Physics Formulas: Gyration, Poisson Ratio, Relativity & Elasticity

Posted on Jan 15, 2026 in Physics

Key Physics Concepts and Equations

(a) Radius of Gyration (k)

The radius of gyration of a body about a given axis is the radial distance from the axis of rotation to a point where the entire mass of the body could be concentrated without changing its moment of inertia.

Mathematically, it is defined by the relation:

I = M k^2

Where:

  • I is the moment of inertia.
  • M is the total mass of the body.
  • k is the radius of gyration (SI unit: meters).

(b) Poisson’s Ratio: Can It Be Negative?

Yes, the Poisson’s ratio (ν) of a material can be negative.

Reason:

Most common materials have a positive Poisson’s ratio (between 0.0 and 0.5), meaning they get thinner when stretched. However, materials with a negative Poisson’s ratio are called auxetic materials.

  • Mechanism: Due to their unique internal hinge-like structures, when these materials are stretched in one direction, they actually become thicker in the perpendicular direction.
  • Examples: Certain specially engineered polymers, foams, and specific crystalline structures like zeolites.

(c) Speed of a Zero Rest Mass Particle

In the theory of relativity, the total energy (E) of a particle is given by the energy-momentum relation:

E^2 = m_0^2 c^4 + p^2 c^2

Where m0 is the rest mass, p is momentum, and c is the speed of light.

If a particle has zero rest mass (m0 = 0), the equation simplifies to:

E = p c

We also know the relativistic relation for momentum and energy: p = E v / c^2 (this follows from E = γ m c^2 and p = γ m v, and the relation holds generally even in limiting cases).

Substituting E = p c into p = E v / c^2:

p = (p c) v / c^2 = p (v / c)

Dividing both sides by p (assuming p ≠ 0):

1 = v / c

Conclusion: Therefore, any particle with zero rest mass (like a photon) must always move at the speed of light in free space (v = c).

(d) Relation between g and G

Let M be the mass of the Earth and R be its radius. Consider a body of mass m on the surface of the Earth.

According to Newton’s Law of Gravitation, the force of attraction is:

F = G M m / R^2

According to Newton’s Second Law of Motion, the force acting on the body is its weight:

F = m g

Equating the two expressions:

m g = G M m / R^2

By canceling m from both sides, we get the relation:

g = G M / R^2

Theorems on Moments of Inertia

In classical mechanics, these two theorems are essential for calculating the moment of inertia of bodies with complex shapes or when the axis of rotation is shifted.

(i) Theorem of Perpendicular Axes

Statement:

The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane (I_z) is equal to the sum of its moments of inertia about two mutually perpendicular axes (I_x and I_y) in its own plane, all three axes being concurrent (meeting at the same point).

Mathematically:

I_z = I_x + I_y

Proof:

Consider a plane lamina in the XY plane. Let there be a particle of mass m at a point P(x, y).

  • The distance of the particle from the X-axis is y.
  • The distance of the particle from the Y-axis is x.
  • The distance of the particle from the Z-axis (origin O) is r, where r^2 = x^2 + y^2.
  • Moment of inertia about the X-axis: I_x = Σ m y^2
  • Moment of inertia about the Y-axis: I_y = Σ m x^2
  • Moment of inertia about the Z-axis: I_z = Σ m r^2

Substituting r^2 = x^2 + y^2 gives I_z = I_x + I_y. (Proved)

(ii) Theorem of Parallel Axes

Statement:

The moment of inertia of a body about any axis (I) is equal to the sum of its moment of inertia about a parallel axis passing through its center of mass (Icm) and the product of the mass of the body (M) and the square of the distance (h) between the two parallel axes.

Mathematically:

I = I_{cm} + M h^2

Proof:

Consider a body of total mass M. Let AB be the axis passing through the center of mass (C) and EF be the parallel axis at a distance h. Consider a small mass element m at point P, at a distance x from the axis AB.

  • The distance of mass m from axis EF is (x + h).
  • The moment of inertia about axis EF is Σ m (x + h)^2.

Expanding the bracket and using Σ m x^2 = I_{cm}, Σ m = M, and Σ m x = 0 (by definition of center of mass), we obtain:

I = I_{cm} + M h^2 (Proved)

Lorentz Transformation

The Lorentz transformation is a set of equations that relate the space and time coordinates of an event in one inertial frame to those in another inertial frame moving relative to the first frame with a constant velocity. It is a fundamental concept in special relativity.

Lorentz Transformation Equations

The Lorentz transformation equations for position and time are:

x' = γ (x - v t)
y' = y
z' = z
t' = γ (t - v x / c^2)

where γ = 1 / sqrt(1 - v^2 / c^2), v is the relative velocity, c is the speed of light, and (x, y, z, t) and (x’, y’, z’, t’) are the coordinates in the two frames.

Lorentz Transformation for Momentum and Energy

The Lorentz transformation for momentum and energy can be derived from the Lorentz transformation for position and time.

Let’s consider a particle with rest mass m0 moving with velocity u in the x-direction. The momentum and energy of the particle in the rest frame are:

p = m_0 u / sqrt(1 - u^2 / c^2)
E = m_0 c^2 / sqrt(1 - u^2 / c^2)

Using the Lorentz transformation, we can derive the momentum and energy in a moving frame:

p' = γ (p - v E / c^2)
E' = γ (E - v p)

These equations show how momentum and energy transform under Lorentz transformations and demonstrate how they are intertwined in special relativity.

Elasticity

Elasticity is the ability of a material to return to its original shape and size after an external force is removed. Materials that exhibit elasticity can withstand stress and strain without undergoing permanent deformation.

Key Terms

  1. Young’s Modulus (E): A measure of a material’s stiffness, defined as the ratio of stress to strain within the proportional limit. It is a measure of the material’s ability to resist deformation under tensile or compressive stress.

    E = Stress / Strain = (F / A) / (ΔL / L)

  2. Bulk Modulus (K): A measure of a material’s resistance to compression, defined as the ratio of volumetric stress to volumetric strain. It is a measure of the material’s ability to withstand changes in volume under hydrostatic pressure.

    K = Volumetric Stress / Volumetric Strain = ΔP / (ΔV / V)

  3. Modulus of Rigidity (G): A measure of a material’s resistance to shear stress, defined as the ratio of shear stress to shear strain. It is a measure of the material’s ability to withstand deformation under shear stress.

    G = Shear Stress / Shear Strain = (F / A) / (Δx / h)

  4. Poisson’s Ratio (ν): A measure of a material’s lateral strain response to a longitudinal tensile or compressive load. It is defined as the ratio of lateral strain to longitudinal strain.

    ν = Lateral Strain / Longitudinal Strain = (Δd / d) / (ΔL / L)

Normal Coordinates and Normal Modes of Vibration

In a coupled system, normal coordinates and normal modes of vibration are essential concepts in understanding the complex motion of the system.

Normal Coordinates

Normal coordinates are a set of independent coordinates that describe the motion of a system in a way that each coordinate corresponds to a single frequency of vibration. In other words, normal coordinates are a set of decoupled coordinates that simplify the description of the system’s motion.

Normal Modes of Vibration

A normal mode of vibration is a pattern of motion in which all parts of the system oscillate at the same frequency. In a normal mode, each part of the system moves with the same frequency, but with a specific amplitude and phase.

Properties of a Normal Mode

  1. Single Frequency: Each normal mode has a single frequency of vibration.
  2. Fixed Phase Relationship: The motion of different parts of the system has a fixed phase relationship.
  3. Independent Motion: Normal modes are independent of each other, meaning that the motion in one mode does not affect the motion in another mode.

Significance of Normal Modes

  1. Simplifies Complex Motion: Normal modes simplify the description of complex motion in coupled systems.
  2. Predicts System Behavior: Normal modes can be used to predict the behavior of a system under different types of excitation.
  3. Analyzes System Stability: Normal modes can be used to analyze the stability of a system.