A mass m is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius r0 and angular speed ω0 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. <img src="/images/question-image/engineering-physics/classical-mechanics/1689400906-a-mass-m-is-constrained-to.png" title="Classical Mechanics mcq question image" alt="Classical Mechanics mcq question image"> The magnitude of the radial velocity of the mass

The Hamiltonian of a particle is given by $$H = \frac{{{p^2}}}{{2m}} + V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right) + \phi \left( { + \left| {\overrightarrow {\bf{r}} } \right|} \right)\overrightarrow {\bf{L}} .\overrightarrow {\bf{S}} ,$$ where $$\overrightarrow {\bf{S}} $$ is the spin, $$V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$ and $$\phi \left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$ are potential functions and $$\overrightarrow {\bf{L}} \left( { = \overrightarrow {\bf{r}} \times \overrightarrow {\bf{p}} } \right)$$ is the angular momentum. The Hamiltonian does not commute with

A

$$\overrightarrow {\bf{L}} + \overrightarrow {\bf{S}} $$

B

$$\overrightarrow {{{\bf{S}}^2}} $$

C

$${L_z}$$

D

$$\overrightarrow {{{\bf{L}}^2}} $$

A square hole is made in a circular lamina, the diagonal of the square is equal to the radius of the circle as shown in below figure the shift in the centre of gravity is

A

$$\frac{{{\text{r}}\left( {\pi - 0.75} \right)}}{{\left( {\pi - 0.5} \right)}}$$

B

$$\frac{{{\text{r}}\left( {\pi - 0.25} \right)}}{{\left( {\pi - 0.75} \right)}}$$

C

$$\frac{{{\text{r}}\left( {\pi - 0.5} \right)}}{{\left( {\pi - 0.75} \right)}}$$

D

$$\frac{{{\text{r}}\left( {\pi - 0.5} \right)}}{{\left( {\pi - 0.25} \right)}}$$

In a cubic system with cell edge a, two phonons with wave vectors $${\overrightarrow {\bf{q}} _1}$$ and $${\overrightarrow {\bf{q}} _2}$$ collide and produce a third phonon with a wave. vector $${\overrightarrow {\bf{q}} _3}$$ such that $${\overrightarrow {\bf{q}} _1} + {\overrightarrow {\bf{q}} _2} = {\overrightarrow {\bf{q}} _3} + \overrightarrow {\bf{R}} $$ where, $$\overrightarrow {\bf{R}} $$ is a lattice vector. Such a collision process will lead to(a)

A

finite thermal resistance

B

zero thermal resistance

C

an infinite thermal resistance

D

a finite thermal resistance for certain $$\left| {\overrightarrow {\bf{R}} } \right|$$ only

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/sec2

B

0.75 m/sec2

C

1.00 m/sec2

D

1.25 m/sec2

If for a system of N particles of different masses m₁, m₂, . . . m_N with position vectors $${\overrightarrow {\bf{r}} _1},\,{\overrightarrow {\bf{r}} _2},\,.\,.\,.\,{\overrightarrow {\bf{r}} _N}$$ and corresponding velocities $${\overrightarrow {\bf{v}} _1},\,{\overrightarrow {\bf{v}} _2},\,.\,.\,.\,{\overrightarrow {\bf{v}} _N}$$ respectively such that $$\sum\limits_i {\overrightarrow {{{\bf{v}}_i}} = 0,} $$ then

A

total momentum must be zero

B

total angular momentum must be independent of the choice of the origin

C

the total force on the system must be zero

D

the total torque on the system must be zero

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.

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

$$\frac{{3\pi \rho {R^5}}}{4}$$

The member which does not carry zero force in the structure shown in below figure, is

A

ED

B

DC

C

BC

D

BD

In the structure shown in below figure, the member which carries zero force, is

A

AB

B

BC

C

BE

D

All the above

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 circular hole of 50 mm diameter is cut out from a circular disc of 100 mm diameter as shown in the below figure. The center of gravity of the section will lie

A

In the shaded area

B

In the hole

C

At ‘O’

D

None of these