2001 Paper 3 Q11

Year: 2001
Paper: 3
Question Number: 11

Course: zNo longer examinable
Section: Moments of inertia

Difficulty: 1700.0 Banger: 1500.0

moments of inertia rotational dynamics energy conservation cylinder hanging mass angular velocity trigonometric equation optimisation

Problem

A uniform cylinder of radius \(a\) rotates freely about its axis, which is fixed and horizontal. The moment of inertia of the cylinder about its axis is \(I\,\). A light string is wrapped around the cylinder and supports a mass \(m\) which hangs freely. A particle of mass \(M\) is fixed to the surface of the cylinder. The system is held at rest with the particle vertically below the axis of the cylinder, and then released. Find, in terms of \(I\), \(a\), \(M\), \(m\), \(g\) and \(\theta\), the angular velocity of the cylinder when it has rotated through angle \(\theta\,\). Show that the cylinder will rotate without coming to a halt if \(m/M>\sin\alpha\,\), where \(\alpha\) satisifes \(\alpha=\tan \frac12\alpha\) and \(0<\alpha<\pi\,\).

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1500.0

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Problem source

A uniform cylinder of radius $a$ rotates freely about its axis,
which is fixed and horizontal. The moment of inertia of the cylinder about 
its axis is $I\,$. A light string is wrapped around the
cylinder and supports a mass $m$ which hangs freely. A particle of mass
$M$ is fixed to the surface of the cylinder. The system is held at
rest with the particle  vertically below the axis of the cylinder, and
then released. Find, in terms of $I$, $a$, $M$, $m$,  $g$ and $\theta$, 
the angular velocity of
the cylinder when it has rotated through angle $\theta\,$.

Show that the cylinder will 
 rotate without coming to a halt  if
$m/M>\sin\alpha\,$, where $\alpha$ satisifes $\alpha=\tan \frac12\alpha$ and 
$0<\alpha<\pi\,$.

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