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7
MCQ (Single Correct Answer)

The density of a non-uniform rod of length 1 m is given by $$\rho(x)=a\left(1+b x^2\right)$$ where, a and b are constants and $$0 \leq x \leq 1$$. The centre of mass of the rod will be at

A
$$\frac{3(2+b)}{4(3+b)}$$
B
$$\frac{4(2+b)}{3(3+b)}$$
C
$$\frac{3(3+b)}{4(2+b)}$$
D
$$\frac{4(3+b)}{3(2+b)}$$
8
MCQ (Single Correct Answer)

A merry-go-round, made of a ring-like platform of radius $$R$$ and mass $$M$$, is revolving with angular speed $$\omega$$. 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 of afterwards is

A
$$2 \omega$$
B
$$\omega$$
C
$$\frac{\omega}{2}$$
D
0
9
MCQ (Multiple Correct Answer)

Choose the correct alternatives

A
For a general rotational motion, angular momentum $$\mathbf{L}$$ and angular velocity $$\omega$$ need not be parallel.
B
For a rotational motion about a fixed axis, angular momentum $$\mathbf{L}$$ and angular velocity $$\omega$$ are always parallel.
C
For a general translational motion, momentum $$\mathbf{p}$$ and velocity $$\mathbf{v}$$ are always parallel.
D
For a general translational motion, acceleration $$\mathbf{a}$$ and velocity $$\mathbf{v}$$ are always parallel.
10
MCQ (Multiple Correct Answer)

Figure shows two identical particles 1 and 2, each of mass $$m$$, moving in opposite directions with same speed $$\mathbf{v}$$ along parallel lines. At a particular instant $$\mathbf{r}_1$$ and $$\mathbf{r}_2$$ are their respective position vectors drawn from point $$A$$ which is in the plane of the parallel lines. Choose the correct options.

A
Angular momentum $$I_1$$ of particle 1 about $$A$$ is $$I=m v\left(d_1\right)$$
B
Angular momentum $$I_2$$ of particle 2 about $$A$$ is $$I_2=\mathrm{mvr}_2$$
C
Total angular momentum of the system about $$A$$ is $$I=m v\left(\mathbf{r}_1+\mathbf{r}_2\right)$$
D
Total angular momentum of the system about $$A$$ is $$I=m v\left(d_2-d_1\right) \otimes$$
11
MCQ (Multiple Correct Answer)

The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?

A
The forces may be acting radially from a point on the axis
B
The forces may be acting on the axis of rotation
C
The forces may be acting parallel to the axis of rotation
D
The torque caused by some forces may be equal and opposite to that caused by other forces
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