This
is in fact an expression of Hooke's Law – Force is proportional to
extension.
A Level Maths Notes: M3 – Simple Harmonic Motion for a Horizontally Stretched String (1)
The tension in a stretched string is usually expressed in terms of
it's modulus of elasticityThis
is in fact an expression of Hooke's Law – Force is proportional to
extension.
Suppose that a string of length l is stretched between two points A and B. A particle P is attached to the string at C between the points A and B.
Suppose the lengths AC and CB are in the ratio
then
the natural length of AC is
and the extension of AC is
Similarly
the natural length of CB is
and
the extension of CB is
Suppose the particle P is pulled an extra distance x to the right.
The extension of AC is now
so
Similarly
It will experience a net force to the left, towards the equilibrium position C of magnitude
Now apply
with
which
has the form of simple harmonic motion
with