2025-08-21 12 min
Radius of Gyration Calculator
Calculate radius of gyration from mass moment of inertia and mass: k = sqrt(I/M).
Calculator
Mass Moment of Inertia (I)(kg*m^2) Help
kg*m^2
Must be positive
Total Mass (M)(kg) Help
kg
Must be positive
Decimal Precision
Results update live as you type
Results
Normal
Radius of Gyration
0.3536m
Moment of Inertia
1.2500kg*m^2
Mass
10.0000kg
Visualizations
Results Overview
0.4mRadius of Gyration
1.3mMoment of Inertia
10mMass
Formula
The radius of gyration is the square root of the ratio of mass moment of inertia to total mass.
k = sqrt(I / M)
Alternative Form
I = M * k^2
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| k | Radius of Gyration | m | Equivalent distance for mass concentration |
| I | Mass Moment of Inertia | kg*m^2 | Resistance to rotational acceleration |
| M | Total Mass | kg | Mass of the object |
Explanation
k represents the distance from the axis at which all mass could be concentrated to give the same moment of inertia.
Derivation
From I = Mk^2, solve for k: k = sqrt(I/M).
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Step 1
Determine the mass moment of I about the axis
Result: I entered
Step 2
Enter the total mass of the object
Result: M entered
Step 3
Calculate k = sqrt(I/M)
Result: k computed
Final Answer
0.3536
Worked Examples
See how the calculator works with different scenarios.
Scenario
A 10 kg cylinder with R = 0.5 m, about longitudinal axis.
Inputs
moi:1.25
mass:10
Calculation
I = 0.5 * 10 * 0.5^2 = 1.25 kg*m^2, k = sqrt(1.25/10) = 0.354 m
Result
k = 0.354 m = 0.707R
Interpretation
Equivalent mass concentration at 70.7% of radius.
Understanding Your Result
Your Result: Radius of Gyration: 0.3536 m
Typical range: Solid cylinder: k = 0.707R. Hollow sphere: k = 0.816R. I-beam: k ~ 0.4 * depth.
What This Means
Radius of gyration is the equivalent distance at which all mass could be placed to produce the same moment of inertia.
Why It Matters
Critical for column buckling design, flywheel optimization, rotating machinery, and sports equipment design.
Professional Interpretation
Higher k means better buckling resistance for columns and more energy storage for flywheels.
Recommendations
- •Maximize k for buckling resistance
- •Minimize k for fast rotation
- •Use k to compare mass distributions
Educational Content
Introduction
Radius of gyration is a measure of how mass is distributed relative to an axis of rotation.
Concept Explanation
k = sqrt(I/M). It converts MOI into an equivalent distance, making it easier to compare mass distributions of different objects.
Applications
▸Column buckling analysis
▸Flywheel design
▸Sports equipment
▸Satellite attitude control
▸Rotor dynamics
Benefits
- •Simplifies MOI comparison
- •Useful for buckling calculations
- •Intuitive measure of mass distribution
Limitations
- •Requires known I and M
- •Does not describe shape directly
- •Axis-dependent
Best Practices
- •Use k for slenderness ratios
- •Maximize k for structural efficiency
- •Consider k in rotating equipment design
Common Misconceptions
Myth: k is the physical radius of the object
Truth: k is an equivalent distance, often smaller than the actual radius.
Practical Tips
- •I-beams maximize k for buckling resistance
- •Flywheels maximize k for energy storage
- •Sports equipment minimizes k for fast swings
- •k is always less than or equal to the outer radius
Common Mistakes
Using k instead of actual radius
k is an equivalent distance, not the physical dimension.
Ignoring axis dependence
k changes with the axis of rotation. Always specify the axis.
Assumptions
This calculator makes the following assumptions to provide estimates:
Rigid Body: Object is treated as rigid (no deformation).
Known I: Mass moment of inertia must be known or calculable.
Limitations
Be aware of these limitations when using this calculator:
Axis-Specific: k depends on the chosen rotation axis.
No Shape Info: k alone does not determine the object shape.
Frequently Asked Questions
Answers to common questions about this calculator.
What is radius of gyration?
The radius of gyration (k) is the distance from the axis of rotation at which the entire mass of an object could be concentrated without changing the moment of inertia: k = sqrt(I/M). It provides an intuitive measure of how mass is distributed relative to the axis.
How is radius of gyration related to moment of inertia?
Radius of gyration is defined as k = sqrt(I/M), where I is the mass moment of inertia and M is the total mass. Equivalently, I = Mk^2. A larger k means mass is distributed farther from the axis.
What are the units of radius of gyration?
Radius of gyration has units of length: meters (m) in SI, feet (ft) in Imperial. Despite the name, it is not a radius in the geometric sense but an equivalent distance for mass concentration.
What is the radius of gyration of a solid cylinder?
For a solid cylinder rotating about its longitudinal axis, k = R/sqrt(2) = 0.707R, where R is the radius. This means the equivalent mass concentration is at 70.7% of the radius.
What is the radius of gyration of a thin rod?
For a thin rod of length L rotating about its center, k = L/sqrt(12) = 0.289L. About one end, k = L/sqrt(3) = 0.577L. The end axis gives a larger k because mass is distributed farther from that axis.
What is the radius of gyration of a solid sphere?
For a solid sphere, k = sqrt(2/5) * R = 0.632R. For a hollow sphere, k = sqrt(2/3) * R = 0.816R. The hollow sphere has a larger k because its mass is all at the outer surface.
How does radius of gyration relate to column buckling?
In structural engineering, the slenderness ratio is L/k, where L is the column length and k is the radius of gyration of the cross-section. Higher k means better resistance to buckling. Engineers maximize k by placing material far from the centroid (e.g., I-beams).
What is the radius of gyration of an I-beam?
An I-beam is designed to maximize k (and thus buckling resistance) for a given mass. The flanges carry most of the material far from the neutral axis, giving k approximately 0.35-0.45 times the beam depth, much higher than a solid rectangle.
How is radius of gyration used in sports equipment?
In sports, lower k means faster rotation (easier to swing). Golf drivers have low k for faster clubhead speed. Figure skates pull arms in to reduce k and spin faster. Baseball bats are designed with specific k for optimal swing feel.
What is the radius of gyration of a thin ring?
For a thin ring (hoop) of radius R, k = R. All the mass is at distance R from the axis, so the equivalent concentration point is at R itself. This is the maximum possible k for a given radius.
What is the radius of gyration of a rectangular plate?
For a thin rectangular plate of width a and height b, about an axis through the center perpendicular to the plate: k = sqrt((a^2 + b^2)/12). About an axis parallel to side a: k = b/sqrt(12).
How does radius of gyration affect flywheel design?
Flywheels store energy as KE = 0.5 * M * k^2 * omega^2. For maximum energy storage, designers maximize k by placing mass at the rim (ring geometry). A larger k means more energy stored per unit mass at a given speed.
What is the radius of gyration of a solid cone?
For a solid cone of base radius R about its central axis, k = sqrt(3/10) * R = 0.548R. The mass is concentrated near the axis at the apex and spread at the base, giving an intermediate k value.
How do you measure radius of gyration experimentally?
Measure the period of a physical pendulum: T = 2*pi*sqrt((k^2 + d^2)/(g*d)), where d is the distance from pivot to CM. Alternatively, use a torsion pendulum and solve for k from the oscillation period.
What is the radius of gyration of a composite body?
For a composite body, k = sqrt(I_total / M_total), where I_total is the total MOI (sum of component MOIs using parallel axis theorem if needed) and M_total is the total mass.
How does k relate to the parallel axis theorem?
The parallel axis theorem states I = I_cm + Md^2. Dividing by M: k^2 = k_cm^2 + d^2. So the radius of gyration about any axis is always greater than or equal to the radius of gyration about a parallel axis through the CM.
What is the radius of gyration of a hollow cylinder?
For a hollow cylinder with outer radius R2 and inner radius R1, k = sqrt((R2^2 + R1^2)/2). For a thin-walled pipe (R1 ~ R2 = R), k ~ R.
How is radius of gyration used in earthquake engineering?
The radius of gyration of a building cross-section determines its resistance to torsional response during earthquakes. A larger k reduces torsional amplification of seismic forces.
What is the radius of gyration of a human body?
For a standing human about the ankles, k is approximately 1.0-1.1 m (about 60% of height). About the hips, k is approximately 0.3 m. These values change dramatically with body posture.
How does k affect rotor balancing?
Rotor unbalance creates forces proportional to M * e * omega^2, where e is the eccentricity. The radius of gyration determines the rotor response to unbalance. Higher k rotors are more sensitive to bearing placement.
What is the radius of gyration of a half-disk?
For a half-disk of mass M and radius R about the axis through the center of the full disk perpendicular to the plane, k = R/sqrt(2), same as the full disk for this axis by symmetry.
How does k relate to angular momentum?
Angular momentum L = I * omega = M * k^2 * omega. For a given angular momentum, a body with larger k rotates more slowly. This is why ice skaters extend arms (increasing k) to slow rotation.
What is the radius of gyration of an ellipse?
For a uniform solid ellipse with semi-axes a and b, about the a-axis: k = b/sqrt(5). About the b-axis: k = a/sqrt(5). For a circle (a=b=R), k = R/sqrt(5) = 0.447R.
How is radius of gyration used in vehicle dynamics?
The yaw radius of gyration (about the vertical axis) affects turning response. Lower yaw k means quicker directional changes. Sports cars have lower yaw k than SUVs for better agility.
What is the difference between radius of gyration and gyration radius?
They are the same quantity. Both refer to k = sqrt(I/M). The term "radius of gyration" is more common in structural engineering, while "gyration radius" is used in rotational dynamics.
How does temperature affect radius of gyration?
Temperature changes affect k through thermal expansion. As an object expands, dimensions increase and k grows proportionally. The effect is typically small: dk/k = alpha * dT/2, where alpha is the coefficient of thermal expansion.
What is the radius of gyration of a quarter-circular arc?
For a quarter-circular wire of radius R, the radius of gyration about the center is k = R. All mass is at distance R from the center, same as a full ring.
How does k affect the design of rotating machinery?
In rotating machinery, k determines the natural frequencies of lateral vibration (critical speeds). Designers must ensure operating speed avoids critical speeds. Higher k raises critical speeds.
What is the radius of gyration of a uniform wire bent into a circle?
For a uniform wire bent into a circle of radius R, rotating about an axis through the center perpendicular to the plane, k = R. All mass is at distance R from the axis.
How is radius of gyration used in satellite attitude control?
Satellite attitude control uses k to calculate the torque needed for rotation: T = I * alpha = M * k^2 * alpha. Lower k means less torque needed for the same angular acceleration, saving fuel.
What is the radius of gyration of a thin triangular plate?
For a uniform thin triangular plate about an axis through the centroid parallel to the base, k = h/sqrt(18) where h is the height. The CG location and area distribution determine k.
How does k relate to the mass moment of inertia factor in planetary science?
The MOI factor (I/MR^2) equals (k/R)^2. For Earth, this factor is 0.3307, meaning k = 0.575R. A uniform sphere would have k/R = sqrt(2/5) = 0.632. The difference reveals Earth mass concentration toward the core.
What is the radius of gyration of a torus about its central axis?
For a torus with major radius R and minor radius r (thin-walled), k = sqrt(R^2 + r^2/2). For a thin torus (r << R), k approximately equals R. All mass is near distance R from the axis.
How is radius of gyration used in pendulum design?
The period of a physical pendulum is T = 2*pi*sqrt((k^2 + d^2)/(g*d)). Maximizing k for a given pivot-to-CM distance d increases the period, useful for clock pendulums requiring slow oscillation.
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Glossary
Technical terms used in mortgage calculations explained in plain language.
- Radius of Gyration
- k = sqrt(I/M). Equivalent distance for mass concentration.
- Mass Moment of Inertia
- I = integral(r^2 dm). Resistance to rotational acceleration.
- Slenderness Ratio
- L/k for columns. Higher values mean more buckling risk.
- Parallel Axis Theorem
- I = I_cm + Md^2. Relates MOI about parallel axes.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
AISC 360Eurocode 3ISO 10137
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Page Information
Last Updated:
2025-08-21
Version:
2.0.0
License:
Free to use for educational and professional purposes.