question_answer

Figure shows a student, sitting on a stool that can rotate freely about a vertical axis. The Student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose moment of inertia is I about its central axis. The wheel is rotating at an angular speed en; from an overhead perspective, the rotation is counter clockwise. The axis of the wheel point?s vertical, and the angular momentum l, of the wheel points vertically upward. The student now inverts the wheel; as a result, the student and stool rotate about the stool axis. With what angular speed and direction does the student then rotate? (The moment of inertia of the student + stool + wheel system about the stool axis is in).

                         

A) \[\frac{2{{\overrightarrow{L}}_{i}}}{{{I}_{0}}},\,\]Counter clockwise

B) \[\frac{2{{\overrightarrow{L}}_{i}}}{{{I}_{0}}},\,\]Clockwise

C) \[\frac{{{\overrightarrow{L}}_{i}}}{{{I}_{0}}},\,\]Counter clockwise

D) \[\frac{{{\overrightarrow{L}}_{i}}}{{{I}_{0}}},\,\]Clockwise

Correct Answer: A

Solution :

There is not torque acting on the system (Student+ stool+ wheel), so the angular momentum of the system about vertical axis remains constant. Let \[\Delta {{\overrightarrow{L}}_{1}}\] and  \[\Delta {{\overrightarrow{L}}_{2}}\] are the change in angular momentum of (student+ stool) and wheel respectively, then we have

\[\Delta {{\overrightarrow{L}}_{1}}+\Delta {{\overrightarrow{L}}_{2}}=0\]

\[\therefore \Delta {{\overrightarrow{L}}_{1}}=-\Delta {{\overrightarrow{L}}_{2}}\]

\[\begin{align}   & As\,\Delta {{\overrightarrow{L}}_{2}}=-\Delta {{\overrightarrow{L}}_{i}}-\Delta {{\overrightarrow{L}}_{i}} \\  & =-2{{\overrightarrow{L}}_{i}} \\ \end{align}\]

\[\therefore \Delta {{\overrightarrow{L}}_{1}}=2{{\overrightarrow{L}}_{i}}\]

As initial angular momentum of (student+ stool) is zero, so

\[2{{\overrightarrow{L}}_{i}}={{I}_{0}}\overrightarrow{\omega }\]

\[or\,\overrightarrow{\omega }=\frac{2\overrightarrow{{{L}_{i}}}}{{{I}_{0}}}\]

The positive result tells that the angular velocity of rotation is counterclockwise.