Acceleration of a Particle
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Acceleration of a Particle in a Moving Reference Frame
Centripetal Acceleration

Acceleration of a Particle in a Moving Reference Frame

The XYZ system shown in Figure 1 is an inertial reference frame while the X'Y'Z' frame is a moving (translation + rotation) reference frame. The position of point P in the figure can be expressed in terms of these two reference frames:

    Figure 1

   [1]

where ro = the position of the origin of the rotating frame, and r' = the position of P relative to the origin of the moving frame.

From [1] and [6] of Rotating Reference Frame, the displacement of point P can then be expressed as

   [2]

where dro = the displacement due to translation of the origin of the rotating frame, dr' = the displacement of point P relative to the origin of the rotating frame, dq x r' = the displacement due to rotation of the moving frame, and (dr')rot = the displacement of point P observed in the rotating frame (or the displacement of P relative to the moving frame).

From [2], the velocity of point P can be expressed as

    [3]

where v'r = the velocity of point P relative to the moving reference frame. Thus, the velocity of point P relative to the origin of the moving reference frame can be divided into the velocity due to rotation of the frame and the velocity observed in the rotating frame.

Acceleration of point P can then be obtained similarly:

    [4]

where a = the acceleration of point P, ao = the acceleration of the origin of the moving frame, and a'r = the acceleration of point P relative to the moving frame. Note here that all vectors involved in the velocity equation shown in [3] are subject to derivation and [9] of Rotating Reference Frame was applied to all cases. [14] of Rotating Reference Frame shows that the time-derivative of the angular velocity vector is not affected by the rotation of the moving frame. As a result, the acceleration of point P shows 5 different terms.

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Centripetal Acceleration

The 1st & the 2nd terms in [4] are the results of the translational and rotational acceleration of the moving reference frame, respectively. The 3rd term, w x (w x r'), is due to the rotation of the moving frame and, as shown in Figure 2, this vector heads toward the center of rotation (point C). This acceleration is called the centripetal acceleration. The 4th term is due to the motion of the point within the moving frame and is called the Coriolis acceleration. The last term is due to the acceleration of the point within the moving reference frame.

    Figure 2

If the moving frame does not translate but rotates with a constant angular velocity and point P does not move within the moving frame, [4] can be simplified to

   [5]

since

   [6]

In order words, the centripetal acceleration is the basic requirement for a constant circular motion of a body around an axis of rotation. According to Newton's 2nd Law of Motion the force acting on a particle is equal to the its mass times acceleration. Therefore, from [6]:

   [7]

This force is the centripetal force which causes a constant circular motion of a body. Note that this force acts in the direction of the center of rotation, point C shown in Figure 2.

If the angular velocity of the rotating reference frame is not constant, [4] reduces to

   [8]

since

   [9]

In the case of planar rotation where the direction of the angular velocity does not change, the 1st term of [8] can be labeled as the tangential acceleration since it is in the direction of w x r' shown in Figure 2. (The angular velocity vector and the angular acceleration vector show the same direction in this case.) The 2nd term then becomes the radial acceleration.

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Young-Hoo Kwon, 1998-