefficient of linear thermal expansion of $PQ$ is $1.2\times {10}^{-5} {\mathrm{K}}^{-1}$, the change in length of the wire $PQ$ is