Environmental Engineering Reference
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where ω is the angular frequency of the oscillation. It is related to the time taken
to complete one cycle (i.e. the period) T by ω
2 π
T . What can we deduce from
this result? Differentiating with respect to time gives us
=
d x
d t =−
v(t)
=
sin ωt
(2.7)
and
d 2 x
d t 2
2 cos ωt.
a(t)
=
=−
(2.8)
So we can write
ω 2 x.
a(t)
=−
(2.9)
We can connect the force with the motion by simply substituting for x using
Hooke's Law, i.e.
k
ω 2 a.
F
=
(2.10)
Here k is constant and ω is measured in the experiment. We can reasonably ask
the question “on what does ω depend?”. To answer this we need to do some more
experiments. We can imagine using different calibrated springs and different masses
on the end. We could for example replace the single block on the end of the spring
with N identical blocks of the same material all glued together, i.e. we increase
the amount of 'stuff' on the end of the spring by N times. Doing this experiment
results in a decrease in ω by a factor N . We repeat this process with springs of
different k and we start to observe a remarkable pattern in the data. The coefficient
k
ω 2 is proportional to the number of blocks no matter which spring we choose, i.e.
k
ω 2
=
Nm 0 ,
(2.11)
where m 0 is independent of k . What this means is that our experiments have
revealed a property of the motion that has nothing to do with the particular spring
we use but which depends on whatever is on the end of the spring. Not only that,
but the property depends linearly on the amount of 'stuff' in motion. We define
Nm 0
=
m to be the inertial mass of what is on the end of the spring. In which
case we can rewrite Eq. (2.10) as
F
=
ma.
(2.12)
We are now in a position where we can try to write down a law of motion. But
first we shall introduce the momentum, p , of a classical particle:
=
p
m v ,
(2.13)
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