1989 Paper 3 Q11

Year: 1989
Paper: 3
Question Number: 11

Course: UFM Mechanics
Section: Momentum and Collisions 2

Difficulty: 1700.0 Banger: 1499.3
momentum and collisions coefficient of restitution oblique impact spheres string constraint rigid rod constraint Newton's experimental law impulse along line of centres geometry suspended sphere

Problem

A smooth uniform sphere, with centre \(A\), radius \(2a\) and mass \(3m,\) is suspended from a fixed point \(O\) by means of a light inextensible string, of length \(3a,\) attached to its surface at \(C\). A second smooth unifom sphere, with centre \(B,\) radius \(3a\) and mass \(25m,\) is held with its surface touching \(O\) and with \(OB\) horizontal. The second sphere is released from rest, falls and strikes the first sphere. The coefficient of restitution between the spheres is \(3/4.\) Find the speed \(U\) of \(A\) immediately after the impact in terms of the speed \(V\) of \(B\) immediately before impact. The same system is now set up with a light rigid rod replacing the string and rigidly attached to the sphere so that \(OCA\) is a straight line. The rod can turn freely about \(O\). The sphere with centre \(B\) is dropped as before. Show that the speed of \(A\) immediately after impact is \(125U/127.\)

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1499.3

Banger Comparisons: 2

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Problem source
A smooth uniform sphere, with centre $A$, radius $2a$ and mass $3m,$ is suspended from a fixed point $O$ by means of a light inextensible string, of length $3a,$ attached to its surface at $C$. A second smooth unifom sphere, with centre $B,$ radius $3a$ and mass $25m,$ is held with its surface touching $O$ and with $OB$ horizontal.
The second sphere is released from rest, falls and strikes the first sphere. The coefficient of restitution between the spheres is $3/4.$
Find the speed $U$ of $A$ immediately after the impact in terms of the speed $V$ of $B$ immediately before impact. 
The same system is now set up with a light rigid rod replacing the string and rigidly attached to the sphere so that $OCA$ is a straight line. The rod can turn freely about $O$. The sphere with centre $B$ is dropped as before. Show that the speed of $A$ immediately after impact is $125U/127.$
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