Physics, asked by BrainlyHelper, 1 year ago

Separating motion of a system of particles into motion of the centre of mass and motion about the centre of mass, show L = L' + R x MVwhere L' = Σri' + pi' is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember ri' = ri - R; rest of the notation is in the standard notation used in the chapter. Note L' and MR x V can be said to be angular momentum, respectively about and of the centre of mass of the system of particles.

Answers

Answered by abhi178
1
Let assume that , position vector of ith particle with respect to origin is r_i, position vector of ith particle with respect to centre of mass is r'_i and position vector of centre of mass with respect to origin is R.
It is given that, r'_i=r_i-R
so, r_i=r'_i+R

It is also given, that,
p_i=p'_i+m_iV

Taking cross product of this relation with r_i ,
r_i\times p_i=r_i\times p'_i+r'_i\times m_iV\\r_i\times p_i=(r'_i+R)\times p'_i+(r'_i+R)\times m_iV\\r_i\times p_i=r'_i\times p'_i+R\times p'_i+r'_i\times m_iV+R\times m_iV\\L=L'+R\times p'_i+r'_i\times m_iV+R\times m_iV

where, R\times p'_i=0
And r'_i m_iV=0

So, L=L'+R\times m_iV

In simply you can say that, L = L' + R × MV
Answered by MRSmartBoy
0

Answer:

Let assume that , position vector of ith particle with respect to origin is r_ir

i

, position vector of ith particle with respect to centre of mass is r'_ir

i

and position vector of centre of mass with respect to origin is R.

It is given that, r'_i=r_i-Rr

i

=r

i

−R

so, r_i=r'_i+Rr

i

=r

i

+R

It is also given, that,

p_i=p'_i+m_iVp

i

=p

i

+m

i

V

Taking cross product of this relation with r_ir

i

,

\begin{gathered}r_i\times p_i=r_i\times p'_i+r'_i\times m_iV\\r_i\times p_i=(r'_i+R)\times p'_i+(r'_i+R)\times m_iV\\r_i\times p_i=r'_i\times p'_i+R\times p'_i+r'_i\times m_iV+R\times m_iV\\L=L'+R\times p'_i+r'_i\times m_iV+R\times m_iV\end{gathered}

r

i

×p

i

=r

i

×p

i

+r

i

×m

i

V

r

i

×p

i

=(r

i

+R)×p

i

+(r

i

+R)×m

i

V

r

i

×p

i

=r

i

×p

i

+R×p

i

+r

i

×m

i

V+R×m

i

V

L=L

+R×p

i

+r

i

×m

i

V+R×m

i

V

where, R\times p'_i=0R×p

i

=0

And r'_i m_iV=0r

i

m

i

V=0

So, L=L'+R\times m_iVL=L

+R×m

i

V

In simply you can say that, L = L' + R × MV

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