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Chapter 6= Problem 17 & 18
Chapter 9= problem 3 only
6.17 A smooth wire has the form of the helix x - a cosé, y = a sine, : = be, where 6 is
a real parameter, and de b are positive constants. The wire is fixed with the axis Or
pointing
vertically upwards. A particle P. which can slide freely on the wire, is released from rest at the
point (a, 0, 2mb). Find the speed of P when it reaches the point (a,0,0) and the time taken
for il to do so.
6.18 A smooth wire has the form of the parabola : = x /2b, y = 0. where b is a positive
constant. The wire is fixed with the axis Oz pointing vertically upwards. A particle P. which
can slide freely on the wire, is performing oscillations with x in the range - sisa Show
that the period T of these oscillations is given by
b² + x 2
1/2
dx.
T=
(gb)
a? - x 2
By the above integral, obtain a new formula small. for T. Use
this formula to tind a two-term approximation to T. valid when the ratio a/b is
making the substitution x = a sin * in
9.3 The internal potential energy function for a diatomic molecule is approximated by the
Morse potential
V(r) = Vo 2 - Vo.
where / is the distance of separation of the two atoms, and Vo. a, b are positive constants.
lake 3 sketch of the Morse potential.
Suppose the molecule is restricted to vibrational motion in which the centre of mass G of
the molecule is fixed, and the atoms move on a fixed straight line through G. Show that there
is a single equilibrium configuration for the molecule and that it is stable. If the atoms each
have mass m, find the angular frequency of small vibrational oscillations of the molecule.

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