What is Radius of Gyration? Definition and Formula
Physics · System Of Particles And Rotational Motion · NEET
The radius of gyration (k) is the distance from the axis of rotation at which the whole mass M of a body can be imagined to be concentrated, so that its moment of inertia stays the same: I = M k squared. So the formula is k = sqrt(I / M), and its unit is the metre (it has the dimension of length). Memory hook: think of k as the "one spot" where you could squeeze all the mass into a single point and the body would still be equally hard to spin.
The real body has mass spread at many distances (left). The radius of gyration k is the single distance (right) at which the whole mass M would give the same moment of inertia, so I = M k squared and k = sqrt(I/M).
Your doubts, answered
Is the radius of gyration the same as the actual radius R of the body?
No. R is a fixed geometric size of the object, but k depends on how the mass is spread about the chosen axis. For a ring about its central axis, every part is at distance R, so I = M R squared and k = R (they match). But for a disc about the same axis, I = (1/2) M R squared, so k = R / sqrt(2), which is smaller than R. So k equals R only in special cases like the ring.
Does the radius of gyration depend on the axis of rotation?
Yes, strongly. Since k = sqrt(I / M) and I changes with the axis, k also changes when you pick a different axis. For a disc, k = R/sqrt(2) about the central perpendicular axis, but k = R/2 about a diameter. Same body, same mass, different axis gives a different k. Always state the axis when you give a value of k.
What are the unit and dimension of radius of gyration?
The SI unit is the metre (m) and the dimension is [L], length. This is easy to prove: I has units kg m squared and M has units kg, so I/M has units m squared, and taking the square root gives m. NCERT points this out directly: writing I = M k squared shows k must have the dimension of length.
Why use radius of gyration when moment of inertia already exists?
k gives a single length that tells you how far, on average (in a mass-weighted, squared sense), the mass sits from the axis. It lets you compare shapes cleanly: a body with a larger k has its mass spread farther out and is harder to spin. It also simplifies formulas, since once you know k you get I = M k squared for any mass instantly, and rotational KE becomes (1/2) M k squared omega squared.
⚠️ The NEET trap ✗ Radius of gyration = radius of the body, so k = R for every shape. ✓ k = sqrt(I/M), which depends on the shape and the axis. It equals R only for a ring/hoop about its central axis; for a disc it is R/sqrt(2) (central axis) or R/2 (diameter). 🧠 Ring is the only shape where all mass sits at R, so only there does k = R. For everything else, compute k = sqrt(I/M).
Real NEET questions
NEET 2022
The ratio of the radius of gyration of a thin uniform disc about an axis through its centre and normal to its plane to that about its diameter is:
A · 2 : 1
B · sqrt(2) : 1 ✓
C · 4 : 1
D · 1 : sqrt(2)
Solution: Use k squared = I/M for each axis.
Step 1 (normal axis through centre): I1 = (1/2) M R squared, so k1 squared = I1/M = R squared / 2.
Step 2 (diameter): I2 = (1/4) M R squared, so k2 squared = I2/M = R squared / 4.
Step 3 (take ratio): k1 squared / k2 squared = (R squared/2) / (R squared/4) = 2.
Step 4: k1 / k2 = sqrt(2) = sqrt(2) : 1. Answer: B.
NEET 2023
The ratio of the radius of gyration of a solid sphere (about its own axis) to that of a thin hollow sphere of the same mass and radius (about its axis) is:
A · 3 : 5
B · 5 : 3
C · sqrt(3) : sqrt(5) ✓
D · sqrt(5) : sqrt(3)
Solution: Use k squared = I/M for each sphere (same M and R).
Step 1 (solid sphere): I = (2/5) M R squared, so k_solid squared = (2/5) R squared.
Step 2 (hollow sphere): I = (2/3) M R squared, so k_hollow squared = (2/3) R squared.
Step 3 (ratio of squares): k_solid squared / k_hollow squared = (2/5) / (2/3) = 3/5.
Step 4: k_solid / k_hollow = sqrt(3/5) = sqrt(3) : sqrt(5). Answer: C.
Solved System Of Particles And Rotational Motion NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It is the distance k from the axis where all the mass M can be assumed to be concentrated so that I = M k squared holds, giving k = sqrt(I/M).
What is the formula for radius of gyration?
k = sqrt(I / M), where I is the moment of inertia about the axis and M is the total mass. Equivalently, I = M k squared.
What is the unit of radius of gyration?
The metre (m). Its dimension is length [L], because I/M has units of m squared and the square root gives m.
Is radius of gyration a vector or scalar?
It is a scalar. It is just a length; it has magnitude but no direction, even though it is measured from the axis of rotation.
What is the radius of gyration of a disc and a ring?
For a ring about its central axis, k = R. For a disc about its central perpendicular axis, k = R/sqrt(2); about a diameter, k = R/2. The value depends on the shape and the axis.
Can radius of gyration be greater than the body's radius?
Yes, for some axes. For example, a disc about a tangent in its plane has I = (5/4) M R squared, giving k = R sqrt(5)/2, which is larger than R because the mass is far from that axis.
