SME
University Physics I: Mechanics · Problem type
How much of a floating body sinks
Use this for “What fraction of an ice block (917 kg/m³) floats under seawater (1,025 kg/m³)?”
Diagram
Floating:
Vsub
V
=
ρbody
ρ
=
917
1,025
=
0.895.
FB = ρgVsub = 1,025 × 9.81 × 1.789 = 17,990 N, the block’s weight.
This is an example. Type your own values to start a new problem.
f
%
=
ρbody (
)
ρ (
)
V
Body volume
example
Vsub
Volume under the surface
calculated
FB
Buoyant force
calculated
W
Weight of the body
worked out (not typed)
Formulas
f
=
100
×
ρbody
ρ
89.4634
=
100
×
917
1,025
Vsub
=
f
×
V
100
1.7893
=
89.4634
×
2
100
FB
=
ρ
×
9.81
×
Vsub
17,991.54 = 1,025 × 9.81 × 1.7893
FB
=
W
17,991.54 = 17,991.54
W
=
ρbody
×
9.81
×
V
17,991.54 = 917 × 9.81 × 2
Assumptions
•
The body floats at rest, so the buoyant force FB equals its weight (Archimedes’ principle).
•
FB = ρgVsub is the weight of the fluid pushed aside, with g = 9.81 m/s².
•
The body has one density all through; one denser than the fluid sinks, so ρbody ≤ ρ.
Step-by-step
WE KNOW
ρbody = 917 kg/m³, ρ = 1,025 kg/m³, V = 2 m³
FIND
share under the surface (f), volume under the surface (Vsub), buoyant force (FB), weight of the body (W)
Step 1 · Find share under the surface (f)
Use f = 100 × ρbody ÷ ρ
Weight
ρbody
gV
equals
buoyant
force
ρgVsub,
so
the
share
under
is
Vsub
V
=
ρbody
ρ
.
f
=
100
×
917
1,025
f
=
91,700
1,025
f
=
89.4634%
Step 2 · Find volume under the surface (Vsub)
Use Vsub = f × V ÷ 100
The part under the surface is that share of the whole volume.
Vsub
=
89.4634
×
2
100
Vsub
=
178.9268
100
Vsub = 1.789268 m³
Step 3 · Find buoyant force (FB)
Use FB = ρ × 9.81 × Vsub
The buoyant force is the weight of the fluid the body pushes aside: ρg times Vsub.
FB = 1,025 × 9.81 × 1.789268
FB = 10,055.25 × 1.7893
FB = 17,991.54 N
Step 4 · Find weight of the body (W)
Use FB = W
Floating at rest, the forces balance: the weight equals FB.
W
=
FB
W
=
17,991.54
N
Check
Put the numbers back in. Both sides should match.
89.4634
=
100
×
917
1,025
✓
1.789268
=
89.4634
×
2
100
✓
17,991.54 = 1,025 × 9.81 × 1.789268 ✓
17,991.54 = 17,991.54 ✓
17,991.54 = 917 × 9.81 × 2 ✓
Topic check
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