Systems of Particles and Rotational Motion

PARAGRAPH -2One twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity ω
_{0}. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is μ and the acceleration due to gravity is g. |

The total kinetic energy of the ring is :-

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JEE Advanced 2017

PARAGRAPH -2One twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity ω
_{0}. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is μ and the acceleration due to gravity is g. |

The minimum value of ω

_{0}below which the ring will drop down is :-

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JEE Advanced 2017

PARAGRAPH 1Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed ω about its vertical axis through its centre. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis $\left(\overrightarrow{\mathit{\omega}}=\omega \hat{k}\right)$ . A small block of mass m is gently placed in the slot at $\overrightarrow{\mathit{r}}=\left(R/2\right)\hat{\mathit{i}}$ at t = 0 and is constrained to move only along the slot. |

The distance

*r*of the block at time

*t*is

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JEE Advanced 2016

PARAGRAPH 1m moving on the rotating disc and the force ${\overrightarrow{\mathit{F}}}_{\mathrm{in}}$ experienced by the particle in an inertial frame of reference is ${\overrightarrow{F}}_{\mathrm{rot}}={\overrightarrow{F}}_{\mathrm{in}}+2m\left({\overrightarrow{\nu}}_{\mathrm{rot}}\times \overrightarrow{\omega}\right)+m\left(\overrightarrow{\mathit{\omega}}\times \overrightarrow{\mathit{r}}\right)\times \overrightarrow{\mathit{\omega}}$, where ${\overrightarrow{\nu}}_{\mathrm{rot}}$ is the velocity of the particle in the rotating frame of reference and $\overrightarrow{\mathit{r}}$ is the position vector of the particle with respect to the centre of the disc.Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed ω about its vertical axis through its centre. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis $\left(\overrightarrow{\omega}=\omega \mathit{}\hat{\mathit{k}}\right)$ . A small block of mass m is gently placed in the slot at $\overrightarrow{r}=\left(R/2\right)\hat{\mathit{i}}$ at t = 0 and is constrained to move only along the slot. |

The net reaction of the disc on the block is

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JEE Advanced 2016

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AIPMT 2013