A merry-go-round, made of a ring-like platform of radius \(R\) and mass \(M,\) is revolving with the angular speed . A person of mass \(M\) is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round afterwards is:
1. \(\omega\)
2. \(2\omega\)
3. \(\omega/2\)
4. \(0\)
1. \(\omega\)
2. \(2\omega\)
3. \(\omega/2\)
4. \(0\)
| (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. |
Choose the correct option:
| 1. | (a), (c) | 2. | (b), (c) |
| 3. | (c), (d) | 4. | (a), (b), (c) |
Consider the following statements.
| (a) | angular momentum \(l_1\) of particle \(1\) about \(A\) is \(l_1=mv(d_1)\) ⊙ |
| (b) | angular momentum \(l_1\) of particle \(2\) about \(A\) is \(l_1=mv(r_2)\) ⊙ |
| (c) | total angular momentum of the system about \(A\) is \(l=mv(r_1+r_2)\) ⊙ |
| (d) | total angular momentum of the system about \(A\) is \(l=mv(d_2-d_1)\) ⊗ |
Choose the correct option from the given ones:
| 1. | (a), (c) only |
| 2. | (a), (d) only |
| 3. | (b), (d) only |
| 4. | (b), (c) only |
The figure shows a lamina in \(\text{XY}\)-plane. Two axes \(\text{z}\) and \(z'\) pass perpendicular to its plane. A force \(\vec{F}\) acts in the plane of the lamina at point P as shown. (The point \(\text{p}\) is closer to the \(z'\)-axis than the \(\text{z}\)-axis.)
| (a) | torque \(\vec{\tau}\) caused by \(\vec{F}\) about \(\text{z}\)-axis is along - \(\hat{k}\) |
| (b) | torque \(\vec{\tau}'\) caused by \(\vec{F}\) about \(z'\)-axis is along - \(\hat{k}\) |
| (c) | torque caused by \(\vec{F}\) about the \(\text{z}\)-axis is greater in magnitude than that about the z'-axis |
| (d) | total torque is given by \(\vec{\tau}_{net}=\vec{\tau}+\vec{\tau}'\) |
Choose the correct option:
1. (c, d)
2. (a, c)
3. (b, c)
4. (a, b)
The centre of mass of an extended body on the surface of the earth and its centre of gravity:
| (a) | are always at the same point for any size of the body. |
| (b) | are always at the same point only for spherical bodies. |
| (c) | can never be at the same point. |
| (d) | is close to each other for objects, say of sizes less than 100 m. |
| (e) | both can change if the object is taken deep inside the earth. |
Choose the correct alternatives:
2. (a), (d)
3. (d) only
4. (c), (d)
A meter scale is moving with uniform velocity. This implies:
| 1. | the force acting on the scale is zero, but a torque about the centre of mass can act on the scale. |
| 2. | the force acting on the scale is zero and the torque acting about the centre of mass of the scale is also zero. |
| 3. | the total force acting on it need not zero but the torque on it is zero. |
| 4. | neither the force nor the torque needs to be zero. |
The density of a non-uniform rod of length 1m is given by \(\rho ( x) = a \left( 1 + bx^{2} \right)\) where, \(a\), and \(b\) are constants and \(0 \leq x \leq 1\). The centre of mass of the rod will be at:
| 1. | \(\dfrac{3(2+b)}{4(3+b)}\) | 2. | \(\dfrac{4(2+b)}{3(3+b)}\) |
| 3. | \(\dfrac{3(3+b)}{4(2+b)}\) | 4. | \(\dfrac{4(3+b)}{3(2+b)}\) |
A uniform square plate has a small piece \(Q\) of an irregular shape removed and glued to the center of the plate leaving a hole behind in the figure. The \(COM\) of the plate is now in the following quadrant of the \(x\text-y\) plane.
1. \(\text{I}\)
2. \(\text{II}\)
3. \(\text{III}\)
4. \(\text{IV}\)
A uniform square plate has a small piece \(Q\) of an irregular shape removed and glued to the center of the plate leaving a hole behind in the figure. The moment of inertia about the \(\mathrm{z}\)-axis is then
| 1. | increased |
| 2. | decreased |
| 3. | the same |
| 4. | changed in unpredicted manner |
When a disc rotates with uniform angular velocity, which of the following is not true?
| 1. | the sense of rotation remains the same. |
| 2. | the orientation of the axis of rotation remains the same. |
| 3. | the speed of rotation is non-zero and remains the same. |
| 4. | the angular acceleration is non-zero and remains the same. |
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