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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 – III

General Motion in Two Dimensions

where on right hand side first term expresses change in the direction,

while second term expresses change in its magnitude.

Taking dot product of (1) with the unit vector e we have

dA de dA e A e

d  d  d 

Now,  

2 1 1 1

... 2 2 2

dA dA d dA dA A A A A d  A d  A d  A d  d 

i.e.,..

dA A dA dA e

d  A d  d 

Then from (2) and (3) we have

de e

d 

 as A^ ^0 (4)

Now we can conclude following two results:

(1) The magnitude of an unit vector rotating in two dimension about its

initial point does not change with respect to .

(2) If e be a unit vector, then

de

d 

is another unit vector, but

perpendicular to e.

Thus we have

di j d 

 and ˆ

dj i d 

where negative sign occurs because the sense of change in ˆ j will be

opposite to that in i ˆ.

Cartesian Components of Motion

Let OXY be a Cartesian plane a fixed frame of reference. Let a particle

describe a path in the plane. Let us consider two positions on the path P

and Q very close to each other. Two perpendicular directions at both

positions are parallel to x -axis and y -axis respectively.

Thus as the position changes on the path, the

two perpendicular directions do not change.

Let position vector of P be r x y ( ,^ ). If i ˆ and

ˆ j are unit vectors along x -axis and y -axis,

Similarly, if a be acceleration with a  a and  be the angle between the

acceleration vector and positive x -axis, then

ˆ ˆ^2 2

  ,   and tan 

y a xi yj a x y x

So, it is clear from above that the direction of the velocity and acceleration

at any point on the path are different.

Radial and Transverse Components (Polar Components)

Two perpendicular directions in polar system of a two dimensional frame

of reference are radial and transverse directions (cross radial direction).

Let the position of a moving particle at time on its path be P r ( , ).

Further, let unit vector along the radial

direction OP be i ˆ and that along transverse

direction at P along  increasing direction be

ˆ j.

It may be noted that the radial directions at

So, the two velocity components at P r  ,  are

(i) Radial velocity r along the radial direction (^)   i ˆ.

(ii) Transverse velocity r  along the transverse direction (^)  ˆ j .

If the velocity vector makes an angle  with the radial direction then

tan

r d r r dr

   and  

2 2 2

v  r  r  , where v  v.

Differentiating (1) with respect to time with reference to fixed frame we

get

 

dv di dj a ri r r j r r dt dt dt

i.e.,  

di d dj d a ri r r j r r d dt d dt

i.e.,    

a  ri  r   r  j  r  j  r   i

i.e.,    

a  r  r  i  2 r   r  j

Thus radial and transverse acceleration components are

2

r  r  and 2 r   r , i.e.,

2

r  r  and  

1 d (^) 2 r r dt

Also we have    

2 2 2 2

a  r  r   2 r   r  and 2

tan

r r

r r

where a  a and  is the angle which the direction of a makes with the

radial direction.

Now, the linear velocity at P will be the rate of change of arc length w.r.t.

time and it is

ds s dt

 along the tangent.

No normal velocity component is present.

Thus, the velocity (linear velocity) v is at

P may be written as

v  si ˆ , i.e., v  v  s.

So, the acceleration of the particle will be

  ˆ^   ˆ

dv di di d ds a si s si s dt dt d ds dt

i.e.,

2 1  ˆ^  ˆ^  ˆ^  ˆ

v a si sj s si j

where

ds

d

 is the radius of curvature of the curve at P and it may be

obtained from intrinsic equation s  f    of the curve.

Thus, the tangential and normal accelerations are s and

2 v

If the acceleration makes angle  with the tangential direction and a  a

then

2 2 2 2 v a s

and

2

tan 

v

s

Let v be the velocity. Then  

2 2 2 2 2 as

r r v r r  r r   

i.e.,

1 2

2

v 1 r 

Now from (2) we have

2 r d (^) 2 r r r r r r dt r

2 2 2

r d r rr rr r r r dt r

2 2

2

r r r r r r

2 2

2

r r r r r r

2

2

r r r

 

  

2 r Ar B r

  , where 2

A 1 , B

 

  

2 r r C r

  (where

B

C

A

r r C r r

Integrating we have log^ r^ ^ C^ log^ r^ log D  

C r Dr

Therefore,

1 2

2

C v Dr

Thus, the velocity is proportional to any arbitrary power of r.