Fluids in Accelerating Containers
In a lift, gravity seems stronger; in an accelerating truck, the water surface tilts.
Lesson at a glanceShow the key points
- In a container accelerating upward at a₀, the pressure difference over a height z is ρz(g + a₀) (13.6).
- The buoyant force becomes B = m(g + a₀), where m is the mass of displaced liquid (13.7).
- With a horizontal acceleration a₀, two points a distance l apart along the acceleration differ in pressure by lρa₀ (13.8).
- The free surface tilts at angle θ with tan θ = a₀/g, rising toward the back.
Carry a full cup of tea while walking quickly and it slops over the back edge. Our hydrostatic results assumed the liquid was at rest in an inertial frame. What changes when the container accelerates?
A liquid in a lift
Pressure difference
A beaker of liquid rides in a lift accelerating upward at a₀. Take a vertical cylinder of liquid of cross-section ΔS between a point A and a point B a height z above it. The forces on it are up, down and its weight down, and it accelerates upward with the lift:
Buoyant force
Replace an immersed body by liquid again. The substitute liquid accelerates up with everything else, so :
A bucket of water is in a lift accelerating upward at 2.0 m/s². Find the gauge pressure 10 cm below the surface, and the buoyant force on a 50 cm³ object held fully under water (g = 10 m/s²).
A liquid accelerating sideways
Now let the container accelerate horizontally at a₀. Take two points A and B on the same horizontal line, a distance l apart along the acceleration (A behind, B in front). The horizontal cylinder of liquid between them must be pushed forward:
So points at the same level no longer have equal pressure. There is no vertical acceleration, so vertically still holds: the liquid must be deeper above A than above B. Writing the depths h₁ and h₂,
A tanker truck partly full of water accelerates at 3.0 m/s². At what angle does the water surface settle?
- 1Vertical acceleration a₀: pressure difference , buoyant force .
- 2Horizontal acceleration: along a level line; surface tilts with .
- 3The free surface is always perpendicular to the effective gravity.
Check yourself
Keep exploring
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