classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...

, the stretch rule (sometimes referred to as Routh's rule) states that the

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...

of a

rigid object In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...

is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that the distribution of mass remains unchanged except in the direction parallel to the axis. This operation leaves

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...

s oriented parallel to the axis unchanged in radius. This rule can be applied with the

parallel axis theorem The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body ...

and the

perpendicular axes rule The perpendicular axis theorem (or plane figure theorem) states that the moment of inertia of a planar lamina (i.e. 2-D body) about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about th ...

to find moments of inertia for a variety of shapes.

Derivation

The (scalar) moment of inertia of a rigid body around the z-axis is given by: :

I_z = \int_V d^3 r \, \rho(\mathbf)\,r^2

Where

r

is the distance of a point from the z-axis. We can expand as follows, since we are dealing with stretching over the z-axis only: :

I_z = \int_^ dz \int_ dx dy \, \rho(x, y, z)\,r^2

Here,

L

is the body's height. Stretching the object by a factor of

a

along the z-axis is equivalent to dividing the mass density by

a

(meaning

\rho'(x, y, z) = \rho(x, y, z/a)/a

), as well as integrating over new limits

0

and

aL

(the new height of the object), thus leaving the total mass unchanged. This means the new moment of inertia will be: :

I_' = \int_^ dz \int_ dx dy \, \rho'(x, y, z) \,r^2

= \int_^ adz' \int_ dx dy \, \frac \,r^2

= \int_^ dz' \int_ dx dy \, \rho(x, y, z') \,r^2 = I_z

References

{{DEFAULTSORT:Stretch Rule Classical mechanics Moment (physics)