7.7 Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.

Let at certain instant two particles be at points P and Q, as shown in the figure.

Angular momentum of the system about point P

$\stackrel{?}{L_p}$ = mv x 0 + mv x d = mvd ……. (i)

Angular momentum of the system about point Q

$\stackrel{?}{L_q}$ = mv x d + mv x 0 = mvd ……. (ii)

Consider a point R, which is at a distance y from point Q such that QR = y

PR = d – y

Angular momentum of the system about point R

$\stackrel{?}{L_R}$  = mv x (d – y) + mv x y = mvd – mvy + mvy = mvd ……. (iii)

Comparing equations (i), (ii) and (iii) we get

$\stackrel{?}{L_p}$ = $\stackrel{?}{L_q}$ = $\stackrel{?}{L_R}$ …… (iv)

<p>Let at certain instant two particles be at points P and Q, as shown in the figure.</p><p>Angular momentum of the system about point P</p><p><span title="Click to copy mathml"><span contenteditable="false"> <math> <mover accent="true"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mi>p</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo>? </mo> </mover> </math> </span></span>= mv x 0 + mv x d = mvd ……. (i)</p><p>Angular momentum of the system about point Q</p><p><span title="Click to copy mathml"><span contenteditable="false"> <math> <mover accent="true"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mi>q</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo>? </mo> </mover> </math> </span></span>= mv x d + mv x 0 = mvd ……. (ii)</p><p>Consider a point R, which is at a distance y from point Q such that QR = y</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1729061062phpMjuLwQ_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1729061062phpMjuLwQ.jpeg" alt="" width="291" height="171"></picture></div></div><p>PR = d – y</p><p>Angular momentum of the system about point R</p><p><span title="Click to copy mathml"><span contenteditable="false"> <math> <mover accent="true"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo>? </mo> </mover> </math> </span></span>&nbsp;= mv x (d – y) + mv x y = mvd – mvy + mvy = mvd ……. (iii)</p><p>Comparing equations (i), (ii) and (iii) we get</p><p><span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><msub><mrow><mrow><mi>L</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub></mrow></mrow><mo>? </mo></mover></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><msub><mrow><mrow><mi>L</mi></mrow></mrow><mrow><mrow><mi>q</mi></mrow></mrow></msub></mrow></mrow><mo>? </mo></mover></math></span> =<span contenteditable="false"> <math> <mover accent="true"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo>? </mo> </mover> </math> </span>…… (iv)</p>

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