Objective Theory

If we define angular frequency πœ”πœ” as Eq. (4)the solution of , ... Department of PHYSICS YONSEI University Lab Manual Simple Harmonic Motion Ver.201705...

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Lab Manual

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

Simple Harmonic Motion

Ver.20170512

[International Campus Lab]

Simple Harmonic Motion

Objective

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

Theory

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

Reference

-------------------------th

Young & Freedman, University Physics (14 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) -----------------------------------------------------------------------------

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. Oscillation 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 restoring force 𝐹𝐹π‘₯π‘₯ is directly proportional to the displacement from equilibrium π‘₯π‘₯. The constant of proportionality between 𝐹𝐹π‘₯π‘₯ and π‘₯π‘₯ is the spring constant π‘˜π‘˜. Then, 𝐹𝐹π‘₯π‘₯ = βˆ’π‘˜π‘˜π‘˜π‘˜ Lab Office (Int’l Campus)

(1)

Fig. 1

Model for periodic motion. When the body is displaced from its equilibrium position at π‘₯π‘₯ = 0, the spring exerts a restoring force back toward the equilibrium position.

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Lab Manual

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

Simple Harmonic Motion

Ver.20170512

When the restoring force is directly proportional to the dis-

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

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

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

tion is called simple harmonic motion, abbreviated SHM.

cy as though it were horizontal.

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 displacement if the displacement is sufficiently small. Thus if the amplitude is small enough, we can use SHM as an approximate model for many different periodic motions. By substituting the Newton’s second law 𝐹𝐹π‘₯π‘₯ = π‘šπ‘šπ‘Žπ‘Žπ‘₯π‘₯ into Eq.

(1) and rearranging the terms, the acceleration π‘Žπ‘Žπ‘₯π‘₯ of a body in SHM is given by

𝑑𝑑2 π‘₯π‘₯ π‘˜π‘˜ = βˆ’ π‘₯π‘₯ 2 𝑑𝑑𝑑𝑑 π‘šπ‘š

π‘Žπ‘Žπ‘₯π‘₯ =

(2)

ing body as a function of time 𝑑𝑑. 𝑑𝑑𝑑𝑑 2

+

and suspend from it a body with mass π‘šπ‘š. 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 π‘˜π‘˜Ξ”π‘™π‘™ on the body balances its weight π‘šπ‘šπ‘šπ‘š, so π‘˜π‘˜Ξ”π‘™π‘™ = π‘šπ‘šπ‘šπ‘š.

Take π‘₯π‘₯ = 0 to be this equilibrium position and take the posi-

tive π‘₯π‘₯-direction to be upward. When the body is a distance π‘₯π‘₯

above its equilibrium position, the extension of the spring is Δ𝑙𝑙 βˆ’ π‘₯π‘₯ . The upward force it exerts on the body is then

π‘˜π‘˜(Δ𝑙𝑙 βˆ’ π‘₯π‘₯), and the net π‘₯π‘₯-component of force on the body is 𝐹𝐹net = π‘˜π‘˜(Δ𝑙𝑙 βˆ’ π‘₯π‘₯) + (βˆ’π‘šπ‘šπ‘šπ‘š) = βˆ’π‘˜π‘˜π‘˜π‘˜.

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

And now we can express the displacement π‘₯π‘₯ of the oscillat𝑑𝑑2 π‘₯π‘₯

Suppose we hang a spring with force constant π‘˜π‘˜ (Fig. 2)

horizontal SHM. The only real change is that the equilibrium position π‘₯π‘₯ = 0 no longer corresponds to the point at which the spring is unstretched.

π‘˜π‘˜ π‘₯π‘₯ = 0 π‘šπ‘š

(3)

If we define angular frequency πœ”πœ” as Eq. (4), the solution of

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

πœ”πœ” = οΏ½

π‘˜π‘˜ π‘šπ‘š

π‘₯π‘₯ = π‘Žπ‘Ž cos πœ”πœ”πœ”πœ” + 𝑏𝑏 sin πœ”πœ”πœ”πœ” π‘₯π‘₯ = 𝐴𝐴 cos(πœ”πœ”πœ”πœ” + πœ™πœ™)

(4)

(5) (6)

The corresponding frequency 𝑓𝑓 and period 𝑇𝑇 relationships

are

𝑓𝑓 = 𝑇𝑇 = Lab Office (Int’l Campus)

πœ”πœ” 1 π‘˜π‘˜ οΏ½ = 2πœ‹πœ‹ 2πœ‹πœ‹ π‘šπ‘š

1 2πœ‹πœ‹ β€²π‘šπ‘š = = 2πœ‹πœ‹οΏ½ 𝑓𝑓 πœ”πœ” β€²π‘˜π‘˜

(7)

(8)

Fig. 2

A body attached to a hanging spring

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Lab Manual

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Simple Harmonic Motion

Ver.20170512

A simple pendulum is an idealized model consisting of a

The corresponding frequency and period relationships are

point mass suspended by a massless, unstretchable string. 𝑓𝑓 =

When the point mass is pulled to one side of its straightdown equilibrium position and released, it oscillates about the equilibrium position.

𝑇𝑇 =

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 π‘₯π‘₯ measured along the arc. If the motion is simple harmonic, the restoring force must be directly proportional to π‘₯π‘₯ or πœƒπœƒ = π‘₯π‘₯⁄𝐿𝐿.

πœ”πœ” 1 β€²π˜¨π˜¨ οΏ½ = 2πœ‹πœ‹ 2πœ‹πœ‹ ′𝐿𝐿

(12)

1 2πœ‹πœ‹ 𝐿𝐿 = = 2πœ‹πœ‹οΏ½ 𝑓𝑓 πœ”πœ” 𝘨𝘨

(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 𝘨𝘨 is determined entirely by its length 𝐿𝐿. A long

pendulum has a longer period than a shorter one.

The restoring force is provided by gravity; the tension 𝑇𝑇

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

approximately simple harmonic. When the amplitude is not

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

small, the departures from simple harmonic motion can be

al components. The restoring force πΉπΉπœƒπœƒ is the tangential com-

ponent of the net force:

We emphasize again that the motion of a pendulum is only

substantial. The period can be expressed by an infinite series; when the maximum angular displacement is Θ, the period is given by

πΉπΉπœƒπœƒ = βˆ’π‘šπ‘šπ‘šπ‘š sin πœƒπœƒ

(9)

The restoring force πΉπΉπœƒπœƒ is proportional not to πœƒπœƒ but to sin πœƒπœƒ,

so the motion is not simple harmonic. However, if the angle πœƒπœƒ

is small, sin πœƒπœƒ is very nearly equal to πœƒπœƒ in radians. With this approximation, Eq. (9) becomes

πΉπΉπœƒπœƒ = βˆ’π‘šπ‘šπ‘šπ‘šπ‘šπ‘š = βˆ’π‘šπ‘šπ‘šπ‘š

π‘₯π‘₯ 𝐿𝐿

or

πΉπΉπœƒπœƒ = βˆ’

π‘šπ‘šπ‘šπ‘š π‘₯π‘₯ 𝐿𝐿

𝐿𝐿 12 Θ 12 β‹… 32 Θ 𝑇𝑇 = 2πœ‹πœ‹οΏ½ οΏ½1 + 2 sin2 + 2 2 sin4 + β‹― οΏ½ 𝘨𝘨 2 2 2 β‹…4 2

(14)

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

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

(10)

The restoring force is then proportional to the coordinate for small displacements, and the force constant is π‘˜π‘˜ = π‘šπ‘šπ‘šπ‘šβ„πΏπΏ .

From Eq. (4) the angular frequency πœ”πœ” of a simple pendulum with small amplitude is

π‘˜π‘˜ π‘šπ‘šπ‘šπ‘šβ„πΏπΏ β€²π˜¨π˜¨ πœ”πœ” = οΏ½ = οΏ½ =οΏ½ π‘šπ‘š π‘šπ‘š ′𝐿𝐿

(11)

Fig. 3

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An idealized simple pendulum

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Lab Manual

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

Simple Harmonic Motion

Ver.20170512

Equipment

1. List

Item(s)

PC / Software

Qty.

Description

1

Records, displays and analyzes videos.

Camera

1

Feeds or streams its image in real time to a computer.

Tripod

1

Supports a camera.

Screen

1

PVC foam board, white, 900 Γ— 1200mm

Meter Stick

1

Measures the length of pendulums.

Spring

1

Video Analysis: SG PRO

Exerts a restoring force back toward an equilibrium position.

Designed to hang several holed weights. Weight Hanger

1

- Polycarbonate base with steel post - Mass: approx. 5g

Weights (Disk)

1 set

Holed weights - Mass: approx. 5g, 10g, 20g(Γ— 2), 50g

Weight (Cylinder)

1

Weight with yellow band

Weight (Ball)

1

Green plastic ball

Thread Scissors

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Suspends a ball weight to form a simple pendulum.

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Item(s)

Simple Harmonic Motion

Ver.20170512

Qty.

Description

Three-Finger Clamp

1

Holds pendulums.

Multi-clamp

1

Provides stable support for experiment set-ups.

A-shaped Base

1

Support Rod (1100mm)

1

Electronic Balance

Provide stable support for experiment set-ups.

Measures mass with a precision to 0.01g.

Setup

Setup1. Equipment setup

Weights Details:

Use

Shape

Disk

Cylinder

Ball

Expt.1 Spring Constant

Expt. 2 Motion of Mass

Expt. 3 Simple Pendulum

β€’ Flat Disk β€’ Center Hole

β€’ with Hook β€’ Yellow Band (for auto-track)

β€’ with Hook β€’ Green Plastic (for auto-track)

Image

Setup2. Software Setup (SG PRO) If you are new to SG PRO software, see β€œMotion of a RigidBody” lab manual.

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Lab Manual

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

Simple Harmonic Motion

Ver.20170512

Procedure

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

(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

Use the electronic balance to measure mass. - Weight Hanger: approx. 5g

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

hanger. Do not forget to include the mass of the weight hanger (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. π‘šπ‘š (kg) 0.040

𝐹𝐹 = π‘šπ‘šπ˜¨π˜¨ (N)

π‘₯π‘₯ (m)

0.045 0.050 0.055 0.060 0.065

(2) Set up equipment.

0.070 0.075

Suspend the spring so that it

0.080

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

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 (3) Specify your reference point of measurement.

small applied forces, as you must first overcome any initial tension before you see any apparent change in length.

For your reference, choose any measuring point such as the bottom end of the spring. (5) Find the spring constant π‘˜π‘˜. Find the slope of 𝐹𝐹-π‘₯π‘₯ graph using the method of least

squares (see the appendix of β€œFree Fall” lab manual). The spring constant is equal to the slope of the 𝐹𝐹-π‘₯π‘₯ graph. (6) Repeat measurement.

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

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Experiment 2. Motion of a Mass on a Spring

(1) Measure the mass of the cylinder weight.

Simple Harmonic Motion

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(6) Analyze your result. β‘  Select [T-Y] (time vs. 𝑦𝑦-axis) graph. β‘‘ Change [ν‘œμ‹œλ°©λ²•](display type) to [μ„ ](line).

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

(2) Set up equipment. β‘  Set the meter stick aside. β‘‘ Place the camera and the screen.

β‘’ Right-click on the graph and select [μ‹­μžμ„  μΆ”κ°€](Show Crosshairs) from the list, and then click anywhere to show

β‘’ Suspend the spring and attach the weight at the end of the spring.

crosshairs. Drag the crosshairs to read off the coordinates of the graph.

(3) Run SG PRO software.

See the β€œMotion 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.

Lab Office (Int’l Campus)

β‘£ Read off the coordinates of every peaks and calculate the period 𝑇𝑇 of the motion.

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Lab Manual

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

Simple Harmonic Motion

Ver.20170512

(7) Repeat the experiment.

(3) Let the pendulum swing. Displace the pendulum about 5Β° from its equilibrium posi-

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

tion and let it swing.

(8) Analyze the result.

(4) Record a video and analyze the result.

Using the spring constant π‘˜π‘˜ (expt.1) and the mass π‘šπ‘š of

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

β€²π‘šπ‘š 𝑇𝑇 = 2πœ‹πœ‹οΏ½ β€²π‘˜π‘˜

Using [T-X] (time vs. π‘₯π‘₯-axis) graph, find the period of oscilla-

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

𝐿𝐿 𝑇𝑇 = 2πœ‹πœ‹οΏ½ 𝘨𝘨

(8)

(13)

(5) Vary the length 𝐿𝐿 of the pendulum and repeat the experiment.

Experiment 3. Simple Pendulum

(6) (Optional) Find the period of the simple pendulum when the amplitude is not small.

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.

When the maximum angular displacement Θ is not small,

verify the period 𝑇𝑇 is given by

𝐿𝐿 12 Θ 12 β‹… 32 Θ 𝑇𝑇 = 2πœ‹πœ‹οΏ½ οΏ½1 + 2 sin2 + 2 2 sin4 + β‹― οΏ½ 𝘨𝘨 2 2 2 β‹…4 2

(14)

(2) Measure the length 𝐿𝐿 of the pendulum.

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

Lab Manual

Simple Harmonic Motion

Ver.20170512

Result & Discussion

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

End of LAB Checklist

Please put your equipment in order as shown below.

β–‘ Delete your data files and empty the trash can from the lab computer. β–‘ Turn off the Computer and the Interface. β–‘ Leave the White Screens together at the front of the laboratory. β–‘ Place the Camera and Tripod assembly on any safe place. β–‘ Take care not to permanently deform the Spring. (Never stretch it beyond the elastic limit.)

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