Physics Laboratory
last update:2003. 4. 7
Exp 4. Motion of oscillator of an object
Purpose of Experiment
All system which has a equilibrium states has a turning back quality about little change. The turning back quality in dynamic system appears as restoring force. Specially, when the change from equilibrium state is quite small, the size of restoring force is proportional to a degree of change. Also, dynamic system has the inertia which maintains a kinetic condition like that. When this restoring force and inertia appear together, the system has simple harmonic motion. The object which is hung in a spring, a pendulum, LC electric circuit, the solid material or oscillation of the atom from intramolecular etc. simple harmonic motion appears about many physical systems. So, the simple harmonic motion is considered seriously in physics. In this experiment we use a pendulum, punched ruler of length 1m.
We measure the period of a simple harmonic motion and compare with theory, then we check the motion of object which is used as a pendulum. As the change becomes quite larger from the equilibrium state, the restoring force is not proportional to change quantity any more and the nonlinear effect appears. We study the nonlinear effect of a simple harmonic motion when the pendulum amplitude is large.
- Halliday & Resnick 일반물리학책에서
Have you ever thought of a swing as a pendulum? If we see different view, we can see the same phenomena newly. Here you can obviously feel that period of the swing is constant but is irrelevant to a weight of swing person or the magnitude of a pendulum. Think how swing people increase his vibration energy. Think how the people who having a swing augment their vibration energy.
ExperimentOutline
Experiment the motion of a pendulum and observe how the period of the simple harmonic motion is determined.
Understand the reason a simple harmonic motion occurs in the nature and the exchanging between the kinetic energy and the potential energy of a object which does a simple harmonic motion.
See the nonlinear effect of the simple harmonic motion of a pendulum.
Experimental Method
These items of equipment are provided in the laboratory. (The number of item is indicated in parentheses.)
Photogate Model SG-1122(1)
Stand for pendulum motion(1)
a protractor(1)
weight whose mass is each different (3)
scales (2, common use)
If you need other items, check with your teaching assistant or the experiment preparation room (19-114), or prepare them yourself.
1. As the picture, Hang the thread to the stand in order to oscillate freely and Measures the period of oscillation motion.
1. Adjust the experiment device perpendicular to the ground.
② Make the pendulum do the oscillation motion by making a pendulum lean the angle 30 to the vertical and releasing. In here, adjust the position and the direction of a photogate to pass through the photogate perpendicular to a beam.
③ Turn on the switch of a photogate detector and place the operation at the PEND mode. And place the Save(MEMORY) switch at OFF(or middle) and the resolving power to 1 ms. Then if you press the RESET button, it displays on the light emitting diode(LED) READ. In here, read a period T.
④ Press the Return button for the new measurement. [You can do this measurement on placing the MEMORY switch at ON. It has a benefit, not pressing the RESET button continuously. But as it is displayed by adding the subsequent measurement value automatically, you must write a value on the display. To know each period, minus from the next value. The cumulative time is continuous. This case, however, consider that the largest order isn't displayed if the time overs 9.999 sec.]
⑤ Calculate the average by repeating measurement more than 5 times for the period T.
⑥ At the right time, measure the weight of an using weight by the common use scales.
2) Check by comparing with the period which is obtained by the theoretical express . And calculate the gravitational acceleration g by using the obtained period in this experiment.
3) As the above method, perform the each experiment for 1. the change of a period for the length of a thread / 2. the change of a period for the weight of a pendulum / 3. the change of a period for the amplitude of a oscillation. Find the factor which effects on a period. And think why the result is.
To write the experiment note, that method is encouraged.
1) the change of a period for the length of a thread
The length of thread of a pendulum = 1. 2. 3. (m)
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Measurement times / Period T (s) = 1. 2. 3.
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1
2
3
4
5
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Average : TAV =
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[Calculate the gravitational acceleration g by using calculated period T and theoretical expression.]
2) the change of a period for the weight of a pendulum
The weight of weight = 1. 2. 3.
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Measurement times / Period T (s) = 1. 2. 3.
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1
2
3
4
5
------
Average : TAV =
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3) the change of a period for the amplitude of a oscillation
Magnitude of an oscillation = 1. 2. 3. (degree)
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Measurement times / Period T (s) = 1. 2. 3.
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1
2
3
4
5
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Average : TAV =
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Background Theory
If we oscillate from left to right an apple, suspended by wire that is fixed one end, the apple does the periodic motion with moving back and forth. Is this motion the simple harmonic motion? How long is the period T of vibration? It consists of a particle of mass m and a negligible mass string of length L, as figure 1.a. The pendulum oscillates freely left and right of vertical which passes hanging points over a surface. The force on a pendulum consists of the tension T and the gravitational force Fg. when the angle of between a string and the vertical is θ, let's divide the gravitational force into the radial component Fg cosθ and the tangential component Fg sinθ. As the tangential component tends to reduce the angular displacement, it gives the torque, restoring the circular motion to equilibrium for the axis which passes hanging point of a pendulum.
[Fig1]
The torque is defined by
The negative sign means that the torque tends to reduce the angle.
L is the magnitude of a moment arm which is defined by the magnitude when the force applied.
If we solve by substituting Eq.1 into , we obtain
where I is the moment of inertia for pivot point and α is the angular acceleration.
In Eq.2 if the angular displacement is small, we can approximate sinθ with θ.( in radian measure)
For example, if , . So the difference ratio of two values is 0.1%. Therefore If we approximately simplify the expression,
While this equation shows that the angular acceleration of a pendulum is proportional to the angular displacement and its sign is different. As the pendulum moves right as Fig1.a, the acceleration increases. So it stops for a moment and moves left again.
Exactly, the motion of a pendulum which moves within a small angle is approximately simple harmonic oscillation. So the maximum angular position must be small. As comparing with Eq.3, angular acceleration on a simple harmonic motion, and displacement relation, we can see that angular frequency is .
Also using by Eq the period of a frequency of a pendulum is.
In a simple pendulum, we assume that all mass m is localized where it is distance from pivot point as a pendulum.
In this case the moment of inertia is represented by from Equation , and if we substitute this value to equation, we can indicate the a period of a simple pendulum with oscillation in small angle.
Things to Think About
We already know the period of a oscillation of which a point of mass m is chained to a thread of the length L is
If rewrite the period as
, we can see that a period of a oscillation is determined by the ratio between the rotational inertia which tends to continue the motion and the restoring torque which tends to restore stable state.
In this experiment, we call it the physical pendulum when we use the object which is not concentrated
Generally, the period of a oscillation T is represented by
where I is the inertia for the oscillation of a pendulum, M is the mass, h is the distance from pivot point to center of mass of a pendulum, and g is gravitational acceleration.
In this experiment, when we assume M is the mass of rod, L is the length of rod, and it is uniform object, the moment of inertia I for the main axis of a pendulum is represented by
by using parallel axis theorem.
In here, Icm is the moment of inertia for passing through the center of mass.
and it is represented by .
Therefore, the moment of inertia of this pendulum is
)
and the period is
in Eq(1).
If we use a rod of the length L=1m and the gravitational acceleration g = 9.8 m/s2, the period of a pendulum is
,where the distance from the pivot point to the center is h(m).
Therefore the period of a pendulum is maximum Tmin = 1.52 s s at h = 0.29 m. At h=0,that is, if it close to the center, it will be infinitely long. Being the period infinitely long means that the object don't come back, because the suspended rod at hole in the center continuously turns only one direction when it is no friction between the axis and the air.
If we consider case of a pendulum of a mass m of suspended on a distance d from a pivot point
in Eq(3), the moment of inertia is
.
and the new center of mass is located
away from the pivot point
The period of oscillation of pendulums is
,so it depends on the mass of pendulum but also the position of pendulum.
Things to Think About 2
we call the indistinguishable center, not being differentiated between back and forth or up and down, the center of a object. And we call the point, where the effect by the mass of a object is localized in an any point, the center. How long Is the length of thread of a pendulum, having same period with pendulum of an object and localizing in one point ?
If we use the rod of mass M and length L, where the upper end of the rod is suspended, the distance from the hanging point to the center of mass is h = L/2.
So From Eq(7)
. From Eq(1) the length of an thread L0, having the same period, is
. That is,
.Another way if we make the hanging point moved near a center of mass slowly, the period of a pendulum will be decreasing in the beginning(∵ h2 decreases in a numerator of Eq(7)) and increasing at the later (∵ h decreases in a denominator of Eq(7)) Therefore it exists a distance h, being same with original period T. In order to obtain the distance h, if we replace h with h' and make same with using period and T,
.And from here,
.That is, it is h' = L/6 and L/2. In case of h' = L/2, the period is T = ∞. So the finding solution is h' = L/6. That is, the period of pendulum which is suspended on the distance (2/3)L from one end as
of Eq(A3) is same with the period of a pendulum which is suspended one end. This position is called the center of oscillation.
Reference
- Measurement method of the period of oscillation motion - photogate
- Treatment of measurement data
- Analysis method based on the graph
- Christian Huygens - A pioneers of the wave mechanics who invented the pendulum clock
- Simple history of a pendulum
- Nonlinear effect in a pendulum