Force and Acceleration

From force and acceleration of a particle we know that we can determine the force on a particle from Newton's second law F=ma. This is also true for rigid bodies. However, there is one additional aspect that needs to be considered for rigid bodies, which is rotation about a fixed axis. To determine the force from a body rotating around a fixed axis equation 1 would be used.

(1)

The variable I in equation 1 represents the mass moment of inertia.

Mass Moment of Inertia

Mass moment of inertia is the resistance an object has to rotation. It is based off of the objects mass and how far that mass is from the center of rotation. A general equation that can be used to calculate the mass moment of inertia can be seen in equation 2.

(2)

Now what if the object its self is not rotating but instead is rotating around a specific point out in space? To calculate the mass moment of inertia for this case, the parallel axis theorem can be used. To use the parallel axis theorem the mass moment of inertia of the object as if it was rotating around its own centroid needs to be calculated, and the distance to the point in space that the rigid body will be rotating about must also be known. If those two variables are known equation 3 can be used to calculate the mass moment of inertia of a rigid body rotating around a point in space. You can also consider the mass as a lumped mass, which will remove make it possible to remove the first part of equation 3.

(3)

A mass moment of inertia is sometimes represented as the radius of gyration. What the radius of gyration does is it transforms a mass moment of inertia into a unit length used equation 4.

(4)

For a list of how to calculate different mass moments of inertia follow this link.

Translational Motion and its Forces

Recall from kinematics of a rigid body, there is no rotation if the motion of the rigid body has translational motion. This means that if a line segment is drawn between two particles, that line segment should remain parallel as that rigid body moves from point a to point b, which results in no rotational force since there is no rotation about a fixed axis. So to calculate the forces on the rigid body from certain acceleration you would only have to consider different variables of F=ma, which can be seen in the equations below. You may also need to consider moments when dealing with a rigid body. However, the moments for a rigid body will sum to zero.

(5)

(6)

(7)

(8)

(9)

(10)

Equation 5-10 represents the forces in the x, y, z, normal, and tangential directions.

Roation about a Fixed Axis and its Forces

If a rigid body is rotating around a fixed axis the tangential and normal components would have to be considered along with the effects of the mass moment. The equations that would be used to solve a problem that has rotation about a fixed axis can be seen below.

(11)

(12)

(13)

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