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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
12
MCQ (Multiple Correct Answer)

Figure shows a lamina in $$x y$$-plane. Two axes $$z$$ and $$z^{\prime}$$ pass perpendicular to its plane. A force $$\mathbf{F}$$ acts in the plane of lamina at point $$P$$ as shown. Which of the following are true? (The point $$P$$ is closer to $$z^{\prime}$$-axis than the $$z$$-axis.)

A
Torque $$\tau$$ caused by $$\mathbf{F}$$ about $$z$$-axis is along $$-\hat{\mathbf{k}}$$
B
Torque $$\tau^{\prime}$$ caused by $$\mathbf{F}$$ about $$z^{\prime}$$-axis is along $$-\hat{\mathbf{k}}$$
C
Torque $$\tau$$ caused by $$\mathbf{F}$$ about $$z$$-axis is greater in magnitude than that about $$z$$-axis
D
Total torque is given be $$\tau=\tau+\tau^{\prime}$$
13
MCQ (Multiple Correct Answer)

With reference to figure of a cube of edge $$a$$ and mass $$m$$, state whether the following are true or false. (0 is the centre of the cube.)

A
The moment of inertia of cube about z-axis is $$I_{\mathrm{z}}=I_x+I_{\mathrm{y}}$$
B
The moment of inertia of cube about $$z^{\prime}$$-axis is $$I_z=I_z+\frac{m a^2}{2}$$
C
The moment of inertia of cube about $$z^{\prime \prime}$$-axis is $$=I_z+\frac{m a^2}{2}$$
D
$$I_x=I_y$$
14
Subjective

The centre of gravity of a body on the earth coincides with its centre of mass for a small object whereas for an extended object it may not. What is the qualitative meaning of small and extended in this regard? For which of the following two coincides? A building, a pond, a lake, a mountain?

Explanation

When the vertical height of the object is very small as compared to the earth's radius, we call the object small, otherwise it is extended.

(i) Building and pond are small objects.

(ii) A deep lake and a mountain are examples of extended objects.

15
Subjective

Why does a solid sphere have smaller moment of inertia than a hollow cylinder of same mass and radius, about an axis passing through their axes of symmetry?

Explanation

The moment of inertia of a body is given by $$I=\Sigma m_i r_i^2$$ [sum of moment of inertia of each constituent particles]

All the mass in a cylinder lies at distance $$R$$ from the axis of symmetry but most of the mass of a solid sphere lies at a smaller distance than $$R$$.

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