2025-08-21 12 min

Center of Gravity Calculator

Calculate the center of gravity for up to 5 weighted positions.

Calculator

Decimal Precision

Mass 1

Weight(N)

X Position(m)

Y Position(m)

Mass 2

Weight(N)

X Position(m)

Y Position(m)

Mass 3

Weight(N)

X Position(m)

Y Position(m)

Mass 4

Weight(N)

X Position(m)

Y Position(m)

Mass 5

Weight(N)

X Position(m)

Y Position(m)

Results update live as you type

Results

Normal

X-CG

2.44m

Y-CG

1.00m

Total Weight

450.00N

Active Masses

Visualizations

Results Overview

2.4mX-CG

1mY-CG

450mTotal Weight

3mActive Masses

Formula

The center of gravity is the weight-weighted average position of all mass components.

x_cg = sum(w_i * x_i) / sum(w_i)

Alternative Form

y_cg = sum(w_i * y_i) / sum(w_i)

Variables

Symbol	Name	Unit	Description
w_i	Weight	N	Weight of component i
x_i	X-position	m	X-coordinate of component i
y_i	Y-position	m	Y-coordinate of component i

Explanation

Each coordinate of the CG is the ratio of the sum of weight-position products to the total weight.

Derivation

From sum(w_i * (x_i - x_cg)) = 0, solve for x_cg.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Enter weight and position for each mass component

Result: Inputs entered

Step 2

Calculate weighted sum: sum(w_i * x_i)

Result: Sum computed

Step 3

Calculate total weight: sum(w_i)

Result: Total computed

Step 4

Divide weighted sum by total weight to get CG

Result: CG computed

Final Answer

2.44

Worked Examples

See how the calculator works with different scenarios.

Scenario

100 N at x=2m and 200 N at x=8m.

Inputs

w1:100

x1:2

w2:200

x2:8

Calculation

x_cg = (100*2 + 200*8) / 300 = 6 m

Result

CG at x = 6 m

Interpretation

CG is always closer to the heavier mass.

Understanding Your Result

Your Result: X-CG: 2.44 m

Typical range: Car CG: 0.45-0.7 m. Person CG: 55-57% of height.

What This Means

The CG is the effective point of application of the total weight.

Why It Matters

Critical for vehicle stability, structural analysis, and crane operations.

Professional Interpretation

Use CG analysis for stability assessment and load planning.

Recommendations

•Account for all mass components
•Consider CG changes with loading
•Use safety margins for stability

Educational Content

Introduction

Center of gravity analysis is fundamental to understanding stability and weight distribution.

Concept Explanation

The CG is the weight-weighted average position of all components.

Applications

▸Vehicle design

▸Structural engineering

▸Crane operations

▸Aerospace

▸Robotics

Benefits

•Predicts stability limits
•Enables weight optimization
•Essential for safety analysis

Limitations

•Assumes uniform gravity
•Point calculation only
•No dynamic effects

Best Practices

•Include all significant masses
•Check CG in all loading conditions
•Use CG for stability margins

Common Misconceptions

Myth: CG is always inside the object

Truth: CG can be outside the physical material (e.g., boomerang).

Practical Tips

•CG shifts toward heavier components
•Lower CG means more stability
•Wider base increases tipping angle
•Roof loads raise CG significantly

Common Mistakes

Ignoring small masses

Even light components affect CG if far from other masses.

Forgetting the y-coordinate

Always compute both x and y CG for 2D systems.

Assumptions

This calculator makes the following assumptions to provide estimates:

Uniform Gravity: Gravitational field is constant.

Static Analysis: No dynamic effects.

Limitations

Be aware of these limitations when using this calculator:

2D Analysis: Handles x-y plane CG only.

Point Masses: Components treated as point masses.

Frequently Asked Questions

Answers to common questions about this calculator.

What is center of gravity?

The center of gravity (CG) is the point where the entire weight of a body can be considered to act. For a uniform gravitational field, it coincides with the center of mass. The CG is calculated as x_cg = sum(w_i * x_i) / sum(w_i).

What is the difference between center of gravity and center of mass?

Center of mass depends only on mass distribution. Center of gravity depends on mass distribution AND gravitational field. In uniform gravity (most engineering), they coincide.

How is center of gravity calculated for multiple masses?

For n masses: x_cg = (w1*x1 + w2*x2 + ... + wn*xn) / (w1 + w2 + ... + wn), and similarly for y_cg.

Why is center of gravity important in vehicle design?

CG height directly affects rollover risk. Lower CG reduces rollover tendency. Fore-aft CG position affects weight distribution for traction. Sports cars target CG heights below 0.5m.

How does center of gravity affect stability?

An object is stable when its CG is above the base of support. Tipping occurs when the CG projection falls outside the base. The tipping angle is theta = arctan(b/h).

What is the CG of a car?

Typical passenger car CG height is 0.5-0.6 m above ground. Sports cars are lower (~0.45 m), SUVs are higher (~0.7 m).

How does load placement affect CG?

Roof loads raise the CG, increasing rollover risk. Heavy items in the trunk move CG rearward. Side loads shift CG laterally.

What is the CG of a person?

A standing person CG is at approximately 55-57% of height (around the navel). It shifts with posture.

How does CG affect aircraft design?

Aircraft CG must stay within limits for controllable flight. Forward CG increases stability. Aft CG improves efficiency.

What is the relationship between CG and tipping?

Tipping occurs when the weight line of action passes outside the base. Tipping angle = arctan(b / (2 * h_cg)).

How do you experimentally find CG?

Suspend the object from a point; the CG lies along the vertical line through the suspension point. Repeat from a different point; the CG is at the intersection.

What is the CG of a triangle?

For a uniform triangular plate, the CG is at the centroid: the intersection of the medians. For vertices at (x1,y1), (x2,y2), (x3,y3), the CG is at ((x1+x2+x3)/3, (y1+y2+y3)/3).

How does fuel affect vehicle CG?

Fuel in a floor-mounted tank has minimal CG effect. As fuel is consumed, CG shifts. Aircraft carefully track fuel CG effects.

What is the CG of a semicircle?

For a uniform semicircular plate of radius R, the CG is on the axis of symmetry at distance 4R/(3*pi) from the flat edge.

How does CG affect crane operation?

Cranes have load charts based on CG. The tipping moment must not exceed the restoring moment.

What is the CG of a composite body?

For a composite body, calculate the weighted average of each component CG: x_cg = sum(m_i * x_i) / sum(m_i).

How does CG affect suspension design?

Suspension geometry manages CG movement during cornering, braking, and acceleration.

What is the CG of a sphere?

For a uniform solid sphere, the CG is at the geometric center, regardless of size.

How does CG affect ship stability?

Ships have a metacentric height (GM) that depends on CG height. Higher CG reduces GM, making the ship less stable.

What is the CG of a rectangle?

For a uniform rectangular plate, the CG is at the geometric center: (width/2, height/2).

How does CG affect motorcycle dynamics?

Lower CG improves low-speed stability. CG position relative to contact patches determines self-steering tendency.

What is the CG of a hollow cylinder?

For a thin-walled hollow cylinder, the CG is on the axis of symmetry at the geometric center.

How is CG used in structural engineering?

In earthquake engineering, CG of a building determines seismic force distribution. Torsional effects arise when CG does not align with center of rigidity.

What is the CG of a cone?

For a solid cone of height h, the CG is on the axis at h/4 from the base.

How does CG affect bicycle design?

Lower CG improves stability, higher CG improves responsiveness. Rider CG is much higher than bike CG.

What is the CG of a hemisphere?

For a solid hemisphere of radius R, the CG is on the axis at 3R/8 from the flat face.

How does CG affect prosthetic design?

Prosthetic limb design must match the natural CG location for comfortable walking.

What is the CG of a torus?

For a torus with major radius R and minor radius r, the CG is at the center of the torus on the axis of symmetry.

How does CG affect satellite orientation?

Satellites orient using reaction wheels that torque about the CG. Fuel consumption shifts CG.

What is the CG of a parabolic plate?

For a uniform parabolic plate with vertex at origin and axis along y, the CG is at (0, 2h/5).

How does CG affect roller coaster design?

Roller coasters are designed for specific CG paths through loops. Rider CG affects g-forces experienced.

What is the CG of a system with added mass?

When mass m_new is added at position (x_new, y_new) to a system with CG at (x_cg, y_cg) and total mass M, the new CG is: x_cg_new = (M * x_cg + m_new * x_new) / (M + m_new).

How does CG affect robot balance?

Walking robots must keep their CG over the support polygon. Dynamic walking involves controlled falling.

What is the CG of a trapezoid?

For a uniform trapezoid with parallel sides a and b, height h, the CG is at height h(2a+b)/(3(a+b)) from the base.

How does CG affect diving?

In diving, athletes tuck to bring mass closer to the rotation axis, spin faster, then extend to slow rotation.

What happens when CG shifts during motion?

Dynamic CG shifts affect handling in vehicles, balance in robots, and stability in rotating machinery. Engineers account for CG migration in design.

How does CG relate to the parallel axis theorem?

The parallel axis theorem relates MOI about any axis to MOI about a parallel axis through the CG: I = I_cg + Md^2.

What is the CG of a spherical shell?

For a uniform spherical shell, the CG is at the geometric center, same as a solid sphere, due to spherical symmetry.

How does CG affect seismic base shear distribution?

Base shear distributes to floors based on mass and height. Higher floors with larger CG receive more seismic force: F_i = V * (w_i * h_i) / sum(w_j * h_j).

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Center of Gravity: The point where total weight acts. Weight-weighted average position.
Weight: Force due to gravity: W = mg. Measured in Newtons.
Moment: Tendency of a force to rotate about a point: M = F * d.
Stability: Ability to return to equilibrium after disturbance.

Authoritative References

Trusted sources and standards referenced by this calculator.

Engineering Toolbox MIT OCW - Statics

Applicable Standards

ISO 12100ASME B30.20

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Page Information

Last Updated:

2025-08-21

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.