MOTION IN A MOVING FRAME, Study notes of Mathematics

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B.Sc. Mathematics – 2 nd^ Semester

MTB 202 – Statics and Dynamics

by

Dr. Krishnendu Bhattacharyya

Department of Mathematics, Institute of Science, Banaras Hindu University

Part – VII

Motion in a Moving Frame of Reference

Motion in a Linearly Moving Frame A) Linear motion with uniform velocity: Let us consider an object of mass m moving in a plane O AB . This frame itself is moving with uniform linear velocity with reference to a fixed plane of reference OXY. Let any time t , the coordinates of the object at position P , with reference to fixed plane and moving plane be r^  x y ,  and r^ ^  x^ ,^ y 

respectively, where as those of O  with

reference to O be ro (^)  xo , y 0 . Then

r  r  ro^ , or^ x  x ^  xo , y  y  yo.

Differentiating the above expressions w.r.t time variable w.r.t the fixed frame of reference we have v  v  vo , or x  x ^  xo , y  y  yo , (i)

where the terms from the left to right represent velocities of the particle with reference to fixed plane OXY moving plane O AB  and uniform velocity of the moving plane with reference to the fixed plane, respectively.

B) Linear Motion with an acceleration: Let the plane O AB  be moving with an acceleration a with reference to the fixed plane. Then differentiating (i) w.r.t time (w.r.t fixed plane), we have a  a  ao , or x  x ^  xo , y  y  y 0 ,

where first two terms from left to right represent the accelerations of the particle with reference to fixed plane and moving plane, respectively. So, the equation of motion of the object w.r.t the fixed plane is

ma  F , i.e., ma   F  mao.

Since the term on the left hand side represents the product of mass and acceleration (effective force) of the object with reference to moving plane, therefore the equation represent the equation of motion of the object with reference to O AB . Comparing with the previous case, we observe that an additional force (^)   mao  comes into effect. Obviously, it is due to the

linear motion of O AB  with an acceleration. So we may conclude that a frame moving linearly with acceleration is not a Newtonian Frame of Reference. The force is a fictitious force and it vanishes when the acceleration vanishes.

OAB be x = OQ and y = PQ.

Let any time the axis OA make angle  with the fixed axis OA.

Let i and j be unit vectors along two axes OA and OB respectively.

Then we have r  xi  yj.

The rate of change of r with time reference to the fixed frame OXY will be

drdt  r  xi  yj  x didt (^)  ydjdt.

We know that didt^  ddi d  dt^^ ^  j and djdt^  ddj d  dt^^ ^   i ,

where  is the magnitude of the angular velocity vector along the axis of

rotation and is the rate of change with respect to time variable w.r.t the fixed frame of the angular displacement of the rotating frame.

So, r   x   y i    y  x  j ,

i.e., the velocity components of the object at P are  x   y  and  y   x 

along OA and OB respectively, when observed from the fixed plane. Again differentiating w.r.t. t , the acceleration of the object with reference to the fixed frame will have the expression

r   x   y i    y   x  j   x   y   j   y   x   i  ,

If OAB plane is at rest and coincides with OXY the above equation of motion reduces to those in accordance with Newton’s laws of motion, i.e., reduce to mx  X my ,  Y.

But, the plane OAB is rotating w.r.t fixed plane OXY , hence the equations of motion of the object in the rotating plane w.r.t fixed frame may be written as mx  X  X (^) 1  X 2 and my  Y  Y 1 (^)  Y 2 ,

where F 1^  X^ 1 , Y 1  and F 2^  X^ 2 , Y 2  may be interpreted as two additional

fictitious forces introduced due to the rotating of the plane.

Thus the plane rotating about its origin, even with uniform angular velocity in its own plane is not a Newtonian frame of reference.

The force F 1 is of magnitude 2 m  v 1 , where v 1  xi  yj is the linear

velocity of the particle along the tangent to the path described by the particle in the rotating plane. It is known as Coriolis force and is perpendicular to direction of v 1 in a sense opposite to that of angular

velocity. Also the second force F 2 has magnitude m ^2 r and acts along the

radial direction. It is referred to as Centrifugal force.