Perpendicular Axis Theorem

Srujana

MOMENT OF INERTIA

. The velocity of a particle in a rotating object is given by

v = r w

. This equation is used to derive the kinetic energy of a rotating object:

K= 1/2 mv² = 1/2 m(r )² = 1/2 mr² ²

. Since the total energy is the sum of the energies of all the particles and as w

is the same for all particles, we have:

K_Total = 1/2 (mr²) ²

. Comparing the given expression with the kinetic energy in case of linear

motion we get mr² analogus to mass in th linear motion

. It turns out that the value of the summation is the rotational equivalent of mass, and is referred to as rotational inertia or moment of inertia ( I):

K_Rotational = 1/2 I ²

. For every particle on the rotating body, there is a specific moment of inertia

for that object and for that specific axis of rotation. As the axis changes I

changes and as the shape changes I changes.

. Thus Moment of inertia is dependent on,

1. Shape of the rotating body(i.e distribution of mass)
2. Axis of rotation

. It is independent of

1. Velocity of rotation
2. Duration of rotation

Derivation

. Before we start the derivation, we should clarify how to find I

. Since mr² is different in different parts of a solid object, we will be integrating

by splitting up an object into pieces of mass each with the same mass of d m .

. Now, each piece of mass is probably at a different distance (r) from the axis

of rotation.

. Thus, I is given by

I = _{m 0}Ir²m = r² dm

. Assuming that the object we are dealing with has a uniform density of r,

we will be concerning ourselves with the infinitely small pieces of volume (dV)

that the infinitely small pieces of masses ( dm) take up:

I = r² dV = r²dV

Since density is constant, it can remain outside the integral.

. Here is a table of the ten most common moments of inertia that are used

Shape and Axis		Rotational Inertia
Hoop About Central Axis		I = MR²
Hoop About Any Diameter		I = 1/2 MR²
	Annular Cylinder (Ring) About Central Axis		I = M(R₁² +R₂²)
	Solid Cylinder (Disk) About Central Axis		I = 1/2 MR²
	Solid Cylinder (Disk) About Central Diameter		I = 1/4 MR²+1/12 ML²
	Thin Rod About Axis Through Center Perpendicular to Length		I = 1/12 ML²
	Thin Rod About Axis Through One End Perpendicular to Length		I = 1/3 ML²
	Solid Sphere About Any Diameter		I = 2/5 ML²
	Thin Spherical Shell About Any Diameter		I = 2/3 ML²
	Slab About Perpendicular Axis Through Center		I = M(a² +b²)

. The Parallel and perpendicular axis theorems help in

determining the moment of inertia about a given axis.

Parallel-Axis Theorem

. Given the moment of inertia I_CM of a body about an axis through its center of mass ,

then the moment of inertia about a new axis parallel to the first but displaced a

distance h can be found through a relation called the parallel-axis theorem.

. The relation is stated as:

I = I_CM + Md²

. The expression added to the center of mass moment of inertia will be recognized as

the moment of inertia of a point mass - the moment of inertia about a parallel axis is

the center of mass moment plus the moment of inertia of the entire object treated as a

point mass at the center of mass.

Perpendicular Axis Theorem

. For a planar object, the moment of inertia about an axis perpendicular to the plane is

the sum of the moments of inertia of two perpendicular axes through the same point in

the plane of the object.

. The utility of this theorem goes beyond that of calculating moments of strictly planar

objects.

. It is a valuable tool in the building up of the moments of inertia of three dimensional

objects such as cylinders by breaking them up into planar disks and summing the

moments of inertia of the composite disks.