Solve the following :
Question: The descending pulley shown in figure has a radius $20 \mathrm{~cm}$ and moment of inertia $0.20 \mathrm{~kg}^{-} \mathrm{m}^{2}$. The fixed pulley is light and the horizontal plane frictionless. Find the acceleration of the block if its mass is $1.0 \mathrm{~kg}$. Solution: Translatory Motion Equation $T_{1}=1, a-(\mathrm{i})$ $\mathrm{mg}-\left(T_{1}+T_{2}\right)=\mathrm{m}^{\frac{a}{2}}-(\mathrm{ii})$ Rotational Motion Equation $\tau=\mathrm{I} \alpha$ $\operatorname{mg}-\left(T_{1}...
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Question: The pulleys in figure are identical each having a radius $\mathrm{R}$ and moment of inertia $\mathrm{I}$. Find the acceleration of block $M$. Solution: $m g-T_{1}=m a-\cdots--(i)$ $\left(T_{1}-T_{2}\right) r_{1}=I \alpha$ $a=\alpha r$ $\left(T_{1}-T_{2}\right)=\frac{I \alpha}{R^{2}}$ For pulley $1-\cdots-($ ii $)$ $\left(T_{2}-T_{3}\right)=\frac{I \alpha}{R^{2}}$ For pulley $2 \cdots-\cdots($ iii $)$ For block of mass $m$ $T_{3}-m g=m a----(i v)$ $\mathrm{T}_{1}-T_{3}=\frac{2 I \alpha}...
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Question: Suppose the smaller pulley of the previous problem has its radius $5.0 \mathrm{~cm}$ and moment of inertia $0.1 \mathrm{kgm}^{2}$. Find the tension in the part of the string joining the pulleys. Solution: $m g-T_{1}=m a-\cdots-(i)$ $\left(T_{1}-T_{2}\right) r_{1}=I_{1} \alpha-----(i i)$ $T_{2} r_{2}=I_{2} \alpha-\cdots--(i i i)$ substitutingthe value of $T_{2}$ in equation (ii) $\left(T_{1}-\frac{I_{2} \alpha}{r_{2}}\right) r_{1}=I_{1} \alpha$ $T_{1}-\frac{I_{2} \alpha}{r_{2}}=\frac{I_...
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Question: A sting is wrapped on a wheel of moment of inertia $I=0.20 \mathrm{kgm}{ }^{2}$ and radius $10 \mathrm{~cm}$ and goes through a light pulley to support a ablock of mass $2.0 \mathrm{~kg}$ as shown in the figure. Find the acceleration of the block Solution: Moment of inertia of the pulley. $I=0.20 \mathrm{kgm}^{2}$ From the diagram we get $m g-T=m a-\cdots-(i)$ $\operatorname{Tr}=\mathrm{I} \alpha$ $a=\alpha r$ $\mathrm{T}=\frac{\mathrm{I} \alpha}{r^{2}}-----(i i)$ $U \sin g$ equation (...
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Question: Figure shows two blocks of masses $m$ and $M$ connected by a string passing over a pulley. The horizontal table over which the mass $m$ slides is smooth. The pulley has a radius $r$ and moment if inertia I about its axis and it can freely rotate about the axis. Find the acceleration of the mass $M$ assuming that the string does not slip on the pulley. Solution: $\mathrm{Mg}_{-} T_{2}=\mathrm{Ma}-(\mathrm{i})$ $\mathrm{T}_{2} r-T_{1} r=I \alpha$ $T_{1}=m a$-(iii) $a=r a-(i v)$ Solving, ...
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Question: Suppose the rod in the previous problem has a mass of $1 \mathrm{~kg}$ distributed uniformly over its length. (a) Find the initial angular acceleration of the rod. (b) Find the tension in the supports to the blocks of mass $2 \mathrm{~kg}$ and $5 \mathrm{~kg}$. Solution: (a) $\tau=I \alpha$ $(50 \times 0.5-20 \times 0.5)=\left[(2)^{(0.5)^{2}}+5^{(0.5)^{2}}+\frac{(1)(1)^{2}}{12}\right] \alpha$ $\alpha=8.1 \frac{\operatorname{rad}}{\sec ^{2}}$ (b) $a_{t}=R \alpha=(0.5)(8) \approx 4$ $T_{...
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Question: A light rod of length $1 \mathrm{~m}$ is pivoted at its centre and two masses of $5 \mathrm{~kg}$ ang $2 \mathrm{~kg}$ are hung from the ends as shown in figure. Find the initial angular acceleration of the rod assuming that it was horizontal in the beginning. Solution: $\tau_{\text {Net }}=50 \times 0.5-20 \times 0.5$ $\tau_{\text {Net }}=15 \mathrm{~N}-\mathrm{m}$ $\tau=I \alpha$ $15=\left[(2)(0.5)^{2}+5(0.5)^{2}\right] \alpha$ $\alpha=8.57 \sec ^{2}$...
Read More →Three samples A, B and C of the same gas (γ=1.5)
Question: Three samples $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of the same gas $(\gamma=1.5)$ have equal volume and temperatures. The volume of each sample is doubled, the process being isothermal for A, adiabatic for B and isobaric for $C$. If the final pressure are equal for the three samples, find the ratio of the initial pressures. Solution:...
Read More →A sample of an ideal gas (γ=1.5) is compressed adiabatically
Question: A sample of an ideal gas $(\gamma=1.5)$ is compressed adiabatically from a volume of $150 \mathrm{~cm}^{3}$ to $50 \mathrm{~cm}^{3}$. The initial pressure and the initial temperature are $150 \mathrm{kPa}$ and $300 \mathrm{~K}$. Find (a) the number of moles of the gas in the sample, (b) the molar heat capacity at constant volume, (c) the final pressure and temperature, (d) the work done by the gas in the process and (e) the change in internal energy of the gas. Solution:...
Read More →Consider a given sample of an ideal gas (Cp/Cv=γ)
Question: Consider a given sample of an ideal gas $\left(C_{p} / C_{v}=\gamma\right)$ having initial pressure $p_{0}$ and volume $V_{0}$. (a) The gas is isothermally taken to a pressure $\mathrm{p}_{0} / 2$ and from there adiabatically to a pressure $\mathrm{p}_{0} / 4$. Find the final volume. (b) The gas is brought back to its initial state. It is adiabatically taken to a pressure $p_{0} / 2$ and from there isothermally to a pressure $p_{0} / 4$. Find the final volume. Solution:...
Read More →The initial pressure and volume of a given mass of
Question: The initial pressure and volume of a given mass of a gas $\left(C_{p} / C_{v}=\gamma\right)$ are $p_{0}$ and $V_{0}$. The gas can exchange heat with the surrounding. (a) It is slowly compressed to $\mathrm{V}_{0} / 4$. Find the final pressure. (b) If the gas is suddenly compressed from the volume $\mathrm{V}_{0}$ to $\mathrm{V}_{0} / 2$ and then slowly compressed to $\mathrm{V}_{0} / 4$, what will be the final pressure? Solution:...
Read More →A gas is enclosed in a cylindrical can fitted with a piston.
Question: A gas is enclosed in a cylindrical can fitted with a piston. The walls of the can and the piston are adiabatic. The initial pressure, volume and temperature of the gas are $100 \mathrm{kPa}, 400 \mathrm{~cm}^{3}$ and $300 \mathrm{~K}$ respectively. The ratio of the specific heat capacities of the gas is $C_{p} / C_{v}=1.5$. Find the pressure and the temperature of the gas if it is (a) suddenly compressed (b) slowly compressed to $100 \mathrm{~cm}^{3}$. Solution:...
Read More →Air (γ = 1.4) is pumped at 2atm pressure in a motor tyre at 20°c.
Question: Air $(\gamma=1.4)$ is pumped at 2 atm pressure in a motor tyre at $20^{\circ} \mathrm{C}$. If the tyre suddenly bursts, what would be the temperature of the air coming out of the tyre. Neglect any mixing with the atmospheric air. Solution:...
Read More →An ideal gas at pressure
Question: An ideal gas at pressure $2.5 \times 10^{5} \mathrm{~Pa}$ and temperature $300 \mathrm{~K}$ occupies $100 \mathrm{cc}$. It is adiabatically compressed to half its original volume. Calculate (a) the final pressure, (b) the final temperature and (c) the work done by the gas in the process. Take $\gamma=1.5$. Solution:...
Read More →The volume of an ideal gas (???? = 1.5) is changed adiabatically
Question: The volume of an ideal gas $(\gamma=1.5)$ is changed adiabatically from $4.00$ liters to $3.00$ liters. Find the ratio of (a) the final pressure to the initial pressure and (b) the final temperature to the initial temperature. Solution:...
Read More →In Joly's differential steam calorimeter,
Question: In Joly's differential steam calorimeter, $3 \mathrm{~g}$ of an ideal gas is contained in a rigid closed sphere at $20^{\circ} \mathrm{c}$. The sphere is heated by steam at $100^{\circ} \mathrm{C}$ and it is found that an extra $0.095 \mathrm{~g}$ steam has condensed into water as the temperature of the gas becomes constant. Calculate the specific heat capacity of the gas in $\mathrm{J} / \mathrm{g}-\mathrm{K}$. The latent heat of vaporization of water $=540 \mathrm{cal} / \mathrm{g}$....
Read More →An ideal gas (γ=1.67) is taken through the process abc shown in fig.
Question: An ideal gas $(\gamma=1.67)$ is taken through the process abc shown in fig. The temperature at the point a is $300 \mathrm{~K}$. Calculate (a) the temperature at $b$ and $c, (b)$ the work done in the process, (c) the amount of heat supplied in the path $a b$ and in the path bc and (d) the change in the internal energy of the gas in the process. Solution:...
Read More →Half mole of an ideal gas (???? = 5/3) is taken through
Question: Half mole of an ideal gas $(\gamma=5 / 3)$ is taken through the cycle abcda as shown in fig. Take $\mathrm{R}=\frac{25}{3} \mathrm{~J} / \mathrm{mol}-\mathrm{K}$ (a) Find the temperature of the gas in the states a,b,c and $d$. (b) Find the amount of heat supplied in the processes ab and bc (c) Find the amount of heat liberated in the processes cd and da. Solution:...
Read More →A mixture contains 1 mole of helium
Question: A mixture contains 1 mole of helium $\left(C_{p}=2.5 R, C_{v}=1.5 R\right)$ and 1 mole of hydrogen $\left(C_{p}=3.5 R, C v=2.5 R\right)$. Calculate the values of $C_{p}, C_{v}$ and $\gamma$ for the mixture. Solution:...
Read More →Two ideal gases have the same value of (c9/cv=????).
Question: Two ideal gases have the same value of $\left(c_{9} / c_{v}=\gamma\right)$. What will be the value of this ratio for a mixture of the two gases in the ratio $1: 2$ ? Solution:...
Read More →An ideal gas (c9/cv=????) is taken through a process in
Question: An ideal gas $\left(c_{9} / c_{\mathrm{V}}=\gamma\right)$ is taken through a process in which the pressure and the volume vary as $p=a V^{b}$. Find the value of $b$ for which the specific heat capacity in the process is zero. Solution:...
Read More →An ideal gas (c9/cv=????) is taken through a process in
Question: An ideal gas $\left(c_{9} / c_{\mathrm{V}}=\gamma\right)$ is taken through a process in which the pressure and the volume vary as $p=a V^{b}$. Find the value of $b$ for which the specific heat capacity in the process is zero. Solution:...
Read More →An ideal gas is taken through a process in
Question: An ideal gas is taken through a process in which the pressure and the volume are changed according to the equation $p=k V$. Show that the molar heat capacity of the gas for the process is given by $C=C_{v}+\frac{R}{2}$. Solution:...
Read More →An amount Q of heat is added to a monatomic ideal gas in a process
Question: An amount $\mathrm{Q}$ of heat is added to a monatomic ideal gas in a process in which the gas performs a work $\mathrm{Q} / 2$ on its surrounding. Find the molar heat capacity for the process. Solution:...
Read More →An ideal gas expands from
Question: An ideal gas expands from $100 \mathrm{~cm}^{3}$ to $200 \mathrm{~cm}^{3}$ at a constant pressure of $2.0 \times 10^{5} \mathrm{~Pa}$ when $50 \mathrm{~J}$ of heat is supplied to it. Calculate (a) the change in internal energy of the gas, (b) the number of moles in the gas if the initial temperature is 300k, (c) the molar heat capacity $\mathrm{C}_{\mathrm{p}}$ at constant pressure and (d) the molar heat capacity $\mathrm{C}_{\mathrm{v}}$ at constant volume. Solution:...
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