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Lesson Why does liquid spray sideways?
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Why does liquid spray sideways?
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Why does liquid spray sideways?
Why does liquid spray sideways when a vessel accelerates?
KEY IDEA

The liquid wants to keep moving. The vessel pulls it forward. Result: surface tilts and at the leading edge it spills.

Vessel · liquid Trolley Liquid spray a

Spring–mass equation of motion

From Hooke's law to x(t) = A cos(ωt + φ)
DERIVATION · 9 LINES · UNDERGRAD
1
Assume
F = -kx
Hooke's law — spring's restoring force.
intuition"stretch more, pulls harder"
2
Newton's 2nd
F = m \ddot x
ẍ is second derivative.
intuition"force ↦ acceleration"
3
Substitute
m \ddot x = -kx
Combine.
intuition"glue them"
4
Rearrange
\ddot x = -\frac{k}{m}\, x
Divide by m.
intuition"isolate ẍ"
5
Define
\omega^2 \;\equiv\; \frac{k}{m}
Introduce ω — angular frequency.
intuition"ω is what oscillates"
6
ODE
\ddot x + \omega^2 x = 0
L7 · L8 · L9  veiled until we cover this line
PROBLEM · IRODOV 1.27

Vessel of liquid on accelerating trolley

A trolley carries a vessel of water. The trolley accelerates horizontally at a = 3 m/s². Find the angle the liquid surface makes with the horizontal.
Step 3 of 7 · Tilt-angle from force balance
STEP 1Identify knownsCORRECT
Identify what's given.
YOUR ANSWER
a = 3 m/s², g = 9.8 m/s²
STEP 2Set up FBDCORRECT
Free-body diagram for a fluid parcel.
STEP 3Tilt-angle from force balanceYOUR TURN
Surface-perpendicular condition: write the relation between a, g, and tilt angle θ.
\tan\theta \;=\; ?
The resultant of g and a defines the effective vertical.
YOUR ANSWER
STEP 4Plug in numbers
QUESTION 3 OF 5CONCEPT · MCQ

A trolley accelerates at 3 m/s². Which expression gives the tilt angle θ?

Atan θ = g/a
Btan θ = a/g
Csin θ = a/g
Dθ = arctan(a · g)
vessel a → tan θ = a / g θ = 17.0° ✓ correct
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