Physics Laboratory

Last update: 2015. 08. 31

Experiment 1 - 6. Motion of spring pendulum

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 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 the 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.

First, use spring, suspending weight, as vertical oscillator in this experiment. And then observe the each motion of them and a pendulum which is suspended by thread which is fixed length. And then observe the motion of object when you use spring, suspending weight, as pendulum. In this case, since the gravitational force acts on a pendulum as the restoring force with the restoring force of spring, two simple harmonic motions appear in 2-dimension plane. Furthermore, since they are not independent but combined, the motion of weight becomes very variously.

As the change becomes quite large from the equilibrium state, the restoring force is not proportional

Purpose of Experiment

to change quantity any more. And the nonlinear effect occurs. We study the nonlinear effect and the coupling scheme of simple harmonic motion.

⊙ 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 doing a simple harmonic motion. ⊙ Measure the two simple harmonic motion of a spring respectively.

▶ First, we think how the restoring force of a spring occurs by measuring the spring constant of a spring.

▶ When we neglect a mass of spring, how do we determine the each period? ▶ when we consider the own mass of spring, how does the each period change?

⊙ Measure the two simple harmonic motion of a pendulum respectively.

These items of equipment are provided in the laboratory. (The number of item is indicated in parentheses.) Spring(1) Stand for spring (1) Computer(1) CCD camera(1) The plate with attaching the teflon rods(1) Protractor (1) Thread (1, common use) Calipers (2, common use)

If you need other items, check with your teaching assistant or the experiment preparation room (19-114), or prepare them yourself.

Outline of Experiment

Experimental Method

The following is a recommended experiment method. 1) Measure the change of length of a spring according to the weight of weight. And calculate the spring constant. ① Hang the plate of weight to the spring, and control the stand in order to be the experimental device perpendicular to ground. ② By using prepared weight, measure the length of a spring for the weight of 12 kinds weight. At this point, if possible, do not let the spring oscillate. And Fix the plate of weight and eye height are same.[In here, the direct measurement of length of a spring is no need. why?] ③ Check whether the length of spring is restoring or not when you remove the weight on the plate of weight. Does the difference has an impact on the future experiment If it has the difference? If so, repeat the previous measurement and calculate the average value.

2) Limit the motion of a weight, suspended by a spring, on the vertical motion. And by using CCD camera and computer, measure the simple harmonic motion of weight. ① Set up the experimental device as videos. After hanging the base of weight to spring, turn on the computer and run "I-CA" program.

Click the [file- camera setting] of menu. And then check that whether it displays the screen of CCD camera or not.

* Tip. Camera setting 1. After running I-CA program, select the camera setting in file menu. 2. Press the menu of the remote until setup menu appears. 3. Select ALS/AES of the third line. 4. Select LEVEL -FIX- OFF on the last line. 5. Select 1/250. 6. After pressing BACK on the remote, return back. 7. Select AGS/SENS. 8. After selecting LIGHT, select NOMAL. 9. Select SENS. 10. Select X32(32times). 11. Out by pressing BACK twice. 12. If you press H, the position of shooting will be changed Left to Right and Right to Left.

13. If you press V, a subject is reversed. Finish the setting of initial experimental equipment by seeing the manual of program and CCD. To be showed the experimental equipment appropriate location on the screen, adjust the position of PC camera. (All equipments don’t need to be showed on the screen. That is, take the motion to display the motion of ball, being suspended by plate of weight, on the screen. *Caution : Before the experiment, Align Horizontal and vertical.

② Put on weight of wanting weight. Wait while weight is stabilized in equilibrium position and

then specify the data saving path. And start the capture of screen. Make weight do horizontal oscillation by releasing after pulling a spring in equilibrium position.

Start the measurement. If the save of Data finishes, analyze the saved data by selecting the [picture-screen analysis]. First, specify the path of saved Data. And then determine the standard point of subject by determining the initial frame and final frame (Reference below Fig)

Start analysis. After finishing analysis, if you save the Data, the image file which is used at analysis

and information file of position of a subject will be showed on the screen.( x1, x2 and y1, y2 coordinate according to the time)

③ You can draw a graph by using Excel.(If you want to road the data by Origin, Open the data by using Excel and select the file format as [Text(Tab-delimited)].And road it after save.) ④ Can you say that the motion of weight is the simple harmonic motion? If not, search the reason and improve the experiment method. (Think why the motion of y-direction occurs as the upper result and what it affects on the result.) Measure the period T for more 3 kinds weight. [Caution: If the

weight is quite light, it will appear as phenomena that the weight floats from the plate of weight during the spring does the oscillation motion. It is another interesting phenomenon, but excludes that in here.] ⑤ By using the common use scale, measure the weight of plate of weight and spring. ⑥ Compare this result with the calculated period from theoretical expression by using spring constant of (1), and check whether they are different or not. If they are different, review the reason. Think the magnitude of restoring force, the directional velocity, and the magnitude and direction of acceleration in the maximum and minimum amplitude. In the experiment result, do you observe the damping of amplitude by friction of air? (In the graph of experiment result, the maximum point and minimum point of amplitude maybe show as the envelop forms. what is the reason? For this case, draw the value of (highest + lowest amplitude)/2 for the time in order to see the effect of air friction.)

3) Hang weight to a thread which having appropriate length. Use it as pendulum and then measure

the period. ① Make the pendulum by using a thread which having appropriate length. ② Make weight do the oscillation motion on the vertical plane by leaning weight from equilibrium

position and releasing. Then measure its motion. [Caution: In the oscillation motion of pendulum, the motion may not occur in a vertical plane, but may occur the precession. Make the amplitude(angle) of a oscillation be small possibly.]

③ Do horizontal plane and vertical plane have the same period? What is the reason? (For the same case or different case)

Measure the period by changing the weight for more than 3 kinds weight. For the weight, check whether the period changes or not. And obtain the period after drawing the position change graph for the time by using graph processing program. Then compare them. ④ Repeat this measurement for more than 3 kinds length of yarn. How does the value of

gravitational acceleration change according to the length of yarn? What is the reason? ⑤ Analyze as ⑥ of (2).

4) Do the experiment of oscillation motion of pendulum by using weight whose mass is

unknown. ① The code number is written on weight. Experiment assistants can see the mass by using that

code, but don’t tell the information to students. Repeat the previous experiment for two weight whose mass is unknown. And calculate the mass of weight by using theoretical expression, then submit the report which written the mass and code number. (Reflect the process which is obtaining mass in the score.)

5) Construction of Lissajous figures. ① Pull down weight which is suspended a little by a spring, and make it do motion by moving

constant angle from horizontal position. Make it do periodic motion about each axis. Draw the Lissajous figures by using obtained result. If you draw it about each section, you can see

that the shape of figures is how changes.

First we will study the vertical motion of a pendulum, suspended by a spring under gravity. If we assume that the gravitational acceleration is g, the spring constant is k, the mass of a object is M, and the increased length is x, the restoring force of acting on a object will be –kx. We can calculate the net force from Newton’s first law because spring is under gravity.

(1)

Background Theory

Below expression is the general solution of this differential equation. (We can check that below Eq(2) is the solution of 2nd differential equation as follows. We can see

that they are equal by substituting Eq(2) into left and double-differential of Eq(2) into Right.)

(2) In here, the angular velocity ω and the period T each have

ω = (k/m)1/2 (3) T = 2π(M/k)1/2 (4)

. The amplitude xm and the phase φ is determined by initial condition (that is, the increased length x and the velocity v on t=0)

(5) (6)

But the upper result is applied when neglecting the mass of a spring and thinking of a suspended object as a material particle. If we remove this assumption, how does the upper result change?

We can deal with the effect of mass of a spring as follow. Think an uniform spring of mass m. When we suspend a pendulum of mass M vertically, the increasing length of a spring changes according to the position. Its value is

x(y) = (g/k)[m + M(L-y)/L] (7)

,where k, L, y are the spring constant , the length of a spring, the length from the fixed position without acting the gravitation. We can think of the upper expression as the case of a spring that changes the spring constant according to the position without mass. That is, the spring constant of this imaginary spring is

k'(y) = km/[m + M(L-y)/L] (8) And the average spring constant of this spring is

(9) , In case of m << M, we can approximate the upper expression

(10) Therefore the angular frequency and the period of the suspended pendulum of mass M vertically each have

(11)

(12) So we can see that the dependence of a spring for the mass M varies a little.

[주 :If we do the more correct calculation, it is close to

(13) And the real problem is very complicated. [Someone, being interested in, reference J. M. Nunes da Silva, Am. J. Phys. 62, 423 (1994) and the reference books in there.] While In the case of a pendulum that being the length l, the restoring force of a pendulum is

F = - Mgsinθ (14) where the pendulum of mass M is tilted angle θ from the vertical. And Its direction is the tangential

of the arc of the radius l.

when the angle is small(θ<<1), the law of motion is

(15) .So the solution of this differential equation is ()

(16) . where the angular velocity ω' and the period T' each have

ω'= (g/L)1/2 (17) T' = 2π(L/g)1/2 (18)

. The amplitude θm and the phase φ' are determined by the intial condition(that is, the titled angle θ at t = 0 and the angular velocity dθ/dt). When the angle θ is small(θ<<1), we can obtain the solution by approximating the motion of pendulum as the motion which is proportional to the variable θ.

(19)

(20) The angular velocity and the period of Eq(17) and (18) don't relate with the mass M.

But this result is applied when neglecting the mass of a spring and thinking of a suspended object as

a particle. the object, not being localized the mass in a point, is called the physical pendulum. Generally the period of a pendulum is given by

T = 2π(I/mgh)1/2 (21) I is the moment of inertia for the pivot point according to the oscillation and h is the distance from

the pivot point to the center of mass. In conservative system, the mechanical energy is conserved when the external factor as friction isn’t.

In simple harmonic motion, the potential energy and kinetic energy are represented by U = (1/2)kx2 (22)

K = (1/2)mv2 = (1/2)m(d2x/dt2)2 (23) The total mechanical energy is conserved.

E = U + K (24) If you compose two vertical motion of simple harmonic, you can see the Lissajous figures.

(25) (26)

If the frequency and phase are same, you can obtain below expression by eliminating t in Eq(25)

and Eq(26).

(27) That is , the combination of two simple harmonics is represented by a straight line. But if the phase

becomes different, it is represented by circle and ellipse. Finally, let’s think the effect of frictional force for air. If we assume that it is under frictional force

which is proportional to velocity, Eq(2) is represented as (28)

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

Reference

Nonlinear effect in a pendulum

Moment of inertia

※site with various apps about simple harmonic oscillator http://www.scienceall.com/content/c072/physics/mechanical_energy.htm

http://www.scienceall.com/content/c072/physics/oscillation_spring_horizon.htm

http://www.scienceall.com/content/c072/physics/oscillation_spring_vertical.htm

http://www.scienceall.com/content/c072/physics/oscillation.htm

※ site with apps about lissajous

http://www.scienceall.com/content/c072/algorithm/lissajous.htm