Chapter 10
Rotational of a Rigid Object about a Fixed Axis
10.0 Rotational Motion and Angular Displacement
Rotational motion can be compared to the motion of translation (motion in a straight line). When a rigid body rotates about a fixed axis, the angular displacement is the angle q swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if it is counterclockwise and negative if it is clockwise.
10.1 Angular displacement q in case of circular motion is defined as:
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Fig. 10.1 Rotational motion.
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(angular displacement) (10.1)
where w is the angular velocity and t is time.
Hence, we can write the position vector of the particle at any time along the circular path (see Fig. 10.1) as:
(10.2)
its velocity is
(10.3)
its acceleration is given by
Þ
(10.4)
Note that the terms in the square bracket [
], are the same as the position vector,
,
hence, Eq. (10.4) reduces to:
(vector form, radial acceleration)
(10.5a)
Or
(magnitude form, radial acceleration)
(10.5b)
where the negative sign (-) in Eq. (10.5a), means that acceleration is directed towards the center of the circular path.
Recall that w = v/r, so that Eq. (10.5b) gives
(magnitude form, radial acceleration)
(10.6b)
Or in vector from:
(10.6a)
10.2 Rotational of a Rigid Object about a Fixed Axis
Let us consider a rotation of a rigid object about a fixed axis through point 0 perpendicular to the plane of the Fig. 10.2. The position of an object in rotational motion can be described by an angle around an axis. It is important to specify the axis of rotation of the object. One way to find this axis is by using the right hand rule. First, curl your fingers pointing in the direction of rotation. Next look at the direction that your thumb eventually points. This direction is the axis of rotation.
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Define the distance s from (r,0) to (r,q) as:
(10.7)
Or
(unis:
radian (rad)) (10.8)
We usually measure q in radians due to certain advantages that it has over degrees. Using radians it is easy to calculate the arc length of a circle. Arc length is equal to the radius, r, times the angle q. This is convenient since one radian is equal to 2p.
In general:
where one radian is equal to 2p.
10.2.1 Angular displacement
The position of an object in rotational motion can be described by an angle around an axis. Define angular displacement, from Fig. 10.3, as:
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Fig. 10.3 Angular motion of a rigid body.
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(angular displacement)
(10.9)
10.2.2 Angular velocity
If the angular velocity at point P is not constant, then the point has a changing angular velocity. The direction of the angular velocity vector is equal to the direction of the axis of rotation, which can be found using the right hand rule as described above.
As with linear motion, angular velocity is similar to
linear velocity, and is denoted by the Greek letter omega (symbol:
w). The average angular
velocity, 
,
is defined as the change in angular position divided by the change in time:
average angular velocity:
(units: rad/s) (10.10)
The unit of angular velocity is usually measured in radians per second (rad/s). In some cases, you may find that it is also measured in rotations per second.
10.2.2.1 Instantaneous angular velocity
Instantaneous angular velocity can be found by dividing the change in angular position (Dq) by very small changes in time, Dt. Therefore, instantaneous velocity is equal to the limit of the change in t as it approaches 0 (i.e., as Dt ® 0) of the change in q over the change in t:
(units: rad/s) (10.11)
