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The moment of inertia of a uniform sphere of radius, r about an axis passing through its centre is given by $$\frac{2}{5}\left( {\frac{{4\pi }}{3}{r^5}\rho } \right).$$ A rigid sphere of uniform mass density $$\rho $$ and radius R has two smaller spheres of radii $$\frac{R}{2}$$ hollowed out of it as shown in the figure given below.<br><img src="/images/question-image/engineering-physics/classical-mechanics/1689403074-the-moment-of-inertia-of-a.png" title="Classical Mechanics mcq question image" alt="Classical Mechanics mcq question image"><br>The moment of inertia of the resulting body about Y-axis is
A
$$\frac{{\pi \rho {R^5}}}{4}$$
B
$$\frac{{5\pi \rho {R^5}}}{{12}}$$
C
$$\frac{{7\pi \rho {R^5}}}{{12}}$$
D
$$\frac{{3\pi \rho {R^5}}}{4}$$
Correct Answer:
$$\frac{{5\pi \rho {R^5}}}{{12}}$$
A particle of mass m is attached to a thin uniform rod of length a and mass 4m. The distance of the particle from the centre of mass of the rod is $$\frac{a}{2}.$$
The moment of inertia of the combination about an axis passing through a normal to the rod is
A
$$\frac{{64}}{{48}}m{a^2}$$
B
$$\frac{{91}}{{48}}m{a^2}$$
C
$$\frac{{27}}{{48}}m{a^2}$$
D
$$\frac{{51}}{{48}}m{a^2}$$
A body A of mass 6.6 kg which is lying on a horizontal platform 4.5 m from its edge is connected to the end of a light string whose other end is supporting a body of mass 3.2 kg as shown in below figure. If the friction between the platform and the body A is $$\frac{1}{3}$$, the acceleration is
A
0.5 m/sec<sup>2</sup>
B
0.75 m/sec<sup>2</sup>
C
1.00 m/sec<sup>2</sup>
D
1.25 m/sec<sup>2</sup>
A rod of length L with uniform charge density $$\lambda $$ per unit length is in the XY-plane and rotating about Z-axis passing through one of its edge with an angularvelocity $$\overrightarrow \omega $$ as shown in the figure below. $$\left( {{\bf{\hat r}},\,\hat \phi ,\,{\bf{\hat z}}} \right)$$ refer to the unit vectors at Q, $$\overrightarrow {\bf{A}} $$ is the vector potential at a distance d from the origin O along Z-axis for d ≪ L and $$\overrightarrow {\bf{J}} $$ is the current density due to the motion of the rod. Which one of the following statements is correct?
A
$$\overrightarrow {\bf{J}} {\text{ along }}{\bf{\hat r}};\overrightarrow {\bf{A}} {\text{ along }}{\bf{\hat z}};\left| {\overrightarrow {\bf{A}} } \right| \propto \frac{1}{d}$$
B
$$\overrightarrow {\bf{J}} {\text{ along }}\hat \phi ;\overrightarrow {\bf{A}} {\text{ along }}\hat \phi ;\left| {\overrightarrow {\bf{A}} } \right| \propto \frac{1}{{{d^2}}}$$
C
$$\overrightarrow {\bf{J}} {\text{ along }}{\bf{\hat r}};\overrightarrow {\bf{A}} {\text{ along }}{\bf{\hat z}};\left| {\overrightarrow {\bf{A}} } \right| \propto \frac{1}{{{d^2}}}$$
D
$$\overrightarrow {\bf{J}} {\text{ along }}\hat \phi ;\overrightarrow {\bf{A}} {\text{ along }}\hat \phi ;\left| {\overrightarrow {\bf{A}} } \right| \propto \frac{1}{d}$$
A mass m is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius r
0
and angular speed ω
0
by an applied force $$\overrightarrow {\bf{F}} $$ communicated through an inextensible thread that passesthrough a hole on the surface as shown in figure given below. Then, this force is suddenly doubled.
The magnitude of the radial velocity of the mass
A
increases till mass falls into hole
B
decreases till mass falls into hole
C
remains constant
D
becomes zero at radius r<sub>1</sub>, where 0 1 0
Moment of inertia of a hollow circular section, as shown in the below figure about an axis perpendicular to the section, is ________ than that about X-X axis.
A
Two times
B
Same
C
Half
D
None of these
Two solid spheres of radius R and mass M each are connected by a thin rigid rod of negligible mass. The distance between the centre is 4R. The moment of inertia about an axis passing through the centre of symmetry and perpendicular to the line joining the sphere is
A
$$\frac{{11}}{5}M{R^2}$$
B
$$\frac{{22}}{5}M{R^2}$$
C
$$\frac{{44}}{5}M{R^2}$$
D
$$\frac{{88}}{5}M{R^2}$$
The moment of inertia of the shaded portion of the area shown in below figure about the X-axis, is
A
229.34 cm<sup>4</sup>
B
329.34 cm<sup>4</sup>
C
429.34 cm<sup>4</sup>
D
529.34 cm<sup>4</sup>
Moment of inertia of a hollow circular section, as shown in the below figure about X-axis, is
A
$$\frac{\pi }{{16}}\left( {{{\text{D}}^2} - {{\text{d}}^2}} \right)$$
B
$$\frac{\pi }{{16}}\left( {{{\text{D}}^3} - {{\text{d}}^3}} \right)$$
C
$$\frac{\pi }{{32}}\left( {{{\text{D}}^4} - {{\text{d}}^4}} \right)$$
D
$$\frac{\pi }{{64}}\left( {{{\text{D}}^4} - {{\text{d}}^4}} \right)$$
Moment of inertia of a hollow rectangular section as shown in the below figure about X-X axis, is
A
$$\frac{{{\text{B}}{{\text{D}}^3}}}{{12}} - \frac{{{\text{b}}{{\text{d}}^3}}}{{12}}$$
B
$$\frac{{{\text{D}}{{\text{B}}^3}}}{{12}} - \frac{{{\text{d}}{{\text{b}}^3}}}{{12}}$$
C
$$\frac{{{\text{B}}{{\text{D}}^3}}}{{36}} - \frac{{{\text{b}}{{\text{d}}^3}}}{{36}}$$
D
$$\frac{{{\text{D}}{{\text{B}}^3}}}{{36}} - \frac{{{\text{d}}{{\text{b}}^3}}}{{36}}$$
A block of mass m
1
, placed on an inclined smooth plane is connected by a light string passing over a smooth pulley to mass m
2
, which moves vertically downwards as shown in the below figure. The tension in the string is
A
$$\frac{{{{\text{m}}_1}}}{{{{\text{m}}_2}}}$$
B
$${{\text{m}}_1}{\text{g}}\sin \alpha $$
C
$$\frac{{{{\text{m}}_1}{{\text{m}}_2}}}{{{{\text{m}}_1} + {{\text{m}}_2}}}$$
D
$$\frac{{{{\text{m}}_1}{{\text{m}}_2}{\text{g}}\left( {1 + \sin \alpha } \right)}}{{{{\text{m}}_1} + {{\text{m}}_2}}}$$
