Rotational Motion Question 86
Question: A force $ \mathbf{F}=\alpha \mathbf{\hat{i}}+3\mathbf{\hat{j}}+6\mathbf{\hat{k}} $ is acting at a point $ \mathbf{r}=2\mathbf{\hat{i}}-6\mathbf{\hat{j}}-12\mathbf{\hat{k}} $ . The value of a for which angular momentum about origin is conserved is
Options:
A) - 1
B) 2
C) zero
D) 1
Show Answer
Answer:
Correct Answer: A
Solution:
-
Key Concept When the resultant external torque acting on a system is zero, the total angular momentum of a system remains constant.
This is the principle of the conservation of angular momentum.
Given, force $ \mathbf{F}=\alpha \mathbf{\hat{i}}+3\mathbf{\hat{j}}+6\mathbf{\hat{k}} $ is acting at a point $ \mathbf{r}=2\mathbf{\hat{i}}-6\mathbf{\hat{j}}-12\mathbf{\hat{k}} $
As, angular momentum about origin is conserved.
i.e.$ \tau = $ constant
Torque, $ \tau =0\Rightarrow \mathbf{r}\times \mathbf{F}=0 $
$ \begin{vmatrix} {\mathbf{\hat{i}}} & {\mathbf{\hat{j}}} & {\mathbf{\hat{k}}} \\ 2 & -6 & -12 \\ \alpha & 3 & 6 \\ \end{vmatrix} =0 $
$ \Rightarrow $ $ (-36+36)\mathbf{\hat{i}}-(12+12\alpha )\mathbf{\hat{j}}+(6+6\alpha )\mathbf{\hat{k}}=0 $
So value of a for angular momentum about origin is conserved, $ \alpha =-1 $ .
