Fish tank

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1

a fish tank 14 in deep and 16 by 27 in in cross section is being transported in a car that experiences accelerations of 6 m2/s. How deep can the fish tank be filled without spilling if the acceleration is in the long direction of the tank? how deep if the acceleration is in the short direction

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This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:

  1. PURPOSE OF THE EXERCISE:
    Demonstrate the application of free‐surface tilt under acceleration in a rectangular container and use the effective gravity concept to determine the maximum allowable fill depth without spillage.

  2. ENGINEERING CONTEXT:
    When transporting an open container (here, a fish tank) subject to linear accelerations, the free surface tilts so that one edge approaches the rim. Understanding how to compute the maximum fill depth enables the proper design and safe transport of liquid‐filled vessels.

  3. PHYSICAL CONTEXT:
    The dominant effect is the effective gravitational field in the accelerating frame; the free surface aligns perpendicular to the vector sum of gravity and the acceleration, with a tilt angle given by \theta=\arctan\!\left(\frac{a}{g}\right).

  4. ASSUMPTIONS:
    Assume steady, incompressible flow with a planar free surface and vertical walls; this is reasonable for water in a rigid fish tank subjected to a constant acceleration of a=6\,\mathrm{m/s^2} and ambient gravity g=9.81\,\mathrm{m/s^2}.

  5. DRAWINGS:
    A sketch would depict a rectangular tank (depth H_\mathrm{tank}=14\,\mathrm{in}) with the acceleration vector along one horizontal direction. Two cases are considered: one with the acceleration acting along the long side of the tank (L=27\,\mathrm{in}) and one along the short side (L=16\,\mathrm{in}). The free surface is drawn as an inclined line making an angle \theta with the horizontal.

  6. SOLUTION TECHNIQUE:
    In the accelerated frame the free surface is perpendicular to the effective gravity. Its plane satisfies

z = z_0 - \frac{a}{g}x\,,

where x is the horizontal coordinate along the acceleration direction. For a container of length L in that direction, the difference in water level between the two ends is

\Delta z = \frac{a}{g}L\,.

When the tank is filled uniformly (in the quiescent state) to a depth h, under acceleration the free surface tilts so that one edge is raised relative to the mean by \Delta z/2 and the other is lowered by \Delta z/2. To avoid spillage the highest point of the free surface must be at or below the tank rim, i.e.,

h + \frac{\Delta z}{2} = H_\mathrm{tank}\,.

Substitute \Delta z=\frac{aL}{g} into the expression:

h = H_\mathrm{tank} - \frac{aL}{2g}\,.

Using the given values and the dimensionless acceleration ratio

\frac{a}{g}=\frac{6}{9.81}\approx0.611\,,

we obtain:

• For acceleration along the long side (L=27\,\mathrm{in}):

h = 14\,\mathrm{in} - \frac{0.611 \times 27\,\mathrm{in}}{2}\approx 14\,\mathrm{in} - 8.25\,\mathrm{in}\approx 5.75\,\mathrm{in}\,.

• For acceleration along the short side (L=16\,\mathrm{in}):

h = 14\,\mathrm{in} - \frac{0.611 \times 16\,\mathrm{in}}{2}\approx 14\,\mathrm{in} - 4.89\,\mathrm{in}\approx 9.11\,\mathrm{in}\,.
  1. REFLECTION:
    The analysis reveals that the maximum fill depth is strongly influenced by the dimension of the tank in the direction of acceleration. When the acceleration is aligned with the longer (27 in) side, the water’s free-surface tilt requires a lower average fill depth (approximately 5.75 in) to prevent spillage compared to when the acceleration is along the shorter (16 in) side (approximately 9.11 in). This result underscores the importance of aligning container geometry appropriately when designing systems for safe transport under acceleration.

In formulating this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 4 Engineering Models: Hydrostatic Equation (HIGHLY RELEVANT)
  2. Chapter 3 Engineering Models: Hydrostatic Equation (HIGHLY RELEVANT)
  3. Chapter 3.2: Hydrostatic Equation (HIGHLY RELEVANT)
  4. Chapter 4.2: Review of Hydrostatics (HIGHLY RELEVANT)
  5. Chapter 4.1: Watercraft Part 1 (HIGHLY RELEVANT)

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