Question
Derive the relation between linear displacement and angular displacement.
Solution
Consider, a particle is undergoing uniform circular motion in the plane of the paper in an anticlockwise direction as shown in the figure.
Let, at any instant t, the particle is at A.
At t + dt, the particle is at B.
In going from A to B, the particle undergoes linear displacement
and angular displacement 
From figure,
Since
is very small, therefore 
Thus,
or
...(1)
In vector form:
Here, angular displacement is a vector quantity and according to the right-hand thumb rule, its direction is along the axis of rotation.
In the figure, the angular displacement
is perpendicular to plane of paper in outward direction, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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is along 
in the plane of paper and 
is in the plane of paper perpendicular to both 
Thus, using cross product rule, equation (1) in vector form can be expressed as
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, is the required relation.
Let, at any instant t, the particle is at A.
At t + dt, the particle is at B.
In going from A to B, the particle undergoes linear displacement
and angular displacement From figure,
Since
is very small, therefore Thus,
or
...(1) In vector form:
Here, angular displacement is a vector quantity and according to the right-hand thumb rule, its direction is along the axis of rotation.
In the figure, the angular displacement
is perpendicular to plane of paper in outward direction, is along in the plane of paper and is in the plane of paper perpendicular to both Thus, using cross product rule, equation (1) in vector form can be expressed as
, is the required relation. 
