Roatation

Q.

A boy is standing on a platform which is free to rotate about its axis. The boy holds an open umbrella in his hand. The axis of the umbrella coincides with that of the platform. The moment of inertia of "the platform plus the boy system" is 3.0 $\times {10}^{-3}{\text{kg-m}}^2$ and that of the umbrella is 2.0 $\times {10}^{-3}{\text{kg-m}}^2$. The boy starts spinning the umbrella about the axis at an angular speed of 2.0 rev/s with respect to himself. Find the angular velocity imparted to the platform

Q. A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $xy-$plane) with its axis fixed along the $y-$axis and only free to rotate about its axis. The platform is given a motion in $x-$direction given by $x=A\cos \left(\omega t\right)$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is

1. $MRA{\omega }^2$
2. $\frac{1}{2}MRA{\omega }^2$
3. $2MRA{\omega }^2$
4. $MRA{\omega }^2\times \cos \omega t$

Q. A cube of side $a$ is placed on an inclined plane of inclination $\theta$. What is the maximum value of $\theta$ in degrees for which the cube will not topple?

Q. A solid cylinder of mass M and radius R is resting on a horizontal platform (which is parallel to the x-y plane) with its axis fixed along Y-axis and free to rotate about its axis. The platform is given a motion in the X-direction given by $x=Acos\left(\omega t\right)$. There is no slipping between the cylinder and the platform. The maximum torque acting on the cylinder during its motion is:

1. $\frac{M{\omega }^{2}AR}{3}$
2. $\frac{M{\omega }^{2}AR}{2}$
3. $\frac{2}{3}\times M{\omega }^{2}AR$
4. The situation is not possible.

Q. A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $xy-$plane) with its axis fixed along the $y-$axis and only free to rotate about its axis. The platform is given a motion in $x-$direction given by $x=A\cos \left(\omega t\right)$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is

1. $\frac{1}{2}MRA{\omega }^2$
2. $MRA{\omega }^2$
3. $2MRA{\omega }^2$
4. $MRA{\omega }^2\times \cos \omega t$

Q. A particle of mass $9\times {10}^{-31}$kg having a negative charge of $1.6\times {10}^{-19}$C is projected horizontally with a velocity of ${10}^{6}m{s}^{-1}$ into a region between two infinite horizontal parallel plates of metal. The distance between the plates is $d=03$cm, and the particle enters 0.1 cm below the top plate. The top and bottom plates are connected, respectively, to the positive and negative terminals of a 30 V battery. Find the components of the velocity of the particle just before it hits one of the plates

Q. A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $xy-$plane) with its axis fixed along the $y-$axis and only free to rotate about its axis. The platform is given a motion in $x-$direction given by $x=A\cos \left(\omega t\right)$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is

1. $\frac{1}{2}MRA{\omega }^2$
2. $MRA{\omega }^2$
3. $2MRA{\omega }^2$
4. $MRA{\omega }^2\times \cos \omega t$

Q. A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $xy-$plane) with its axis fixed along the $y-$axis and only free to rotate about its axis. The platform is given a motion in $x-$direction given by $x=A\cos \left(\omega t\right)$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is

1. $\frac{1}{2}MRA{\omega }^2$
2. $MRA{\omega }^2$
3. $2MRA{\omega }^2$
4. $MRA{\omega }^2\times \cos \omega t$