The instant center is also called the instantaneous center of zero velocity (

There are three basic cases to consider when solving problems using the instant center approach.

Case 1 – Consider a rigid body rotating in a plane. We wish to determine the velocity of a point

Where:

The distances

Note that the lines

Also note that the direction of the velocities

The velocity of point

The velocity of point

Combine equations (1) and (2) to eliminate the variable

Equations (1), (2), and (3) can be used to solve for any two unknowns in the equations, with the remaining variables known.

Find the velocity of point

Set up the solution as drawn below, with new variables introduced as shown. The line

Point

By geometry, the distance

Also,

Substitute equation (5) into equation (4) and we get

The points

With

Therefore,

Note that when

For Case 1, a special case exists where the velocities at points

Case 2 – Consider a rigid body rotating in a plane. We wish to determine the angular velocity of the rigid body given the known velocities of points

Note that

The distance between points

By similar triangles:

The velocity of point

The velocity of point

Combining equations (8)-(10), it follows that the angular velocity is

A wheel of radius

The distance between points

Case 3 – Consider a rigid body rotating in a plane. We wish to determine the angular velocity of the rigid body given the known velocities of points

Note that

The distance between points

By similar triangles:

The velocity of point

The velocity of point

Combining equations (12)-(14), it follows that the angular velocity is

A wheel of radius

The distance between points

Note that the spatial location of the instant center can change with time. So in general its use only applies at the instant considered, which corresponds to the information given in the problem, at the instant considered. In general, if the velocities of points

In general, the instant center approach should not be used for finding the acceleration of points in a rigid body. This is because the instant center (

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