Friday, June 14, 2019

Solutions To Theoretical Mechanics Problems

1.   Using the preceding unit vector relations, obtain an expression for the acceleration.

2.A particle  of mass m moves in a plane under the influence of a force F = - kr, directed toward the origin.  Sketch a polar coordinate system (r, q ) to describe the motion of the particle and thereby obtain the Lagrangian (L = T - V, i.e. difference in kinetic and potential energy).

Solutions:

1) We differentiate:  v  =   dr/ dt  n   +   r  dq  /dt l

With respect to t to obtain:

a =     dr 2/ dt 2) n   +   dr/ dt (dn /dt )  +   dr/ dt (d q /dt ) l   r (dq  /dt )l

r  dq  /dt  (dl/dt)

We next use the 2 equations:

a) :  dn /dt   =    dq  /dt l

And:

b) dl /dt   =   -  dq  /dt n

To separate the components :

a =    [ dr 2/ dt 2   -  r (dq  /dt ) 2] n   +  [r (d2 q  /dt 2)   + 2 dr/dt (dq  /dt )] l

Where the first term is just the radial acceleration:  a r

And the second term is the angular acceleration:     a q

2) We make use here of the sketch below: From which:  =  r n

dr/ dt =   ( dr/ dt  n   +   r dn /dt)

dn /dt   =   dn / dq   (dq  /dt) =   l  (dq /dt)

(Since:  dn / dq   =  l )

Then:    dr/ dt =  ( dr/ dt)  n  +  r  dq  /dt l

L  = T   - V

Where:  T  =    ½ m [( dr 2/ dt 2 )   +  r (d2 q /dt 2) ]

V   =  ò r o     kr dr =     ½ k 2

Then:

L  = T   - V   =

½ m [( dr 2/ dt 2 )   +  (d2 q /dt 2) ]    -   ½ k 2

This can also be written in more concise notation:

L =    T   - V  =  ½ m[ r"   +    2 (  " ) ]    -   ½ k 2