BIZOMAT MAKEOS · Mechanics
Stacked Solid Centre of Gravity (Cylinders, Frustums, Cones, Hemispheres, Tipping Angle)
For a vessel, hopper or column made of stacked round solids (or the material filling them), where is its centre of gravity, what does it weigh, and at what tilt would it tip over its base?
Centre of gravity of an upright stack of up to three round solid parts — cylinders, frustums, cones and hemispheres, either way up — from the bottom upwards: the volume and centroid height of each part (as the centroid of shapes tool), its mass from its density, the combined mass and centre of gravity height including any height of legs or base below, and the angle of tilt at which the centre of gravity passes over the edge of the base, against your least acceptable tipping angle.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Part 1 (from the bottom) | ||
| Shape | choice | Shape of the bottom part. |
| Diameter D₁ | mm | Diameter of part 1 (base, or top for a cone point down). |
| Diameter D₂ (frustum top) | mm | Top diameter of part 1 (frustums only). |
| Height (cylinders, frustums, cones) | mm | Height of part 1. |
| Density | kg/m³ | Density of part 1 (of the contents for a filled vessel). |
| Part 2 (from the bottom) | ||
| Shape | choice | Shape of part 2, sitting on part 1. |
| Diameter D₁ | mm | Diameter of part 2. |
| Diameter D₂ (frustum top) | mm | Top diameter of part 2 (frustums only). |
| Height (cylinders, frustums, cones) | mm | Height of part 2. |
| Density | kg/m³ | Density of part 2. |
| Part 3 (from the bottom) | ||
| Shape | choice | Shape of part 3, sitting on part 2. |
| Diameter D₁ | mm | Diameter of part 3. |
| Diameter D₂ (frustum top) | mm | Top diameter of part 3 (frustums only). |
| Height (cylinders, frustums, cones) | mm | Height of part 3. |
| Density | kg/m³ | Density of part 3. |
| Base | ||
| Height of the bottom of the stack above the base | mm | Height of legs or plinth below the stack. |
| Base width (smallest, across the tipping edge) | mm | Width of the base across the edge it would tip over. |
| Least tipping angle accepted | ° | Smallest tilt the stack must survive without tipping. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Part 1 (from the bottom)
What this means: Shape
- What it is
- The shape of the bottom part of the stack, and which way up it sits. The list shows what diameter D₁ means for each shape.
- Units and format
- Choose one of 7 options from the list: Not used; Cylinder (D₁ = diameter); Frustum (D₁ = bottom dia, D₂ = top dia); Cone, point up (D₁ = base dia); Cone, point down (D₁ = top dia); Hemisphere, flat face down (D₁ = dia); Hemisphere, dome down (D₁ = dia).
- Example
- The form opens with “Cone, point down (D₁ = top dia)” selected as a worked example. Choose the option that matches your case.
- Get it right
- For a cone point down, D₁ is the top (base) diameter; for a frustum D₁ is the bottom diameter.
What this means: Diameter D₁
- What it is
- The main diameter of part 1: the cylinder or hemisphere diameter, the bottom diameter of a frustum, or the base diameter of a cone (at the top for a cone point down).
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the diameter, not the radius.
What this means: Diameter D₂ (frustum top)
- What it is
- The top diameter of part 1 when it is a frustum. Other shapes ignore it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only for a frustum; a frustum wider at the top is allowed.
What this means: Height (cylinders, frustums, cones)
- What it is
- The vertical height of part 1 when it is a cylinder, frustum or cone. A hemisphere’s height is its radius, worked out from D₁.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored for hemispheres.
What this means: Density
- What it is
- The density of part 1, or of the contents if the part represents a filled vessel, used to turn its volume into mass.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0 kg/m³ to 25000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1500 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a vessel, the shell is not included; add it as a separate part or mass if it matters.
Part 2 (from the bottom)
What this means: Shape
- What it is
- The shape of part 2, which sits on top of part 1, and which way up it sits. Choose Not used to leave it out.
- Units and format
- Choose one of 7 options from the list: Not used; Cylinder (D₁ = diameter); Frustum (D₁ = bottom dia, D₂ = top dia); Cone, point up (D₁ = base dia); Cone, point down (D₁ = top dia); Hemisphere, flat face down (D₁ = dia); Hemisphere, dome down (D₁ = dia).
- Example
- The form opens with “Cylinder (D₁ = diameter)” selected as a worked example. Choose the option that matches your case.
- Get it right
- For a cone point down, D₁ is the top diameter; for a frustum D₁ is the bottom diameter.
What this means: Diameter D₁
- What it is
- The main diameter of part 2: the cylinder or hemisphere diameter, the bottom diameter of a frustum, or the base diameter of a cone.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the diameter, not the radius.
What this means: Diameter D₂ (frustum top)
- What it is
- The top diameter of part 2 when it is a frustum. Other shapes ignore it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only for a frustum.
What this means: Height (cylinders, frustums, cones)
- What it is
- The vertical height of part 2 when it is a cylinder, frustum or cone. A hemisphere’s height is its radius.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored for hemispheres.
What this means: Density
- What it is
- The density of part 2, or of the contents if it represents a filled vessel section.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0 kg/m³ to 25000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1500 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from material data or the contents specification.
Part 3 (from the bottom)
What this means: Shape
- What it is
- The shape of part 3, which sits on top of part 2, and which way up it sits. Choose Not used to leave it out.
- Units and format
- Choose one of 7 options from the list: Not used; Cylinder (D₁ = diameter); Frustum (D₁ = bottom dia, D₂ = top dia); Cone, point up (D₁ = base dia); Cone, point down (D₁ = top dia); Hemisphere, flat face down (D₁ = dia); Hemisphere, dome down (D₁ = dia).
- Example
- The form opens with “Not used” selected as a worked example. Choose the option that matches your case.
- Get it right
- For a cone point down, D₁ is the top diameter; for a frustum D₁ is the bottom diameter.
What this means: Diameter D₁
- What it is
- The main diameter of part 3: the cylinder or hemisphere diameter, the bottom diameter of a frustum, or the base diameter of a cone.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the diameter, not the radius.
What this means: Diameter D₂ (frustum top)
- What it is
- The top diameter of part 3 when it is a frustum. Other shapes ignore it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only for a frustum.
What this means: Height (cylinders, frustums, cones)
- What it is
- The vertical height of part 3 when it is a cylinder, frustum or cone. A hemisphere’s height is its radius.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored for hemispheres.
What this means: Density
- What it is
- The density of part 3, or of the contents if it represents a filled vessel section.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0 kg/m³ to 25000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1500 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from material data or the contents specification.
Base
What this means: Height of the bottom of the stack above the base
- What it is
- The height of the bottom of part 1 above the base the stack stands on, such as the height of legs, a skirt or a plinth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 800 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher base raises the centre of gravity and lowers the tipping angle.
- Get it right
- Measure from the ground contact surface to the underside of the lowest part.
What this means: Base width (smallest, across the tipping edge)
- What it is
- The narrowest width of the base footprint, measured across the edge about which the stack would tip. The tool assumes the centre of gravity is centred over it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A wider base raises the tipping angle.
- Get it right
- For legs, measure between the outer contact points of the legs, not across the vessel body.
What this means: Least tipping angle accepted
- What it is
- The smallest angle of tilt the stack must survive without its centre of gravity passing over the base edge.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 89°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from your design standard or transport and site requirements; the tool does not suggest one.
Method and formulas
- Centroid heights: cone h ÷ 4 from its base; frustum h (R² + 2Rr + 3r²) ÷ 4(R² + Rr + r²) from the large end; hemisphere 3r ÷ 8 from the flat face (as the centroid of shapes tool); cylinder h ÷ 2.
- Mass = volume × density; CG height = Σ m z ÷ Σ m plus the base height.
- Tipping angle = atan ((base width ÷ 2) ÷ CG height).
- Example: 1,649.3 kg with its CG 1,892.9 mm up; it tips at 17.588°.
Limitations
- Solid, uniformly dense parts; for a vessel, add the shell as separate parts or masses.
- Static tilt; wind, sloshing and acceleration reduce the margin.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the CG of a full hopper on legs
- Check a column or vessel is stable on its base
- Estimate the weight of a filled bin
Who it is for
- Design engineers
- Riggers
- Plant engineers
What you get
- Volume, mass and centroid of each part
- Combined mass and CG height
- Tipping angle
Interpreting the result
- Raising a full hopper on taller legs lowers the tipping angle quickly; widening the leg spread restores it.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 221–227, 66–76. Topic reference; no source tables or recommended settings are reproduced.
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