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.
Solution and Explanation
Let at a certain instant two particles be at points P and Q, as shown in the following figure.
Angular momentum of the system about point P:
\(\^L_p\) = mv × 0 + mv × d
\(= mvd ....(i)\)
Angular momentum of the system about point Q :
\(\^L_Q→ = mv × d + mv × 0 \)
\(= mvd ....(ii)\)
Consider a point R, which is at a distance y from point Q, i.e., QR = y
∴ PR = d - y
Angular momentum of the system about point R :
\(\^L_R = mv × (d - y) + mv × y \)
\(= mvd - mvy + mvy \)
\(= mvd ...(iii)\)
Comparing equations (i), (ii), and (iii), we get :
\(\^L_P = \^L_Q = \^L_R ...(iv)\)
We infer from equation (iv) that the angular momentum of a system does not depend on the point about which it is taken.
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Concepts Used:
Vectors
The quantities having magnitude as well as direction are known as Vectors or Vector quantities. Vectors are the objects which are found in accumulated form in vector spaces accompanying two types of operations. These operations within the vector space include the addition of two vectors and multiplication of the vector with a scalar quantity. These operations can alter the proportions and order of the vector but the result still remains in the vector space. It is often recognized by symbols such as U ,V, and W
Representation of a Vector :
A line having an arrowhead is known as a directed line. A segment of the directed line has both direction and magnitude. This segment of the directed line is known as a vector. It is represented by a or commonly as AB. In this line segment AB, A is the starting point and B is the terminal point of the line.
Types of Vectors:
Here we will be discussing different types of vectors. There are commonly 10 different types of vectors frequently used in maths. The 10 types of vectors are:
- Zero vector
- Unit Vector
- Position Vector
- Co-initial Vector
- Like and Unlike Vectors
- Coplanar Vector
- Collinear Vector
- Equal Vector
- Displacement Vector
- Negative of a Vector