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Unformatted text preview: Hooke’s law, (28)
for a spring, the spring constant here being 4 (29)
We can in fact take equation (28) as an alternate definition of simple harmonic motion. It says: Simple Harmonic Motion is the motion of a particle
subject to a force that is proportional to the displacement
of the particle but opposite in sign.
The mass-spring system shown in Figure 5 forms a linear simple harmonic oscillator (linear
oscillator, for short), where “linear” indicates that F is proportional to x rather than some other power of
x. Figure 5a This is one diagram of a linear simple
harmonic oscillator that shown represents Hooke’s Law.
images/f10013.jpg Figure 5b An alternate diagram showing linear simple
harmonic motion. Notice that if we have twice the mass,
you have twice the force and twice the displacement.
http://plus.maths.org/issue25/features/wilson/hookes.jpg The angular frequency of the simple harmonic motion of the block is related to the spring
constant k and the mass m of the block by equation (29), which yields (30)
Then by combining equations (9) and (30), we can write the period of the linear oscillator as
1 (32) 1
(34) 2 5 By looking at equations (30) and (34) we can see that a large angular frequency (and thus a small period)
will yield a stiff spring (large k) and a small mass.
Every oscillating system, be it a diving board or violin string, has some element of “springiness”
and some element of “inertia” or mass, and thus resembles a linear oscillator. In the linear oscillator
shown in Figure 5, these elements are located in separate parts of the system. The springiness is entirely
in the spring, which we assume to be massless, and the inertia is entirely in the block, which we
assume to be rigid.
2.5 Pendulums We now turn to a pendulum, which is a class of simple harmonic oscillators such that the
springiness is associated with the gravitational force rather than with the elastic properties of a twisted
wire or a compressed or stretched sprin...
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