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Since (when applying the equations of motion) acceleration must be measured relative to an inertial reference frame, it follows that angular velocity and angular acceleration (such as for a rigid body) must also be measured relative to this frame, since these quantities directly affect the acceleration.

Any forces that are acting on an object are a result of the acceleration of this object relative to an inertial reference frame. Thus, if we want to accurately predict the force

When applying the equations of motion for problems on earth, it is common practice to “attach” a coordinate system to the ground and assume that this is an inertial reference frame. However, it is not quite, since the earth itself rotates and the earth in turn revolves around the sun, etc. However, the rotation rate of a (ground) reference frame attached to the earth is very small, and its acceleration is also very small, so for practical purposes it can be treated as an inertial reference frame.

To show this, let’s determine the rotation rate and acceleration of a ground reference frame attached to the earth (so that it moves with the earth).

The rotation rate of the earth is one full revolution (2

The acceleration of this reference frame is due to rotation of the earth, which causes centripetal acceleration. The maximum centripetal acceleration is at the equator, since the radius (perpendicular to the axis of earth’s rotation) is greatest there.

The radius of the earth at the equator is

The centripetal acceleration is

There is also the affect of the rotation of the earth around the sun, and the rotation of our solar system within our galaxy. But these effects are also negligible.

Therefore, a ground reference frame attached to earth can be treated (with a high degree of accuracy) as an inertial reference frame.

The figure below illustrates a typical set up of a dynamics problem that is solved using a global XYZ axis (fixed to ground). This is a coordinate system which serves as an inertial reference frame.

For example, we can accurately solve for the forces

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