Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}$ Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ :. $\cos ^{-1}\left\{\sqrt{1-\mathrm{x}^{2}}\right\}$ Solution:...
Read More →Standing waves of frequency 5.0 kHz are produced
Question: Standing waves of frequency $5.0 \mathrm{kHz}$ are produced in a tube filled with oxygen at $300 \mathrm{~K}$, The separation between the consecutive nodes is $3.3 \mathrm{~cm}$. Calculate the specific heat capacities $C_{p}$ and $C_{v}$ of the gas. Solution:...
Read More →An ideal gas having density
Question: An ideal gas having density $1.7 \times 10^{-3} \mathrm{~g} / \mathrm{cm}^{3}$ at a pressure $1.5 \times 10^{5} \mathrm{~Pa}$ is filled in a Kundt's tube. When the gas is resonated at a frequency of $3.0 \mathrm{kHz}$, nodes are formed at a separation of $6.0 \mathrm{~cm}$. Calculate the molar heat capacities $C_{p}$ and $C_{v}$ of the gas. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cos ^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\}$ Solution:...
Read More →4.0g of helium occupies
Question: 4.0g of helium occupies $22400 \mathrm{~cm}^{3}$ at STP. The specific heat capacity of helium at constant pressure is $5.0 \mathrm{cal} / \mathrm{mo} ;-\mathrm{K}$. Calculate the speed of sound in helium at STP. Solution:...
Read More →The speed of sound in hydrogen at 0°C is
Question: The speed of sound in hydrogen at $0^{\circ} \mathrm{C}$ is $1280 \mathrm{~m} / \mathrm{s}$. The density of hydrogen at STP is $0.089 \mathrm{~kg} / \mathrm{m}^{3}$. Calculate the molar heat capacities $C_{p}$ and $C_{v}$ of hydrogen. Solution:...
Read More →An adiabatic cylindrical tube of cross-sectional area
Question: An adiabatic cylindrical tube of cross-sectional area $1 \mathrm{~cm}^{3}$ is closed at one end and fitted with a piston at the other end. The tube contains $0.03 \mathrm{~g}$ of an ideal gas. At 1 atm pressure and at the temperature of the surrounding, the length of the gas column is $40 \mathrm{~cm}$. The piston is suddenly pulled out to double the length of the column. The pressure of the gas falls to $0.355 \mathrm{~atm}$. Find the speed of sound in the gas at atmospheric temperatu...
Read More →Fig shows an adiabatic cylindrical tube of volume
Question: Fig shows an adiabatic cylindrical tube of volume $V_{0}$ divided in two parts by a frictionless adiabatic separator. Initially, the separator is kept in the middle, an ideal gas at pressure $\mathrm{p}_{1}$ and temperature $\mathrm{T}_{1}$ is injected into the left part and another ideal gas at pressure $p_{2}$ and temperature $T_{2}$ is injected into the right part $\left(C_{p} / C_{v}=\gamma\right)$ is the same for both the gases. The separator is slid slowly and is released at a po...
Read More →Two vessels A and B of equal V0 are connected by
Question: Two vessels $A$ and $B$ of equal $V_{0}$ are connected by a narrow tube which can be closed by a valve. The vessels are fitted with pistons which can be moved to change the volumes. Initially, the valve is open and the vessels contain an ideal gas $\left(C_{p} / C_{v}=\gamma\right)$ at atmospheric pressure $\mathrm{p}_{0}$ and atmospheric temperature $T_{0}$. The walls of the vessel $A$ are diathermic and those of $B$ are adiabatic. The valve is now closed and the pistons are slowly pu...
Read More →Figure shows two vessels with adiabatic walls,
Question: Figure shows two vessels with adiabatic walls, one containing $0.1 \mathrm{~g}$ of helium $(\gamma=1.67) \mathrm{M}=4 \mathrm{~g} / \mathrm{mol})$ and the other containing some amount of hydrogen $(\gamma=1.4, \mathrm{M}=2 \mathrm{~g} / \mathrm{mol})$. Initially, the temperatures of the two gases are equal. The gases are electrically heated for some time during which equal amounts of heat are given to the two gases. It is found that the temperatures rise through the same amount in the ...
Read More →Figure shows two rigid vessels A and B each of
Question: Figure shows two rigid vessels $A$ and $B$ each of volume $200 \mathrm{~cm}^{3}$ containing an ideal gas $\left(C_{\mathrm{V}}=12.5 \mathrm{~J} / \mathrm{mol}\right.$ K). The vessels are connected to a manometer tube containing mercury. The pressure in both the vessels is $75 \mathrm{~cm}$ of mercury and the temperature is $300 \mathrm{~K}$. (a) Find the number of moles of the gas in the vessel. (b) $5.0 \mathrm{~J}$ of heat is supplied to the gas in the vessel $A$ and $10 \mathrm{~J}$...
Read More →Fig. shows a cylindrical tube with adiabatic walls and fitted with
Question: Fig. shows a cylindrical tube with adiabatic walls and fitted with an adiabatic separator. The separator can be slid into the tube by an external mechanism. An ideal gas $(\gamma=1.5)$ is injected in the two sides at equal pressures and temperatures. The separator remains in equilibrium at the middle. It is now slid to a position where it divides the tube in the ratio $1: 3$. Find the ratio of the temperatures in the two parts of the vessel. Solution:...
Read More →1 liter of an ideal gas (γ=1.5) at 300 K is suddenly compressed
Question: 1 liter of an ideal gas $(\gamma=1.5)$ at $300 \mathrm{~K}$ is suddenly compressed to half its original volume. (a) Find the ratio of the final pressure to the initial pressure. (b) If the original pressure is $100 \mathrm{kPa}$, find the work done by the gas in the process. (c) What is the change in internal energy? (d) What is the final temperature? (e) The gas is now cooled to $300 \mathrm{~K}$ keeping its pressure constant. Calculate the work done during the process. (f) The gas is...
Read More →Two samples A and B of the gas have equal volumes and pressures.
Question: Two samples $A$ and $B$ of the gas have equal volumes and pressures. The gas in sample $A$ is expanded isothermally to double its volume and the gas in $B$ is expanded adiabatically to double its volume. If the work expanded adiabatically to double its volume. If the work done by the gas is the same for the two cases, show that $\gamma_{\text {satisfies the equation }} 1-2^{1-\gamma}=(\gamma-1) \ln 2$ Solution:...
Read More →Solve the following :
Question: A string is wrapped over the edge of a uniform disc and the free end is fixed with the ceiling. The disc moves down, unwinding the string. Find the downward acceleration of the disc. Solution: $\mathrm{mg}-\mathrm{T}=\mathrm{ma}-(\mathrm{i})$ $\mathrm{T}=\mid \alpha$ $T \cdot R=\frac{m R^{2}}{2}\left(\frac{a}{R}\right)$ From (i) and (ii) $a=\frac{2 g}{3}$...
Read More →Solve the following :
Question: A sphere of mass $m$ rolls on a plane surface. Find its kinetic energy at an instant when its centre moves with speed $\mathrm{V}$. Solution: $\mathrm{V}=\mathrm{R} \omega$ (Pure rolling) $\mathrm{K} . \mathrm{E}=\frac{1}{2} I \omega^{2}+\frac{1}{2} m v^{2}$ $\quad \frac{1}{2}\left(\frac{2}{5} m R^{2}\right)\left(\frac{v^{2}}{R^{2}}\right)+\frac{1}{2} m v^{2}$ $\mathrm{~K} . \mathrm{E}=\frac{7}{10} m v^{2}$...
Read More →Solve the following :
Question: A cylinder rolls on a horizontal plane surface. If the speed of the centre is $25 \mathrm{~m} / \mathrm{s}$, what is the speed of the highest point? Solution: For pure rolling $v_{C}=R \omega$ Speed at highest point= $\mathrm{v}+\quad R \omega$ $=2 \mathrm{v}$ $=2(25)$ $=50 \mathrm{~m} / \mathrm{s}$...
Read More →Solve the following :
Question: A uniform rod pivoted at its upper end hangs vertically. It is displaced through an angle of $60^{\circ}$ and then released. Find the magnitude of the force acting on a particle of mass $\mathrm{dm}$ at the tip of the rod when the rod makes an angle of $37^{\circ}$ with the vertical. Solution: By energy conservation $\frac{1}{2} I \omega^{2}=m g h$ $\frac{1}{2}\left(\frac{m L^{2}}{3}\right) \omega^{2}=m g \frac{l}{2}\left(\cos 37^{\circ}-\cos 60^{\circ}\right)$ $\omega^{2}=\frac{9 g}{1...
Read More →Solve the following :
Question: A metre stick weighing $240 \mathrm{~g}$ is pivoted at its upper end in such a way that it can freely rotate in a vertical plane through the end. A particle of mass $100 \mathrm{~g}$ is attached to the upper end of the stick through a light string of length I $\mathrm{m}$. Initially, the rod is kept vertical and the string horizontal when the system is released from rest. The particle collides with the lower end of the stick there. Find the maximum angle through which the stick will ri...
Read More →Solve the following :
Question: A metric stick is held vertically with one end on a rough horizontal floor. It is gently allowed to fall on the floor. Assuming that the end at the floor does not slip, find the angular speed of the rod when it hits the floor. Solution: By energy conservation $\mathrm{mg}^{2}=\frac{1}{2} I \omega^{2}$ $m a^{2} \frac{L}{2}=\frac{1}{m L^{2}}\left(\frac{\omega^{2}}{3}\right) \omega^{2}$ $\omega=\sqrt{\frac{3 g}{l}}=\sqrt{3 \times \frac{9.8}{1}}$ $\omega=5.42 \mathrm{rad} / \mathrm{sec}$...
Read More →Solve the following :
Question: The pulley shown in figure has a radius of $20 \mathrm{~cm}$ and moment of inertia $0.2 \mathrm{~kg}^{-} \mathrm{m}^{2}$. The string going over it is attached at one end to a vertical spring of spring constant $50 \mathrm{~N} / \mathrm{m}$ fixed from below, and supports a $1 \mathrm{~kg}$ mass at the other end. The system is released from rest with the spring at its natural length. Find the speed of the block when it has descended through $10 \mathrm{~cm}$. Take $\mathrm{g}=10 \mathrm{...
Read More →Solve the following :
Question: Two blocks of masses $400 \mathrm{~g}$ and $200 \mathrm{~g}$ are connected through a light string going over a pulley which is free to rotate about its axis. The pulley has a moment of inertia $1.6^{\times 10^{-4}} \mathrm{~kg}-\mathrm{m}^{2}$ and a radius $2.0 \mathrm{~cm}$. Find (a) the kinetic energy of the system as the $400 \mathrm{~g}$ block falls through $50 \mathrm{~cm}$, (b) the speed of the blocks at this instant. Solution: Translatory Motion Equation $0.4 \mathrm{~g}_{-} T_{...
Read More →Solve the following :
Question: Suppose the rod with the balls $A$ and $B$ of the previous problem is clamped at the centre in such a way that it can rotate freely about a horizontal axis through the clamp. The system is kept at rest in the horizontal position. A particle $P$ of the same mass $m$ is dropped from a height $h$ on the ball $B$. The particle collides with B and sticks to it. (a) Find the angular momentum and the angular speed of the system just after the collision. (b) What should be the minimum value of...
Read More →Solve the following :
Question: Two small balls $A$ and $B$, each of mass $m$, are joined rigidly by a light horizontal rod of length $L$. The system translates on a frictionless horizontal surface with a velocity $v_{0}$ in a direction perpendicular to the rod. A particle $P$ of mass $m$ kept at rest on the surface sticks to the ball $A$ as the ball collides with it. Find (a) the linear speeds of the balls $A$ and B after the collision, (b) the velocity of the centre of mass $C$ of the system $A+B+P$ and (c) the sys...
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