Harmonic Oscillator
!X;516072910086F1607204
Purposes of the measurement
1.Get familiar with the air track,Jonly balance,and timing measurement device.
2.Quantitatively study properties of the harmonic oscillator.
Measurement Principle
Oscillation is the repetitive variation,typically in time,of some measure about a central value or between two or more different states.Harmonic oscillation is the simplest form of oscillation. Harmonic oscillation can be achieved easily with a spring system.
Within elastic limit,the extension of the spring is proportional to the load applied to it:
F=−kx(1) This is known as the w,where k is the spring constant.
The spring constant can be measured by the so-called Jonly balance.A sketch of the device is shown below.After mounting the spring on the balance,one loads the spring with different amount of calibrated weights.Based on the displacement measured by the ruler scales on the device,one can measure the spring constant.
Figure1:jonly balance
Figure 2:spring oscillator
The scheme of the experiment is shown in Fig.2.An object with mass m is mounted on a horizontal air track.There are two springs with spring constants k1and k2,attaching to both sides of the object.The end of each spring is attached to one end of the air track.Let the origin of the motion be the position when the object is in equilibrium.If the object is displaced by an amount x,according to Newton .s Second Law of Motion,we have
M d 2x
dt
2+(k 1+k 2)x =0(2)
The solution of takes a standard sinusoidal form (therefore the name /harmonic 0)
x =Acos (ω0t +φ0)
(3)
where A is the amplitude of the oscillation,φ0is the initial phase,and ω0=
k 1+k 2
M
is the intrinsic angular frequency of the system.The period of the oscillation is given by
T =2πω0=2π
M
k 1+k 2
(4)When the mass of the spring cannot be neglected,we can quote the effective mass m 0=m 1+m 2
3
,where m 1m 2are the mass of two springs.Then equation (4)can be written as T =2π
M +m 0
k 1+k 2
(5)If friction is ignored,the mechanical energy of the spring and object system is conserved.the sum of the kinetic energy of the object and the potential energy of the spring is a constant.Since
the potential energy of the spring is given by E P =1
2
kx 2,and the kinetic energy of the object is
E K =12
mv 2
.Let the amplitude of the oscillation be A,and the maximum speed of the object is
vmax (when the spring displacement is zero),we have
k =
mv 2max A 2
(6)
Procedure
1.Measurement of spring constants.
(a)Adjust Jonly balance to vertical.Mount the first spring.Adjust knob /G 0to make the
marker on the mirror,its image,and the marker on the glass tube aligned.Write down the initial displacement reading L1.
