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 \(g=10~( \text {m/s}^2)\)
1. | \({\dfrac 1 6}~\text{kg}\) | 2. | \({\dfrac 1 {12}}~ \text{kg}\) |
3. | \({\dfrac 1 2}~ \text{kg}\) | 4. | \({\dfrac 1 3}~ \text{kg}\) |
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?
a. | angular momentum |
b. | linear momentum |
c. | kinetic energy |
d. | moment of inertia |
1. | (a) and (b) |
2. | (b) and (c) |
3. | (c) and (d) |
4. | (a) and (d) |
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:
1. | \(\dfrac{u}{L}\) | 2. | \(\dfrac{2u}{L}\) |
3. | \(\dfrac{12u}{7L}\) | 4. | \(\dfrac{6u}{L}\) |
(a) | for a general rotational motion, angular momentum \(L\) and angular velocity \(\omega\) need not to be parallel. |
(b) | for a rotational motion about a fixed axis, angular momentum \(L\) and angular velocity \(\omega\) are always parallel. |
(c) | for a general translational motion, momentum \(p\) and velocity \(v\) are always parallel. |
(d) | for a general translational motion, acceleration \(a\) and velocity \(v\) are always parallel. |
1. | (a), (c) | 2. | (b), (c) |
3. | (c), (d) | 4. | (a), (b), (c) |
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. \(D\)
1. | \({\dfrac b 2}\) | 2. | \({ \dfrac b 4}\) |
3. | \({\dfrac b 8}\) | 4. | \(\dfrac b 3\) |
1. | \(\omega_0\) | 2. | \(2\omega_0\) |
3. | \(\dfrac32\omega_0\) | 4. | \(\dfrac52\omega_0\) |