2004 Paper 3 Q9

Year: 2004
Paper: 3
Question Number: 9

Course: UFM Mechanics
Section: Circular Motion 2

Difficulty: 1700.0 Banger: 1455.8
circular motion rigid body equilibrium friction moments hoop particle on circular path torque balance

Problem

A circular hoop of radius \(a\) is free to rotate about a fixed horizontal axis passing through a point \(P\) on its circumference. The plane of the hoop is perpendicular to this axis. The hoop hangs in equilibrium with its centre, \(O\), vertically below \(P\). The point \(A\) on the hoop is vertically below \(O\), so that \(POA\) is a diameter of the hoop. A mouse \(M\) runs at constant speed \(u\) round the rough inner surface of the lower part of the hoop. Show that the mouse can choose its speed so that the hoop remains in equilibrium with diameter \(POA\) vertical. Describe what happens to the hoop when the mouse passes the point at which angle \(AOM = 2 \arctan \mu\,\), where \(\mu\) is the coefficient of friction between mouse and hoop.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1455.8

Banger Comparisons: 3

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Problem source
A circular hoop of radius $a$ is free to rotate about a fixed horizontal 
axis passing through a point $P$ on its circumference. The plane of the hoop
is perpendicular to this axis. 
The hoop hangs in equilibrium with its centre, $O$, vertically below $P$.
The point $A$ on the hoop is vertically below $O$, so that $POA$ is a diameter of the hoop. 

A mouse $M$ runs at constant speed $u$ round the 
rough inner surface of the lower part of the hoop. 
Show that the mouse can choose its speed so that the hoop 
remains in equilibrium with diameter $POA$ vertical.
Describe what happens to the hoop when the mouse passes the point at which angle 
$AOM =  2 \arctan \mu\,$, 
where $\mu$ is the coefficient of friction between mouse and hoop.
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