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.
It is given that two particles have mass m , speed is v and the distance between them is d .
The following diagram shows positions of the two particles.
The expression for angular momentum of the system about point P ( L P ) is,
L P = m v × 0 + m v × d = m v d (1)
The expression for angular momentum of the system about point Q ( L Q ) is,
L Q = m v × d + m v × 0 = m v d (2)
Let the distance y from point Q which is at point R. then,
QR = y
Thus, from the figure
PR = d − y
The expression for angular momentum of the system about point R ( L R ) is,
L R = m v × ( d − y ) + m v × y = m v d − m v y + m v y = m v d (3)
From equations (1), (2) and (3),
L P = L Q = L R
Thus, the vector angular momentum of the given two particles will be same irrespective of the point taken.
It is given that two particles have mass
The following diagram shows positions of the two particles.
The expression for angular momentum of the system about point P
The expression for angular momentum of the system about point Q
Let the distance
Thus, from the figure
The expression for angular momentum of the system about point R
From equations (1), (2) and (3),
Thus, the vector angular momentum of the given two particles will be same irrespective of the point taken.
