If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in Figure 2. Similarly, Figure 3 shows an object bouncing on a spring as it leaves a wavelike "trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.
The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by
x
t
=
X
cos
2
πt
T
,
x
t
=
X
cos
2
πt
T
,
size 12{x left (t right )=X"cos" { {2π`t} over {T} } } {}
(6)where XX size 12{X} {} is amplitude. At t=0t=0 size 12{t=0} {}, the initial position is x0=Xx0=X size 12{x rSub { size 8{0} } =X} {}, and the displacement oscillates back and forth with a period TT. (When t=Tt=T, we get x=Xx=X size 12{x=X} {} again because cos2π=1cos2π=1.). Furthermore, from this expression for xx size 12{x} {}, the velocity vv size 12{v} {} as a function of time is given by:
v(t)=−vmaxsin2πtT,v(t)=−vmaxsin2πtT, size 12{v \( t \) = - v rSub { size 8{"max"} } "sin" left ( { {2π`t} over {T} } right )} {}
(7)where vmax=2πX/T=Xk/mvmax=2πX/T=Xk/m size 12{v rSub { size 8{"max"} } =2πX/T=X sqrt {k/m} } {}. The object has zero velocity at maximum displacement—for example, v=0v=0 size 12{v=0} {} when t=0t=0 size 12{t=0} {}, and at that time x=Xx=X size 12{x=X} {}. The minus sign in the first equation for v(t)v(t) size 12{v \( t \) } {} gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have x(t), v(t), t,x(t), v(t), t, size 12{x \( t \) ,v \( t \) ,t} {} and a(t)a(t) size 12{a \( t \) } {}, the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is a=F/m=kx/ma=F/m=kx/m size 12{a=F/m= ital "kx"/m} {}. So, a(t)a(t) size 12{a \( t \) } {} is also a cosine function:
a
(
t
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=
−
kX
m
cos
2π
t
T
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a
(
t
)
=
−
kX
m
cos
2π
t
T
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size 12{a \( t \) = - { { ital "kX"} over {m} } " cos " { {2π t} over {T} } } {}
(8)Hence, a(t)a(t) size 12{a \( t \) } {} is directly proportional to and in the opposite direction to a(t)a(t) size 12{a \( t \) } {}.
Figure 4 shows the simple harmonic motion of an object on a spring and presents graphs of x(t), v(t), x(t), v(t), size 12{x \( t \) ,v \( t \) `} {} and a(t)a(t) size 12{`a \( t \) } {} versus time.
The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.
Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. Describe what happens to the sound waves in terms of period, frequency and amplitude as the sound decreases in volume.
Frequency and period remain essentially unchanged. Only amplitude decreases as volume decreases.
A babysitter is pushing a child on a swing. At the point where the swing reaches xx size 12{x} {}, where would the corresponding point on a wave of this motion be located?
xx size 12{x} {} is the maximum deformation, which corresponds to the amplitude of the wave. The point on the wave would either be at the very top or the very bottom of the curve.
