Change in the unit vectors

[Manish_529]

Homework Statement

how we can write the change in unit vectors over a finite time interval, the change in unit vectors in an infinitesimally small-time interval 'dt' is given by the magnitude of the really small angle that the unit vector has turned about, but what about a change in the unit vector over a finite time interval how can we write that.

Relevant Equations

r hat= d(theta)/dt * theta hat , theta hat= - d(theta)/dt * r hat

i tried integrating the stuff but it didn't work what to do

 

[Orodruin]

Please provide your actual attempt.

 

[Manish_529]

i have attached it

 

[Manish_529]

i have attached the attempt please check it

 

[Orodruin]

While your handwriting is readable, please note for future posts that attempts should be written out explicitly rather than attached as an image. Among other things, this allows quoting relevant passages etc. If you need to wrote a lot of math, refer to the forum LaTeX guide.

Your equation for $d\hat\theta$ is incorrect. It needs to be orthogonal to $\hat\theta$ or the length will change.

You can also differentiate one of the relations to decouple the equations.

 

[Manish_529]

can you please elaborate or just provide an attachment for your work

 

[Orodruin]

can you please elaborate or just provide an attachment for your work

It is your job as the student to provide your work and ask questions regarding your understanding. Providing students with solutions to homework problems before they have solved them themselves is against PF rules.

 

[Manish_529]

i have tried a lot but didn't really get anywhere so at least tell me what to do and where to start from

 

[Orodruin]

I did tell you where you went wrong and you simply ignored it. If it is still unclear, please specify what is still unclear to you.

 

[Manish_529]

i got the thing about d theta hat but i couldn't get what to differentiate exactly

 

[Orodruin]

i got the thing about d theta hat but i couldn't get what to differentiate exactly

Either of the ODEs. The one for $\hat r$ or the one for $\hat\theta$.