Homework 8: Ps 115l Name Lab Partners

Homework 8 Ps 115l Name Lab Partners

Answer the following questions:

Did you get more or less nodes in with the higher frequency tuning forks? Why do you think this happens?

Our Ventrulli tube works by applying a higher pressure at one end. This way the fluid flows towards the lower pressure. Why do you think that velocity increases with smaller diameter and increases with larger diameters? Is there something conserved?

True or False:

• The height of the water in the U-tube depends on the stagnation pressure (Velocity of the fluid) – True False
• The frequency of an acoustic (sound) wave depends on the wavelength and the translational speed – True False
• Buoyancy is always present when a material is dropped into a fluid (gravity is also present) – True False
• The mass of the object is equal to the mass of the displaced fluid – True False
• A larger volume of air flows through large areas than smaller areas (in our Ventrulli tube experiment) – True False

From the lab, you know that the speed of sound is v = 331.4 + 0.6 °C * T, where T is the current temperature. We know that the frequency of a sound wave at 11°C is 550Hz. What is the wavelength of the acoustic wave?

What is the wavelength of the acoustic wave?

A body has a volume of 0.04 m³ and an average density of 1010 kg/m³. Will the body float or sink in water? If it sinks, what is the acceleration? (Density of water is 1000 kg/m³). (Extra credit) Consider the density of the body to be now 900 kg/m³.

Calculate the equilibrium point where the body is floating without any acceleration. Calculate how much of this body is out of the water (you may assume that the body is a cube). If we submerge it a little more than its equilibrium, what will happen over time? (Density of water is 1000 kg/m³)

Paper For Above instruction

Understanding the principles of acoustics, fluid dynamics, and buoyancy is essential for interpreting the phenomena observed in physics experiments such as the ones involving tuning forks, Venturi tubes, and submerged bodies. This paper explores these topics systematically, providing insights into wave behavior, fluid flow, and the conditions for buoyant equilibrium.

Acoustic Nodes and Frequency

The number of nodes produced in a standing wave pattern within a medium is influenced significantly by the frequency of the driving source. When higher frequency tuning forks are used, more nodes tend to appear. This occurs because the wavelength associated with higher frequencies decreases, leading to more nodes fitting within the same length of the medium. According to the wave equation v = fλ, where v is the wave velocity, f is the frequency, and λ is the wavelength, an increase in frequency results in a decrease in wavelength, thus increasing the number of nodes (White, 2014). These nodes represent points of destructive interference where the medium doesn't vibrate, and their density illustrates the wave's frequency characteristics.

Fluid Velocity and the Venturi Effect

The Venturi tube demonstrates a fundamental principle of fluid dynamics: when fluid flows through a constricted section of pipe, its velocity increases as the cross-sectional area decreases, consistent with the conservation of mass expressed by the continuity equation A₁v₁ = A₂v₂. As the diameter diminishes, the area decreases, compelling the fluid to accelerate to maintain a constant flow rate. Conversely, in larger diameters, the increased cross-sectional area reduces velocity (Fox et al., 2011). The Bernoulli principle indicates a trade-off between velocity and pressure: as the fluid's velocity increases in smaller diameters, the static pressure drops, demonstrating energy conservation within the system.

Behavior of Water Levels and True/False Statements

• The height of the water in the U-tube depends on the stagnation pressure (Velocity of the fluid) – True
• The frequency of an acoustic (sound) wave depends on the wavelength and the translational speed – True
• Buoyancy is always present when a material is dropped into a fluid (gravity is also present) – True
• The mass of the object is equal to the mass of the displaced fluid – False (this is Archimedes' principle, where the buoyant force equals the weight of displaced fluid, not that the masses are equal)
• A larger volume of air flows through large areas than smaller areas (in our Ventrulli tube experiment) – True

The statements reflect core principles: the water height is affected by pressure differences, sound frequency relates to wavelength and wave speed, buoyancy depends on the displaced fluid's weight, and air flow volume correlates with cross-sectional area.

Wavelength Calculation

The speed of sound varies with temperature following v = 331.4 + 0.6 °C T. At 11°C, the speed is v = 331.4 + 0.6 11 = 338.0 m/s. Using the wave equation v = fλ with known f = 550 Hz, the wavelength is computed as:

λ = v / f = 338.0 / 550 ≈ 0.6145 meters

Thus, the wavelength of the acoustic wave is approximately 0.615 meters, which corroborates the physics of sound propagation in air at this temperature (Pierce, 2019).

Buoyancy and Fluid Displacement

To determine whether a body floats or sinks, compare its density to that of water. The density of the body initially is 1010 kg/m³, which exceeds water’s 1000 kg/m³. Hence, the body will sink. To find the acceleration, apply Newton's second law with the buoyant force:

F_b = ρ_water V g and F_gravity = ρ_body V g. The net force is F_net = F_b - F_gravity. The acceleration of the body is:

a = (F_net) / (mass) = [ (ρ_water - ρ_body)  V  g ] / (ρ_body  V) = (ρ_water - ρ_body)  g / ρ_body

Substituting the values:

a = (1000 - 1010) * 9.81 / 1010 ≈ -0.097 m/s2

The negative signifies downward acceleration, confirming the body sinks.

If the density is considered to be 900 kg/m³, the body now has less density than water and would float. The equilibrium point is where the weight equals the buoyant force, leading to a submerged volume ratio of:

Submerged volume ratio = ρ_body / ρ_water = 900 / 1000 = 0.9

Assuming the body is a cube with volume 0.04 m³, the submerged volume is:

V_submerged = 0.9 * 0.04 = 0.036 m3

Since the body is floating, about 90% of its volume is submerged, and 10% remains above water.

The out-of-water (exposed) height of the cube depends on its dimensions. If the body is a cube of volume 0.04 m³, its side length is:

s = (0.04)^{1/3} ≈ 0.341 meters

The submerged height is:

h_submerged = V_submerged / (area of the base) = 0.036 / (s2) ≈ 0.036 / 0.116 ≈ 0.31 meters

The exposed part above water is approximately:

h_exposed = s - h_submerged ≈ 0.341 - 0.31 ≈ 0.031 meters

If the cube is submerged slightly more than this equilibrium point, buoyancy will increase, pushing the body upward until it reaches equilibrium. Conversely, if it is submerged less, gravity will dominate, and the body will sink back to the equilibrium position. Oscillations around the equilibrium point tend to damp out over time due to fluid resistance, stabilizing the body's position (Fitzgerald, 2017).

Conclusion

This exploration underscores the interconnectedness of wave mechanics, fluid flow, and buoyancy in physical systems. The behavior of standing waves in acoustics, fluid velocity in constricted tubes, and conditions for floating or sinking bodies exemplify fundamental physics principles vital to both academic understanding and practical applications.

References

• Fitzgerald, P. (2017). Fluid Mechanics and Hydraulics. Oxford University Press.
• Fox, R. W., McDonald, A. T., & Pritchard, T. J. (2011). Introduction to Fluid Mechanics. John Wiley & Sons.
• Pierce, A. D. (2019). Acoustics: An Introduction to Its Physical Principles and Applications. Springer.
• White, F. M. (2014). Fluid Mechanics. McGraw-Hill Education.
• Resnick, R., Halliday, D., & Krane, K. S. (2008). Physics. John Wiley & Sons.
• Morin, D. (2013). Introduction to Classical Mechanics. Cambridge University Press.
• Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J. V. (1999). Fundamentals of Acoustics. John Wiley & Sons.
• Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics. John Wiley & Sons.
• Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers. W. H. Freeman.
• Serway, R. A., & Jewett, J. W. (2013). Physics for Scientists and Engineers. Brooks Cole.

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