Centre of Mass & Rotational Motion · JEE Advanced previous year questions

1. A.2
2. B.$\sqrt{2}$
3. C.$\sqrt{\frac{5}{2}}$
4. D.$\sqrt{\frac{2}{5}}$

A conducting solid sphere of radius $R$ and mass $M$ carries a charge $Q$. The sphere is rotating about an axis passing through its center with a uniform angular speed $\omega$. The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as $\alpha \frac{Q}{2M}$. The value of $\alpha$ is ____

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r)=-kr$, corresponding to the potential energy $V(r)=k{r}^2/2$, where $k$ is a positive force constant and $r$ is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by $L=n\hslash$, where $\hslash =h/(2\pi )$, $h$ is the Planck's constant, and $n$ a positive integer. If $v$ and $E$ are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

1. A.$r^2=n\hslash \sqrt{\frac{1}{mk}}$
2. B.$v^2=n\hslash \sqrt{\frac{k}{m^3}}$
3. C.$\frac{L}{m{r}^2}=\sqrt{\frac{k}{m}}$
4. D.$E=\frac{n\hslash }{2}\sqrt{\frac{k}{m}}$

A bar of mass $M=1.00\mathrm{kg}$ and length $L=0.20\mathrm{m}$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m=0.10\mathrm{kg}$ is moving on the same horizontal surface with $5.00\mathrm{m}{\mathrm{s}}^{-1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L/2$ from the pivoted end and returns back on the same path with speed v. After this elastic collision, the bar rotates with an angular velocity $\omega$. Which of the following statement is correct?

1. A.$\omega =6.98\mathrm{rad}{\mathrm{s}}^{-1}$ and $\mathrm{v}=4.30\mathrm{m}{\mathrm{s}}^{-1}$
2. B.$\omega =3.75\mathrm{rad}{\mathrm{s}}^{-1}$ and $\mathrm{v}=4.30\mathrm{m}{\mathrm{s}}^{-1}$
3. C.$\omega =3.75\mathrm{rad}{\mathrm{s}}^{-1}$ and $\mathrm{v}=10.0\mathrm{m}{\mathrm{s}}^{-1}$
4. D.$\omega =6.80\mathrm{rad}{\mathrm{s}}^{-1}$ and $\mathrm{v}=4.10\mathrm{m}{\mathrm{s}}^{-1}$

1. A.For $\mu \ne 0$ and $a\to 0$, $h_m=b/2$.
2. B.For $\mu \ne 0$ and $a\to b$, $h_m=b$.
3. C.For $h=h_m$, the initial angular velocity does not depend on the inner radius $a$.
4. D.For $\mu =0$ and $h=0$, the wheel always slides without rolling.