objective theory - yonsei universityphylab.yonsei.ac.kr/exp_ref/107_shm_eng.pdf · 2017-05-13 ·...

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Lab Manual

Simple Harmonic Motion Ver.20170512

Lab Office (Int’l Campus)

Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) / 9

[International Campus Lab]

Simple Harmonic Motion

Investigate simple harmonic motion using an oscillating spring and a simple pendulum.

Periodic motion or oscillation refers to any movement of

an object that is repeated in a given length of time. Fig. 1

shows one of the simplest systems that can have periodic

motion.

Whenever the body is displaced from its equilibrium position,

the spring force tends to restore it to the equilibrium position.

We call a force with this character a restoring force. Oscilla-

tion can occur only when there is a restoring force tending to

return the system to equilibrium.

If the spring is an ideal one that obeys Hooke’s law, the re-

storing force 𝐹𝐹𝑥𝑥 is directly proportional to the displacement

from equilibrium 𝑥𝑥. The constant of proportionality between

𝐹𝐹𝑥𝑥 and 𝑥𝑥 is the spring constant 𝑘𝑘. Then,

𝐹𝐹𝑥𝑥 = −𝑘𝑘𝑥𝑥 (1)

Fig. 1 Model for periodic motion. When the body is displaced from its equilibrium position at 𝑥𝑥 = 0, the spring exerts a restoring force back toward the equilibrium position.

Objective

Theory

----------------------------- Reference --------------------------

Young & Freedman, University Physics (14th ed.), Pearson, 2016

14.2 Simple Harmonic Motion (p.459~466)

14.4 Applications of SHM – Vertical SHM (p.470~471)

14.5 The Simple Pendulum (p.474~475)

-----------------------------------------------------------------------------


Lab Manual




When the restoring force is directly proportional to the dis-

placement from equilibrium, as given by Eq. (1), the oscilla-

tion is called simple harmonic motion, abbreviated SHM.

In general, the restoring force depends on displacement in a

more complicated way than in Eq. (1). But in many systems

the restoring force is approximately proportional to displace-

ment if the displacement is sufficiently small. Thus if the am-

plitude is small enough, we can use SHM as an approximate

model for many different periodic motions.

By substituting the Newton’s second law 𝐹𝐹𝑥𝑥 = 𝑚𝑚𝑎𝑎𝑥𝑥 into Eq.

(1) and rearranging the terms, the acceleration 𝑎𝑎𝑥𝑥 of a body

in SHM is given by

𝑎𝑎𝑥𝑥 =𝑑𝑑2𝑥𝑥𝑑𝑑𝑑𝑑2 = −

𝑘𝑘𝑚𝑚𝑥𝑥 (2)

And now we can express the displacement 𝑥𝑥 of the oscillat-

ing body as a function of time 𝑑𝑑.

𝑑𝑑2𝑥𝑥𝑑𝑑𝑑𝑑2 +

𝑘𝑘𝑚𝑚 𝑥𝑥 = 0 (3)

If we define angular frequency 𝜔𝜔 as Eq. (4), the solution of

Eq. (3) becomes Eq. (5) or (6).

𝜔𝜔 = �𝑘𝑘𝑚𝑚 (4)

𝑥𝑥 = 𝑎𝑎 cos𝜔𝜔𝑑𝑑 + 𝑏𝑏 sin𝜔𝜔𝑑𝑑 (5)

𝑥𝑥 = 𝐴𝐴 cos(𝜔𝜔𝑑𝑑 + 𝜙𝜙) (6)

The corresponding frequency 𝑓𝑓 and period 𝑇𝑇 relationships

are

𝑓𝑓 =𝜔𝜔2𝜋𝜋 =

12𝜋𝜋

�𝑘𝑘𝑚𝑚 (7)

𝑇𝑇 =1𝑓𝑓 =

2𝜋𝜋𝜔𝜔 = 2𝜋𝜋�

′𝑚𝑚′𝑘𝑘 (8)

If we use a hanging spring and a body is set in vertical mo-

tion, it also oscillates in SHM with the same angular frequen-

cy as though it were horizontal.

Suppose we hang a spring with force constant 𝑘𝑘 (Fig. 2)

and suspend from it a body with mass 𝑚𝑚. The body hangs at

rest, in equilibrium. In this position the spring is stretched an

amount Δ𝑙𝑙 just great enough that the spring’s upward vertical

force 𝑘𝑘Δ𝑙𝑙 on the body balances its weight 𝑚𝑚𝑚𝑚, so 𝑘𝑘Δ𝑙𝑙 = 𝑚𝑚𝑚𝑚.

Take 𝑥𝑥 = 0 to be this equilibrium position and take the posi-

tive 𝑥𝑥-direction to be upward. When the body is a distance 𝑥𝑥

above its equilibrium position, the extension of the spring is

Δ𝑙𝑙 − 𝑥𝑥 . The upward force it exerts on the body is then

𝑘𝑘(Δ𝑙𝑙 − 𝑥𝑥), and the net 𝑥𝑥-component of force on the body is

𝐹𝐹net = 𝑘𝑘(Δ𝑙𝑙 − 𝑥𝑥) + (−𝑚𝑚𝑚𝑚) = −𝑘𝑘𝑥𝑥.

So vertical SHM doesn’t differ in any essential way from

horizontal SHM. The only real change is that the equilibrium

position 𝑥𝑥 = 0 no longer corresponds to the point at which

the spring is unstretched.

Fig. 2 A body attached to a hanging spring


Lab Manual




A simple pendulum is an idealized model consisting of a

point mass suspended by a massless, unstretchable string.

When the point mass is pulled to one side of its straight-

down equilibrium position and released, it oscillates about the

equilibrium position.

The path of the point mass is not a straight line but the arc

of a circle with radius 𝐿𝐿 equal to the length of the string (Fig.

3). We use as our coordinate the distance 𝑥𝑥 measured along

the arc. If the motion is simple harmonic, the restoring force

must be directly proportional to 𝑥𝑥 or 𝜃𝜃 = 𝑥𝑥 𝐿𝐿⁄ .

The restoring force is provided by gravity; the tension 𝑇𝑇

merely acts to make the point mass move in an arc. We rep-

resent the forces on the mass in terms of tangential and radi-

al components. The restoring force 𝐹𝐹𝜃𝜃 is the tangential com-

ponent of the net force:

𝐹𝐹𝜃𝜃 = −𝑚𝑚𝑚𝑚 sin 𝜃𝜃 (9)

The restoring force 𝐹𝐹𝜃𝜃 is proportional not to 𝜃𝜃 but to sin𝜃𝜃,

so the motion is not simple harmonic. However, if the angle 𝜃𝜃

is small, sin 𝜃𝜃 is very nearly equal to 𝜃𝜃 in radians. With this

approximation, Eq. (9) becomes

𝐹𝐹𝜃𝜃 = −𝑚𝑚𝑚𝑚𝜃𝜃 = −𝑚𝑚𝑚𝑚𝑥𝑥𝐿𝐿 or 𝐹𝐹𝜃𝜃 = −

𝑚𝑚𝑚𝑚𝐿𝐿 𝑥𝑥 (10)

The restoring force is then proportional to the coordinate for

small displacements, and the force constant is 𝑘𝑘 = 𝑚𝑚𝑚𝑚 𝐿𝐿⁄ .

From Eq. (4) the angular frequency 𝜔𝜔 of a simple pendulum

with small amplitude is

𝜔𝜔 = �𝑘𝑘𝑚𝑚 = �𝑚𝑚𝑚𝑚 𝐿𝐿⁄

𝑚𝑚 = �′𝑚𝑚′𝐿𝐿 (11)

The corresponding frequency and period relationships are

𝑓𝑓 =𝜔𝜔2𝜋𝜋 =

12𝜋𝜋

�′𝑚𝑚′𝐿𝐿 (12)

𝑇𝑇 =1𝑓𝑓 =

2𝜋𝜋𝜔𝜔 = 2𝜋𝜋�

𝐿𝐿𝑚𝑚 (13)

Note that these expressions do not involve the mass of the

particle. For small oscillations, the period of a pendulum for a

given value of 𝑚𝑚 is determined entirely by its length 𝐿𝐿. A long

pendulum has a longer period than a shorter one.

We emphasize again that the motion of a pendulum is only

approximately simple harmonic. When the amplitude is not

small, the departures from simple harmonic motion can be

substantial. The period can be expressed by an infinite series;

when the maximum angular displacement is Θ, the period is

given by

𝑇𝑇 = 2𝜋𝜋�𝐿𝐿𝑚𝑚�1 +

12

22 sin2Θ2 +

12 ⋅ 32

22 ⋅ 42 sin4Θ2 + ⋯� (14)

When Θ = 15°, the true period is longer than that given by

approximate by Eq. (13) by less than 0.5%.

Fig. 3 An idealized simple pendulum


Lab Manual




1. List

Item(s) Qty. Description

PC / Software Video Analysis: SG PRO

1 Records, displays and analyzes videos.

Camera Tripod Screen

1 1 1

Feeds or streams its image in real time to a computer. Supports a camera. PVC foam board, white, 900 × 1200mm

Meter Stick

1 Measures the length of pendulums.

Spring 1 Exerts a restoring force back toward an equilibrium position.

Weight Hanger

1 Designed to hang several holed weights. - Polycarbonate base with steel post

- Mass: approx. 5g

Weights (Disk)

1 set Holed weights - Mass: approx. 5g, 10g, 20g(× 2), 50g

Weight (Cylinder)

1 Weight with yellow band

Weight (Ball)

1 Green plastic ball

Thread Scissors

Suspends a ball weight to form a simple pendulum.

Equipment


Lab Manual




Item(s) Qty. Description

Three-Finger Clamp

1 Holds pendulums.

Multi-clamp

1 Provides stable support for experiment set-ups.

A-shaped Base Support Rod (1100mm)

1 1

Provide stable support for experiment set-ups.

Electronic Balance

Measures mass with a precision to 0.01g.

Setup1. Equipment setup

Weights Details:

Disk Cylinder Ball

Use Expt.1

Spring Constant Expt. 2

Motion of Mass Expt. 3

Simple Pendulum

Shape • Flat Disk

• Center Hole

• with Hook • Yellow Band (for auto-track)

• with Hook • Green Plastic (for auto-track)

Image

Setup2. Software Setup (SG PRO)

If you are new to SG PRO software, see “Motion of a Rigid-

Body” lab manual.

Setup


Lab Manual




Experiment 1. Spring Constant

(1) Measure the mass of the weight hanger and the weights.

Use the electronic balance to measure mass.

- Weight Hanger: approx. 5g

- Weights: approx. 5g , 10g , 20g(× 2), 50g each

(2) Set up equipment. Suspend the spring so that it

hangs vertically. Put the weight hanger on the end of the spring.

(3) Specify your reference point of measurement.

For your reference, choose any measuring point such as the

bottom end of the spring.

(4) Measure the change in length of the spring.

Record the position of the reference point on the meter stick,

when different amounts of mass are added onto the weight

hanger. Do not forget to include the mass of the weight hang-

er (5g) in your calculation of the total mass. Determine the

change in length of the spring for mass varying from 40 to

80g in steps of 5g.

𝑚𝑚 (kg) 𝐹𝐹 = 𝑚𝑚𝑚𝑚 (N) 𝑥𝑥 (m)

0.040

0.045

0.050

0.055

0.060

0.065

0.070

0.075

0.080

(5) Find the spring constant 𝑘𝑘.

Find the slope of 𝐹𝐹-𝑥𝑥 graph using the method of least

squares (see the appendix of “Free Fall” lab manual). The

spring constant is equal to the slope of the 𝐹𝐹-𝑥𝑥 graph.

(6) Repeat measurement.

Repeat steps (4) to (5) more than 3 times.

Procedure

Note

When analyzing your data, you have to take into account

the initial tension in the spring. To ensure consistent rest

lengths, most spring manufacturers design extension

springs with an initial tension, which keeps the coils

pressed tightly together. Hooke’s law may not work for

small applied forces, as you must first overcome any ini-

tial tension before you see any apparent change in length.


Lab Manual




Experiment 2. Motion of a Mass on a Spring

(1) Measure the mass of the cylinder weight.

Use the electronic balance to measure mass. (Ignore the marked value on the weight.)

(2) Set up equipment.

① Set the meter stick aside. ② Place the camera and the screen. ③ Suspend the spring and attach the

weight at the end of the spring.

(3) Run SG PRO software.

See the “Motion of a Rigid-Body” lab manual.

You don’t have to calibrate the video scale because you will

measure an elapsed time, not a distance or a length.

(4) Let the weight oscillate vertically.

When displacing the weight, do not stretch the spring more

than about 2cm from its equilibrium position.

(5) Record a video.

Save the video clip including about 5 oscillations.

(6) Analyze your result.

① Select [T-Y] (time vs. 𝑦𝑦-axis) graph.

② Change [표시방법](display type) to [선](line).

③ Right-click on the graph and select [십자선 추가](Show

Crosshairs) from the list, and then click anywhere to show

crosshairs. Drag the crosshairs to read off the coordinates of

the graph.

④ Read off the coordinates of every peaks and calculate

the period 𝑇𝑇 of the motion.


Lab Manual




(7) Repeat the experiment.

Repeat steps (4) to (6) more than 3 times.

(8) Analyze the result.

Using the spring constant 𝑘𝑘 (expt.1) and the mass 𝑚𝑚 of

the weight (step(1)), calculate Eq. (8).

𝑇𝑇 = 2𝜋𝜋�′𝑚𝑚′𝑘𝑘 (8)

Experiment 3. Simple Pendulum

Repeat the procedure of expt. 2 using a simple pendulum.

(1) Set up equipment.

Use a piece of thread and the green ball weight to make a

simple pendulum.

(2) Measure the length 𝐿𝐿 of the pendulum.

(3) Let the pendulum swing.

Displace the pendulum about 5° from its equilibrium posi-

tion and let it swing.

(4) Record a video and analyze the result.

Using [T-X] (time vs. 𝑥𝑥-axis) graph, find the period of oscilla-

tions of the simple pendulum and verify Eq. (13).

𝑇𝑇 = 2𝜋𝜋�𝐿𝐿𝑚𝑚 (13)

(5) Vary the length 𝐿𝐿 of the pendulum and repeat the exper-

iment.

(6) (Optional) Find the period of the simple pendulum when

the amplitude is not small.

When the maximum angular displacement Θ is not small,

verify the period 𝑇𝑇 is given by

𝑇𝑇 = 2𝜋𝜋�𝐿𝐿𝑚𝑚�1 +

12

22 sin2Θ2 +

12 ⋅ 32

22 ⋅ 42 sin4Θ2 + ⋯� (14)


Lab Manual




Your TA will inform you of the guidelines for writing the laboratory report during the lecture.

Please put your equipment in order as shown below.

□ Delete your data files and empty the trash can from the lab computer.

□ Turn off the Computer and the Interface.

□ Leave the White Screens together at the front of the laboratory.

□ Place the Camera and Tripod assembly on any safe place.

□ Take care not to permanently deform the Spring. (Never stretch it beyond the elastic limit.)

Result & Discussion

End of LAB Checklist

objective theory - yonsei universityphylab.yonsei.ac.kr/exp_ref/107_shm_eng.pdf · 2017-05-13 ·...

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