Which of the following points is the likely position of the center of mass of the system shown in the figure?

1. \(A\)
2. \(B\)
3. \(C\)
4. \(D\)

In the \(\mathrm{HCl}\) molecule, the separation between the nuclei of the two atoms is about \(1.27~\mathring{\text A}~(1~\mathring{\text A}=10^{10}~\text m).\) Then the approximate location of the CM of the molecule is:
(Given that a chlorine atom is about \(35.5\) times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus).

For a body, with angular velocity \( \vec{\omega }=\hat{i}-2\hat{j}+3\hat{k}\)  and radius vector \( \vec{r }=\hat{i}+\hat{j}++\hat{k},\)  its velocity will be:
1. \(-5\hat{i}+2\hat{j}+3\hat{k}\)
2. \(-5\hat{i}+2\hat{j}-3\hat{k}\)
3. \(-5\hat{i}-2\hat{j}+3\hat{k}\)
4. \(-5\hat{i}-2\hat{j}-3\hat{k}\)

A body is in pure rotation. The linear speed \(v\) of a particle, the distance \(r\) of the particle from the axis and the angular velocity \(\omega\) of the body are related as \(w=\frac{v}{r}\). Thus:
1. \(w\propto\frac{1}{r}\)
2. \(w\propto\ r\)
3. \(w=0\)
4. \(w\) is independent of \(r\)

If \(\vec F\) is the force acting on a particle having position vector \(\vec r\) and \(\vec \tau\) be the torque of this force about the origin, then:

A particle of mass \(m\) moves in the XY plane with a velocity \(v\) along the straight line AB. If the angular momentum of the particle with respect to the origin \(O\) is \(L_A\) when it is at \(A\) and \(L_B\) when it is at \(B,\) then: