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Chapter 18: Problem 77

A vertical cylinder of radius \(r\) contains an ideal gas and is fitted with a piston of mass \(m\) that is free to move (Fig. \(\mathbf{P 1 8 . 7 7}\) ). The piston and the walls of the cylinder are frictionless, and the entire cylinder is placed in a constant-temperature bath. The outside air pressure is \(p_{0}\). In equilibrium, the piston sits at a height \(h\) above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance \(h+y\) above the bottom of the cylinder, where \(y \ll h\). (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?

Short Answer

Expert verified
The absolute pressure in equilibrium is given by \(P = p_{0} + \frac{mg}{\pi r^2}\). The net force acting on the piston when its base is a distance \(h+y\) is given by the change in gas pressure \(F = \Delta P \pi r^2\). For small oscillations, the frequency is given by \(f = \frac{1}{2 \pi} \sqrt{\frac{\Delta P /\pi r^2}{m}}\). For a large displacement, the oscillations will not be simple harmonic, as the force-displacement relation becomes non-linear.

Step by step solution

01

Calculation of the gas pressure

First, the absolute pressure of the gas trapped below the piston, when in equilibrium, needs to be calculated. This can be found from the principle of equilibrium which states that the sum of the forces acting on a body at rest is zero. Therefore the pressure inside the cylinder is the sum of the atmospheric pressure and the pressure due to the weight of the piston, which is given by \((mg)/A\), where \(m\) is the mass of the piston, \(g\) is the acceleration due to gravity, and \(A\) is the area of the piston. The area \(A\) can be replaced by \(\pi r^2\) where \(r\) is the radius of the piston. Hence the absolute pressure \(P = p_{0} + (mg)/(\pi r^2)\)
02

Calculation of the force on the piston

Next, the net force acting on the piston when it is slightly displaced by a distance \(y\) can be found. The net force is the difference between the gas pressure below and above the piston. The difference in gas pressure is due to the difference in the volume of the gas which changes when the position of the piston changes. Using the ideal gas law \(P = nRT/V\), where \(n\) is the number of moles, \(R\) is the gas constant, \(T\) is the temperature, and \(V\) is the volume, the net force \(F\) can be calculated as \(F = P_{1}\pi r^2 - P_{0}\pi r^2 = \Delta P \pi r^2\), where \(P_{1}\) is the pressure after the piston is displaced and \(\Delta P\) is the change in pressure
03

Calculation of the oscillation frequency

Lastly, the frequency of the small oscillations when the piston is displaced from equilibrium and released needs to be found. This is a problem of simple harmonic motion. The equation for simple harmonic motion is \(F = -kx\), where \(F\) is the net force, \(k\) is the spring constant and \(x\) is the displacement. Comparing this with the force equation from step 2, the spring constant \(k = \Delta P /\pi r^2\). The frequency \(f\) of the oscillation is given by the formula \(f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}\), which can be calculated substituting the value of \(k\). For the oscillations to be simple harmonic, the displacement must be small relative to the total height. A significant displacement will lead to a non-linear relationship between force and displacement, indicating the motion is not simple harmonic.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Simple Harmonic Motion
Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It can be thought of as the motion of a mass on a spring when it is compressed or stretched. Here鈥檚 how it relates to our exercise:

When the piston in the cylinder is pulled up and released, it experiences a force that tries to return it to its equilibrium position. This restoring force, according to Hooke's Law, is given by \( F = -kx \), where \( -k \) is the force constant and \( x \) is the displacement from the equilibrium. When the force is directly proportional to the displacement, and if there is no damping or external force at play, the system will exhibit simple harmonic motion.

However, for the oscillations to maintain SHM characteristics, the displacements \( y \) must be small relative to the equilibrium height \( h \), ensuring that the force remains linearly proportional to the displacement. Under these conditions, the system's behavior can be predicted using SHM equations, and the frequency calculated as outlined in the exercise.
Pressure Equilibrium in Gas Systems
Pressure equilibrium is a condition where the force due to the pressure on one side of an object is balanced by the force due to the pressure on the opposite side. In the context of the exercise, the piston is initially at rest because the force due to both the atmospheric pressure and the gas pressure below the piston balance the gravitational force on the piston.

In achieving pressure equilibrium, the ideal gas law plays a vital role. It relates the pressure \( P \) of an ideal gas with its volume \( V \) and temperature \( T \) in the form \( PV = nRT \), where \( n \) is the number of moles of the gas, and \( R \) is the universal gas constant. When the piston sits at a height \( h \) in equilibrium, the pressure exerted by the gas exactly counterbalances the atmospheric pressure plus the pressure due to the weight of the piston. This equilibrium is outlined by the equation given for the absolute pressure in the exercise.
Determining Oscillation Frequency
The oscillation frequency is how often an object in harmonic motion completes one full cycle of movement, usually measured in hertz (\( Hz \) or cycles per second). The frequency of the oscillation of our piston can be calculated using the formula \( f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant, and \( m \) is the mass of the oscillating body, which in this case, is the piston itself.

This frequency represents how quickly the piston will move up and down after being displaced from its equilibrium position. The spring constant \( k \) in this scenario is derived from the change in pressure \( \Delta P \) as the piston moves away from the equilibrium, which is related to the ideal gas law and the area over which the pressure is acting. It is crucial for students to understand that the spring constant effectively measures the stiffness of the system in response to the applied force and thus determines the rapidity of the oscillations in this example of simple harmonic motion.

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Most popular questions from this chapter

A cylindrical diving bell has a radius of \(750 \mathrm{~cm}\) and a height of \(2.50 \mathrm{~m}\). The bell includes a top compartment that holds an undersea adventurer. A bottom compartment separated from the top by a sturdy grating holds a tank of compressed air with a valve to release air into the bell, a second valve that can release air from the bell into the sea, a third valve that regulates the entry of seawater for ballast, a pump that removes the ballast to increase buoyancy, and an electric heater that maintains a constant temperature of \(20.0^{\circ} \mathrm{C}\). The total mass of the bell and all of its apparatuses is \(4350 \mathrm{~kg}\). The density of seawater is \(1025 \mathrm{~kg} / \mathrm{m}^{3}\). (a) An \(80.0 \mathrm{~kg}\) adventurer enters the bell. How many liters of seawater should be moved into the bell so that it is neutrally buoyant? (b) By carefully regulating ballast, the bell is made to descend into the sea at a rate of \(1.0 \mathrm{~m} / \mathrm{s}\). Compressed air is released from the tank to raise the pressure in the bell to match the pressure of the seawater outside the bell. As the bell descends, at what rate should air be released through the first valve? (Hint: Derive an expression for the number of moles of air in the bell \(n\) as a function of depth \(y ;\) then differentiate this to obtain \(d n / d t\) as a function of \(d y / d t .)\) (c) If the compressed air tank is a fully loaded, specially designed, \(600 \mathrm{ft}^{3}\) tank, which means it contains that volume of air at standard temperature and pressure ( \(0^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) ), how deep can the bell descend?

(a) Show that a projectile with mass \(m\) can "escape" from the surface of a planet if it is launched vertically upward with a kinetic energy greater than \(m g R_{p},\) where \(g\) is the acceleration due to gravity at the planet's surface and \(R_{\mathrm{p}}\) is the planet's radius. Ignore air resistance. (See Problem \(18.70 .\) ) (b) If the planet in question is the earth, at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass \(28.0 \mathrm{~g} / \mathrm{mol}\) ) equal that required to escape? What about a hydrogen molecule (molar mass \(2.02 \mathrm{~g} / \mathrm{mol}\) )? (c) Repeat part (b) for the moon, for which \(g=1.63 \mathrm{~m} / \mathrm{s}^{2}\) and \(R_{\mathrm{p}}=1740 \mathrm{~km}\) (d) While the earth and the moon have similar average surface temperatures, the moon has essentially no atmosphere. Use your results from parts (b) and (c) to explain why.

(a) What is the total translational kinetic energy of the air in an empty room that has dimensions \(8.00 \mathrm{~m} \times 12.00 \mathrm{~m} \times 4.00 \mathrm{~m}\) if the air is treated as an ideal gas at 1.00 atm? (b) What is the speed of a \(2000 \mathrm{~kg}\) automobile if its kinetic energy equals the translational kinetic energy calculated in part (a)?

(a) Calculate the mass of nitrogen present in a volume of \(3000 \mathrm{~cm}^{3}\) if the gas is at \(22.0^{\circ} \mathrm{C}\) and the absolute pressure of \(2.00 \times 10^{-13}\) atm is a partial vacuum easily obtained in laboratories. (b) What is the density (in \(\mathrm{kg} / \mathrm{m}^{3}\) ) of the \(\mathrm{N}_{2}\) ?

A physics lecture room at 1.00 atm and \(27.0^{\circ} \mathrm{C}\) has a volume of \(216 \mathrm{~m}^{3}\). (a) Use the ideal-gas law to estimate the number of air molecules in the room. Assume that all of the air is \(\mathrm{N}_{2} .\) Calculate (b) the particle density - that is, the number of \(\mathrm{N}_{2}\), molecules per cubic centimeter-and (c) the mass of the air in the room.
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