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Question:

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the x-yplane with centre at O and constant angular speed ω. If the angular momentum of the system, calculated about O and P are denoted by →LO and →LP, respectively, then:

→LO and →LP do not vary with time.

→LO varies with time while →LP remains constant.

→LO remains constant while →LP varies with time.

→LO and →LP both vary with time.

Solution:

→LP=m(→PA×→v)
→LP=m([→PO+→OA]×→v)
→LP=m→PO×→v+m→OA×→v
→LP=m→PO×→v+m|OA|v^k
Also, →LO=m(→OA×→v)=m|OA|v^k
Thus →LO=constant
But →PO×→v will continuously change its direction. Thus →LP will change its direction at every instant and thus varies. But |→LP|=constant