Spatial Geometry & Functions · Common Core Grade 7-8 Geometry (Cylinder Volume), Cambridge IGCSE Physics (Density & Displacement Method), High School Physics (Fluid Statics).

Water Displacement & Immersion Visualizer

Explore water displacement dynamics: from simple stone submersion measuring irregular volume to the non-linear surge when a long vertical rod is partially inserted into a cylindrical container, constricting the annular water surface.

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Parameter Controls

container_r

water_h0

rod_r

rod_len

push_depth

Live Derivations & Metrics

current_water_h

Real-time water depth in bucket

delta_h

Height rise above initial h0

submerged_len

Wetted length of the rod

disp_volume

Displaced water volume

Real-Time Formula Derivation

Governing Formula: --

Substitution: --

Regime: --

Worked Examples & Exam Question Deep Dive

Annular Area Contraction: 100 cm² Bucket & 20 cm² Vertical Rod

Problem: A cylindrical bucket with base area 100 cm² contains water to a depth of 16 cm. A sufficiently long iron rod with cross-sectional area 20 cm² is vertically inserted until touching the bottom. What is the new water height?

Analytical Strategy: Water volume is invariant: V = 100 × 16 = 1600 cm³. With the rod touching bottom and protruding, water is confined to the annular region with effective base area S_annular = 100 - 20 = 80 cm². New water height h = 1600 / 80 = 20 cm.

Step-by-Step Derivation:

Step 1: Compute invariant water volume V = 100 × 16 = 1600 cm³.
Step 2: Effective annular surface area S_eff = 100 - 20 = 80 cm².
Step 3: New liquid height h = 1600 / 80 = 20 cm.
Step 4: Water level increase Δh = 20 - 16 = 4 cm.
Step 5: Verify submerged rod length equals 20 cm (consistent with protruding rod condition).

⚠️ Common Misconception Pitfall: Naive non-feedback error: calculating displacement as S_rod × h0 and dividing by S_bucket. The rising water wets additional rod length, which must be solved via effective annular area.

Student Inquiry Worksheet & Practice Problems

Q1. A submerged iron block in a cylinder of radius 10 cm causes water level to rise 3 cm. Find iron volume (π=3.14).

Q2. A rod is pushed down further by 2 cm into a bucket. How does the water level change?

Frequently Asked Questions & Misconceptions

If the rod is pushed so deep that it becomes fully submerged, does water level keep rising at the accelerated rate?

No! Once the rod tip is fully submerged, displaced volume caps out at the rod's total volume. Pushing it deeper causes no further water level rise.