A piston of mass $M$ is hung from a massless spring whose restoring force law goes as $F= -kx^3$, where $k$ is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with 'n' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature $T$) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height $L_0$ to $L_1$, the total energy delivered by the filament is : (Assume spring to be in its natural length before heating)
$nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)$
The problem involves determining the total energy delivered by a heating filament that causes a piston to move upward in a vertical chamber containing an ideal gas. The spring obeys a non-linear restoring force law, and the change occurs isothermally.
Approach:
- Gas Work: For an isothermal process, the work done on the gas can be expressed as: \(W_{\text{gas}} = nRT \ln \left( \frac{V_1}{V_0} \right)\), where \(V_0 = A L_0\) and \(V_1 = A L_1\) are the initial and final volumes respectively. Thus, \(W_{\text{gas}} = nRT \ln \left( \frac{L_1}{L_0} \right)\).
- Gravitational Work: The work done against gravity as the piston moves from height \(L_0\) to \(L_1\) is given by: \(W_{\text{gravity}} = Mg (L_1 - L_0)\).
- Spring Work: For a spring force given by \(F = -kx^3\), the potential energy stored in the spring when the displacement changes from \(L_0\) to \(L_1\) is: \(W_{\text{spring}} = \int_{L_0}^{L_1} kx^3 \, dx = \frac{k}{4} (L_1^4 - L_0^4)\).
Summing these works, the total energy delivered by the filament is:
W_{\text{total}} = nRT \ln \left( \frac{L_1}{L_0} \right) + Mg (L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)
This matches the correct option given:
\(nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)\)
Thus, the correct answer is:
Correct Answer: \(nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)\)
Similar Questions
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- 5 beats/second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be
- A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x=0 at t=0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave ?
A particle is subjected to two simple harmonic motions as :
$x_1 = \sqrt{7} \sin 5t \text{ cm}$ and $x_2 = 2\sqrt{7} \sin \left(5t + \frac{\pi}{3}\right) \text{ cm}$
where x is displacement and t is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \text{ ms}^{-2}$. The value of x is :- Two harmonic waves moving in the same direction superimpose to form awave $x = a \cos (1.5t) \cos (50.5t)$ where $t$ is in seconds. Find the period with which they beat.(close to nearest integer)
- In the resonance experiment, two air columns (closed at one end) of $100 \text{ cm}$ and $120 \text{ cm}$ long, give $15 \text{ beats per second}$ when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is:
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
- The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2 = 50 - x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}~\text{s}$, the value of $x$ is ________.
Two simple harmonic motions, as shown below, are at right angles. They are combined to form lissajous figures.
$x(t) = A \sin (at + \delta)$
$y(t) = B \sin (bt)$
Identify the correct match below :
Important Questions from Oscillations and Waves
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- 5 beats/second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be
- A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x=0 at t=0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave ?
A particle is subjected to two simple harmonic motions as :
$x_1 = \sqrt{7} \sin 5t \text{ cm}$ and $x_2 = 2\sqrt{7} \sin \left(5t + \frac{\pi}{3}\right) \text{ cm}$
where x is displacement and t is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \text{ ms}^{-2}$. The value of x is :- Two harmonic waves moving in the same direction superimpose to form awave $x = a \cos (1.5t) \cos (50.5t)$ where $t$ is in seconds. Find the period with which they beat.(close to nearest integer)
