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Parallel Axis Theorem

Tuesday, August 10, 2010

Parallel Axis Theorem: In physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the moment of inertia of a rigid body about any axis, given the moment of inertia of the object about the parallel axis through the object’s center of mass and the perpendicular distance between the axes.
The moment of inertia about the new axis z is given by:
Iz = Icm + md2
Where:
Icm is the moment of inertia of the object about its center of mass;
m is the object’s mass;
d is the perpendicular distance between the two axes.
This rule can be applied with the stretch rule and perpendicular axis theorem to find moments of inertia for a variety of shapes.
The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D:
Iz = Ix + Ad2,
Where:
Iz is the area moment of inertia of D relative to the parallel axis;
Ix is the are amoment of inertia of D relative to its centroid;
A is the area of the plane region D;
d is the distance from the new axis z to the centroid of the plane region D.
Note: The centroid of D coincides with the center of gravity (CG) of a physical plate with the same shape that has constant density.
In classical mechanics
In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensor Jij from an inertia tensor about a center of mass Iij when the pivot point is a displacement a from the center of mass:
Jij = Iij + m (a2δij – aiaj)
where
a = a1x + a2y + a3z (where x, y, and z are unit vectors)
is the displacement vector from the center of mass to the new axis, and
δij
is the Kronecker delta.
We can see that, for diagonal elements (when i=j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.

Moment of Inertia

Moment of Inertia:
Increasing the mass increases the moment of inertia, symbolized by I, which is sometimes called the rotational inertia of an object. But the distribution of the mass is more important, i.e. distributing the mass further from the centre of rotation increases the moment of inertia by a greater degree. The moment of inertia is measured in kilogram metre2 (kg m2).
The energy required or released during rotation is the torque times the rotation angle; the energy stored in a rotating object is one half of the moment of inertia times the square of the angular velocity. The power required for angular acceleration is the torque times the angular velocity.
Torque
Torque τ is the twisting effect of a force F applied to a rotating object which is at position r from its axis of rotation. Mathematically,
τ = r × F
Where × denotes the cross product. A net torque acting upon an object will produce an angular acceleration of the object according to
τ = Iα
Just as F = ma in linear dynamics.

Rotation around a Fixed Axis

Rotation around a fixed axis:
Rotation around a fixed axis is a special case of rotational motion. It does not involve rotation around more than one axis, and cannot describe such phenomena as wobbling or precession. The kinematics and dynamics of rotation around a fixed axis of a rigid object are mathematically much simpler than those for rotation of a rigid body; there are entirely analogous to those of linear motion along a single fixed direction, which is not true for rotation of a rigid body. The expressions for the kinetic energy of the object, and for the forces on the parts of the object, are also simpler for rotation around a fixed axis, than for general rotational motion. For these reasons, rotation around a fixed axis is typically taught in introductory physics courses after students have mastered linear motion; the full generality of rotational motion is not usually taught in introductory physics classes.
In the beginning study of linear motion, objects are treated as point particles without structure; for such objects it does not matter where a force is applied, only that it is applied. However, for extended objects, the point of application of force does matter. In tennis, for example, if a tennis ball is struck with a strong horizontal force acting through its center of mass, it may travel a long distance before hitting the ground, far out of bounds. Instead, the same force applied in an upward, glancing stroke will yield topspin to the ball, which can cause it to land in the opponent’s court.
The concepts of rotational equilibrium and rotational dynamics are also important in other disciplines. For example, students of architecture benefit from understanding the forces that act on buildings and biology students should understand the forces at work in muscles, bones, and joints. These forces create torques, which tell us how the forces affect an object’s equilibrium and rate of rotation.
An object remains in a state of uniform rotational motion unless acted on by a net torque. This principle is analogous to Newton’s first law of motion. Further, the angular acceleration of an object is proportional to the net torque acting on it, which is the analog of Newton’s Second Law of motion. A net torque acting on an object causes a change in its rotational energy.
Finally, torques applied to an object through a given time interval can change the object’s angular momentum. If there are no external torques, angular momentum is conserved, a property that explains some of the mysterious and formidable properties of pulsars—remmants of supernova explosions that rotate at equatorial speeds approaching that of light.
Translation and rotation
A rigid body is an object of finite extent in which all the distances between the component particles are constant. No truly rigid body exists; external forces can deform any solid. For our purposes, then, a rigid body is a solid which requires large forces to deform it appreciably.
A change in the position of a particle in three-dimensional space can be completely specified by three coordinates. A change in the position of a rigid body is more complicated to describe. It can be regarded as a combination of two distinct types of motion: translational motion and rotational motion.
Purely translational motion occurs when every particle of the body has the same instantaneous velocity as every other particle; then the path traced out by any particle is exactly parallel to the path trace out by every other particle in the body. Under translational motion, the change in the position of a rigid body is specified completely by three coordinates such as x, y, z, and z giving the displacement of any point, such as the center of mass, fixed to the rigid body.
Purely rotational motion occurs if every particle in the body moves in a circle about a single line. This line is called the axis of rotation. Then the radius vectors from the axis to all particles undergo the same angular displacement in the same time. The axis of rotation need not go through the body. In general, any rotation can be specified completely by the three angular displacements with respect to the rectangular-coordinate axes x, y, and z. Any change in the position of the rigid body is thus completely described by three translational and three rotational coordinates.
Any displacement of a rigid body may be arrived at by first subjecting the body to a displacement followed by a rotation, or conversely, to a rotation followed by a displacement. We already know that for any collection of particles—whether at rest with respect to one another, as in a rigid body, or in relative motion, like the exploding fragments of a shell, the acceleration of the center of mass is given by
Fnet = Macm
Where M is the total mass of the system and acm is the acceleration of the center of mass. There remains the matter of describing the rotation of the body about the center of mass and relating it to the external forces acting on the body. The kinematics and dynamics of rotational motion around a single axis resemble the kinematics and dynamics of translational motion; rotational motion around a single axis even has a work-energy theorem analogous to that of particle dynamics.