Define radius of gyration. Write expression for it.

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Accepted Answer

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> [!TLDR]
> Radius of gyration K is defined via I = M K^2, giving K = sqrt(I/M).
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> For a rigid body made up of many particles, the moment of inertia about an axis is I = sum of (m_i r_i^2) over all particles, where r_i is the perpendicular distance of the i-th particle (mass m_i) from the axis.
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> Instead of tracking this detailed mass distribution, it is often convenient to replace the whole body (total mass M) by an imaginary point mass M located at some effective distance K from the axis, such that this point mass alone would have the same moment of inertia as the real body:
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> I = M K^2
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> This distance K is called the radius of gyration. Rearranging:
>
> K = sqrt(I/M)
> …

Linear Quantity	Rotational Analogue
Displacement $x$	Angular displacement $\theta$
Velocity $v$	Angular velocity $\omega$
Acceleration $a$	Angular acceleration $\alpha$
Mass $m$	Moment of inertia $I$
Force $F$	Torque $\tau$
Newton's 2nd law: $F = ma$	$\tau = I\alpha$
Kinetic energy: $\frac{1}{2}mv^2$	$\frac{1}{2}I\omega^2$
Momentum: $p = mv$	Angular momentum: $L = I\omega$

Q.Define radius of gyration. Write expression for it.

Concept understanding — Rotational Dynamics

Rotational Dynamics: The Physics of Spinning Things

The Core Idea

Breaking It Down

Torque — The Rotational "Push"

Moment of Inertia — The Rotational "Mass"

Angular Acceleration — How Fast Rotation Changes

The Complete Picture: Rotational Analogues

The Key Insight: Conservation of Angular Momentum

Q.Define radius of gyration. Write expression for it.

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