A rigid body experiencing a state of general motion in three-dimensions is shown below.

The variables in the figure are defined as follows:

Both coordinate systems XYZ and

The first coordinate system is the global XYZ reference frame. This reference frame is defined as fixed to the ground. The point

The second coordinate system is the local

For both coordinate systems the individual axes shown correspond to the positive directions of those axes. For example, the positive direction of the Y-axes is to the right. The positive direction of the Z-axes is up.

The choice of positive direction for these two coordinate systems is defined in this way because the equations that will be presented here involve vector cross-product multiplication, the mathematics of which is based on this choice of sign convention. So it is important to use the same sign convention when solving problems using the equations presented here. If you use another sign convention, you might get the wrong answer.

We wish to find the general motion equations for velocity and acceleration of point

Note that point

Where:

To find the general motion equation for the velocity of point

At the instant shown, the rigid body is rotating with an angular velocity

The velocity

The direction and magnitude of velocity

Now, let’s say

Similarly, if point

This is the most general equation for velocity of a point

To find the general motion equation for the acceleration of point

Now, by vector addition

Where:

We can rewrite equation (3) as

Differentiate this equation with respect to time and we get

But we know that

and

Thus, equation (4) becomes

Compare this to equation (1) and we see that

The relative velocity

Where:

Therefore,

Now,

where

Also,

The above formulation is a result of vector differentiation. For more information about this see the page on vector derivatives.

Therefore,

and

Therefore,

Substitute this equation and equation (5) into equation (2), and we get

Now,

Therefore,

This is the most general equation for acceleration of a point

For an overview of the general motion equations click here.

To see an example of a problem solved using the general motion equations click here.

If you know the direction of spin of the angular velocity, you must use the right-hand rule to assign the correct direction for its corresponding vector, and thus determine the correct values for its three components (

As mentioned before, equations (1) and (6) (for velocity and acceleration) are based on the sign convention used here (i.e. that is how they are derived).

An example of a sign convention that does not match the one used here is shown below.

This choice of sign convention may result in errors if one uses it to solve problems using equations (1) and (6).

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