SME
Calculus III (Multivariable) · Problem type
Mass and center of mass of a box
Use this for “A block 2 m by 3 m and 4 m tall has density rising evenly from 0 at the base to 600 kg/m³ at the top. Find its mass and center of mass.”
Diagram
L = ⟨2, 0, 0⟩, with length |L| = 2.
W = ⟨0, 3, 0⟩, with length |W| = 3.
H = ⟨0, 0, 4⟩, with length |H| = 4.
L·(W × H) = 24, so the box’s volume is |24| = 24.
This is an example. Type your own values to start a new problem.
m (kg)
=
ρₜₒₚ (kg/m³)
×
L (m)
×
W (m)
×
H (m)
÷
2
z̄
=
2
×
H (m)
÷
3
=
8
3
z̄ (m)
The density grows evenly from 0 to ρₜₒₚ, so the mass integral gives half of ρₜₒₚ × LWH, and the heavier top lifts the center of mass to 2H ÷ 3.
V
Volume of the box
worked out (not typed) · m³
Formulas
V
=
L
×
W
×
H
24 = 2 × 3 × 4
m
=
ρₜₒₚ
×
V
2
7,200
=
600
×
24
2
z̄
=
2
×
H
3
8
3
=
2
×
4
3
Assumptions
•
The density at height z is ρ(z) = ρₜₒₚ × z ÷ H, the same all across each level.
•
Mass is ∭ ρ dV. Each level is a slab of area LW, so m = ∫ from 0 to H of ρ(z) × LW dz.
•
z̄ = (∭ zρ dV) ÷ m. By symmetry the center of mass sits over the middle of the base, at x̄ = L ÷ 2 and ȳ = W ÷ 2.
•
Lengths are in m, density in kg/m³ and mass in kg.
Step-by-step
WE KNOW
L = 2 m, W = 3 m, H = 4 m, ρₜₒₚ = 600 kg/m³
FIND
volume of the box (V), mass of the box (m), height of the center of mass (z̄)
Step 1 · Find volume of the box (V)
Use V = L × W × H
The
box’s
volume:
base
area
LW
times
height
H
.
V
=
2
×
3
×
4
V
=
6
×
4
V
=
24
m³
Step 2 · Find mass of the box (m)
Use m = ρₜₒₚ × V ÷ 2
Integrate
ρₜₒₚ
×
LW
×
z
H
from
0
to
H
:
z
integrates
to
H
2
2,
so
m
is
half
of
what
the
box
would
weigh
at
ρₜₒₚ
all
through.
m
=
600
×
24
2
ρ(z)
×
LW
=
600
×
z
4
×
6
=
900z
4
∫
0
(900
z
) dz
=
[450z²]
from
0
to
4
=
7,200
m
=
7,200
kg
Step 3 · Find height of the center of mass (z̄)
Use z̄ = 2 × H ÷ 3
The
moment
∭
zρ
dV
adds
z
times
the
same
slabs:
z
2
integrates
to
H
3
3
.
Dividing
by
the
mass
leaves
2
H
3,
above
the
middle
because
the
top
is
heavier.
z̄
=
2
×
4
3
4
∫
0
z
2
dz
÷
4
∫
0
z
dz
=
64
3
8
=
8
3
z̄
=
8
3
m
Check
Put the numbers back in. Both sides should match.
24 = 2 × 3 × 4 ✓
7,200
=
600
×
24
2
✓
8
3
=
2
×
4
3
✓
Topic check
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