SME
Statics · Problem type
Centroid of a plate with a hole
Use this for “A 200 × 100 mm plate has a 40 mm hole centered 150 mm from its left edge. Where is the centroid?”
Diagram
x̄
=
ΣAx
ΣA
=
1.812 ×
10
6
18,740
x̄ = 96.65 mm
Plate area A₁ = 20,000 mm²
Hole area A₂ = 1,256.637 mm²
This is an example. Type your own values to start a new problem.
A₁ (mm²)
=
w (mm)
×
h (mm)
A₂ (mm²)
=
π
×
(
d (mm)
)
2
4
x̄ (mm)
×
(
A₁ (mm²)
−
A₂ (mm²)
)
=
A₁ (mm²)
×
w (mm)
2
−
A₂ (mm²)
×
xₕ (mm)
A₁ = w × h is the plate’s area and A₂ = πd² ÷ 4 the hole’s, taken away; the plate’s center is w ÷ 2 from the left edge, and the hole’s is xₕ.
Formulas
A₁
=
w
×
h
20,000 = 200 × 100
A₂
=
π
×
d
2
4
1,256.6371
=
π
×
40
2
4
x̄
=
A₁
× (
w
÷ 2) −
A₂
× xₕ
A₁
−
A₂
96.6478 = (20,000 × (200 ÷ 2) − 1,256.6371 × 150) ÷ (20,000 − 1,256.6371)
Assumptions
•
The hole is a negative area: take its area and its moment away from the whole plate’s.
•
x̄ is measured from the left edge; the hole sits at mid-height, so ȳ stays at h ÷ 2.
•
The plate is thin and even, so its centroid is its balance point.
Step-by-step
WE KNOW
w = 200 mm, h = 100 mm, d = 40 mm, xₕ = 150 mm
FIND
plate area (A₁), hole area (A₂), centroid from the left (x̄)
Step 1 · Find plate area (A₁)
Use A₁ = w × h
The
whole
plate
is
a
rectangle
w
wide
and
h
high.
A₁
=
200
×
100
A₁
=
20,000
mm²
Step 2 · Find hole area (A₂)
Use A₂ = π × d² ÷ 4
A circle’s area is π times its diameter squared, divided by 4.
A₂
=
π
×
40
2
4
A₂
=
π
×
1,600
4
A₂
=
1,600π
4
A₂
=
1,256.637
mm²
Step 3 · Find centroid from the left (x̄)
Use x̄ = (A₁ × (w ÷ 2) − A₂ × xₕ) ÷ (A₁ − A₂)
The plate’s moment about the left edge minus the hole’s, divided by the area left.
x̄
=
(20,000
×
(200
÷
2)
−
1,256.637
×
150)
÷
(20,000
−
1,256.637)
x̄
=
(20,000
×
100
−
1,256.637
×
150)
÷
(20,000
−
1,256.637)
x̄
=
2,000,000 − 188,495.55
18,743.363
x̄
=
1,811,504.45
18,743.363
x̄
=
96.6478
mm
Check
Put the numbers back in. Both sides should match.
20,000 = 200 × 100 ✓
1,256.637
=
π
×
40
2
4
✓
96.6478 = (20,000 × (200 ÷ 2) − 1,256.637 × 150) ÷ (20,000 − 1,256.637) ✓
Topic check
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