Domanda del libro di testo
The 50 kg circular piston shown in FIGURE P18.57 floats on 0.12 mol of compressed air. How far does the piston move if the temperature is increased by 100°C?
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Ch 18: A Macroscopic Description of Matter
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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TOC.tableOfContents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
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Capitolo 18, Problema 53
A 10-cm-diameter, 40-cm-tall gas cylinder, sealed at the top by a frictionless 50 kg piston, is surrounded by a bath of 20°C water. Then 50 kg of sand is slowly poured onto the top of the piston, where it stays. Afterward, what is the height of the piston?
Guida verificata passo dopo passo1
Determine the initial pressure inside the gas cylinder. The pressure is due to the weight of the piston and the atmospheric pressure. Use the formula for pressure: \( P = P_{\text{atm}} + \frac{F}{A} \), where \( F \) is the force due to the weight of the piston (\( F = m g \)) and \( A \) is the cross-sectional area of the cylinder (\( A = \pi r^2 \)).
Calculate the initial volume of the gas. The volume of a cylinder is given by \( V = A h \), where \( A \) is the cross-sectional area and \( h \) is the initial height of the gas column (40 cm).
Apply the ideal gas law to relate the initial and final states of the gas. The law is \( P_1 V_1 = P_2 V_2 \), assuming the temperature remains constant (isothermal process). Here, \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.
Determine the final pressure \( P_2 \). The final pressure includes the weight of the piston, the added sand, and the atmospheric pressure. Use \( P_2 = P_{\text{atm}} + \frac{(m_{\text{piston}} + m_{\text{sand}}) g}{A} \).
Solve for the final height \( h_2 \) of the gas column. Rearrange the ideal gas law to \( h_2 = \frac{P_1 h_1}{P_2} \), substituting the known values for \( P_1 \), \( h_1 \), and \( P_2 \). This gives the final height of the piston.
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Hydrostatic Pressure
Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid and is calculated using the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. In this scenario, the pressure exerted by the sand on the piston must be balanced by the pressure from the water and the gas in the cylinder.
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Percorso guidato
17:04
Pressure and Atmospheric Pressure
Ideal Gas Law
The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. In this case, the gas in the cylinder will respond to changes in pressure and volume as the piston moves. Understanding this relationship is crucial for determining how the height of the piston changes when additional weight is added.
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Percorso guidato
07:21Ideal Gases and the Ideal Gas Law
Equilibrium of Forces
In this scenario, the system reaches equilibrium when the forces acting on the piston are balanced. The weight of the sand and the piston must equal the upward force exerted by the gas pressure in the cylinder. Analyzing the equilibrium of forces allows us to calculate the new height of the piston after the sand is added, ensuring that all forces are accounted for.
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Percorso guidato
05:10Equilibrium
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A diving bell is a 3.0-m-tall cylinder closed at the upper end but open at the lower end. The temperature of the air in the bell is 20°C. The bell is lowered into the ocean until its lower end is 100 m deep. The temperature at that depth is 10°C. How high does the water rise in the bell after enough time has passed for the air inside to reach thermal equilibrium?
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The 50 kg circular piston shown in FIGURE P18.57 floats on 0.12 mol of compressed air. What is the piston height h if the temperature is 30°C?
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