User:Egm6341.s2010.Team1.lei/HW6

For HW6,I am responsible for problems 2 and 6.

=Problem 2-Consider v+dv,show lin.mom.compo. =

Find
At $$ t+dt\,$$,consider $$ v+dv\,$$, show $$ dp_{\bar{y}}=mvd\gamma\,$$ by neglecting hot.

Given
Refer Lecture slide (1) [[media:Egm6341.s10.mtg33.djvu|p.33-2]] for problem statement



[Figure 1]

$$ dp_{\bar{y}}=mvd\gamma\,$$

Solution


[Figure 2]

From Fig.2,it can be seen that

$$ dp_{\bar{y}}=m(v+dv)d\gamma\,$$

$$ =mvd\gamma+ \cancelto{}{mdvd\gamma}\,$$

By neglecting higher order term$$ dvd\gamma\,$$,just retention the lowest term

It can be derived that

$$ dp_{\bar{y}}=mvd\gamma\,$$

Egm6341.s10.team1.lei20:00, 3 April 2010 (UTC)

=Problem 6- Find s=s(t) =

Find
Find $$ s=s(t)\,$$ such that $$ Z(t)=\sum_{i=1}^{4}N_{i}(t)d_{i}\,$$

Given
(1) [[media:Egm6341.s10.mtg35.djvu|p.35-1]]

Local coord. for $$ [t_{i},t_{i+1}]:\,$$

$$ t(s)=(1-s)t_{i}+st_{i+1}\, $$

Solution
Convert

$$ t(s)=(1-s)t_{i}+st_{i+1}\, $$

We can get

$$ s=\frac{t-t_{i}}{t_{i+1}-t_{i}}\,$$

thus

$$ s=s(t)=\frac{t-t_{i}}{t_{i+1}-t_{i}}\,$$

plug$$ s=s(t)\,$$ into

$$ Z(s)=\sum_{i=1}^{4}N_{i}(s)d_{i}\,$$

$$ \Rightarrow Z(T)=\sum_{i=1}^{4}N_{i}(T)d_{i}\,$$

Egm6341.s10.team1.lei20:01, 3 April 2010 (UTC)