Question:
The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $?$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $?/2$. The ring and disc are separated by frictionless ball bearings. The system is in the x-z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of 30$^{\circ}$ with the horizontal. Then with respect to the horizontal surface,
The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $?$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $?/2$. The ring and disc are separated by frictionless ball bearings. The system is in the x-z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of 30$^{\circ}$ with the horizontal. Then with respect to the horizontal surface,
Updated On: Aug 15, 2022
- the point O has a linear velocity $3R?\widehat {i}$
- the point P has a linear velocity $\frac{11}{4}R?\widehat {i}+\frac{\sqrt3}{4}R?\widehat {k}$
- the point P has a linear velocity $\frac{13}{4}R?\widehat {i}-\frac{\sqrt3}{4}R?\widehat {k}$
- the point P has a linear velocity $\bigg(3-\frac{\sqrt3}{4}\bigg)R?\widehat {i}+\frac{1}{4}R?\widehat {k}$
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The Correct Option is B
Solution and Explanation
Velocity of point O is
$v_0=(3r?)\widehat {i}$
$v_{PO}$ is $\frac{R.?}{2}$ in the direction shown in figure.
In vector form
$V_{PO}=-\frac{R?}{4}sin 30^{\circ}\widehat {i}+\frac{R?}{2}cos 30^{\circ}\widehat {k}$
$=-\frac{R?}{4}\widehat {i}+\frac{\sqrt3R?}{4}\widehat {k}$
But $V_{PO}=v_P-v_O$
$\therefore V_P=v_{PO}-v_O$
$=\bigg(-\frac{R?}{4}\widehat {i}+\frac{\sqrt3R?}{4}\widehat {k}\bigg)+3R?\widehat {i}$
$=\frac{11}{4}R?\widehat {i}+\frac{\sqrt3}{4}R?\widehat {k}$
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Concepts Used:
System of Particles and Rotational Motion
- The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
- The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
- The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.