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 uniform rod of length \(200~ \text{cm}\) and mass \(500~ \text g\) is balanced on a wedge placed at \(40~ \text{cm}\) mark. A mass of \(2~\text{kg}\) is suspended from the rod at \(20~ \text{cm}\) and another unknown mass \(m\) is suspended from the rod at \(160~\text{cm}\) mark as shown in the figure. What would be the value of \(m\) such that the rod is in equilibrium?
(Take \(g=10~( \text {m/s}^2)\)

A vector \(\overrightarrow A\) points vertically upward and \(\overrightarrow B\) points towards north. The vector product \(\overrightarrow A\times\overrightarrow B\) is:

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 \(w=\dfrac{v}{r}\). Thus:
1. \(w\propto\dfrac{1}{r}\)
2. \(w\propto\ r\)
3. \(w=0\)
4. \(w\) is independent of \(r\)

If there is no external force acting on a non-rigid body which of the following quantities must remain constant?

Choose the correct option from the given ones:

A uniform rod of mass \(m\) and length \(L\) is struck at both ends by two particles of masses m, each moving with identical speeds \(u,\) but in opposite directions, perpendicular to its length. The particles stick to the rod after colliding with it. The system rotates with an angular speed:

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. \(D\)