000 Circular ENG rev2 20160314 오차수정 - Yonsei...

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

Lab Manual

Circular Motion and Centripetal ForceVer.20160315

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) / 20

[For International Campus Lab ONLY]

Circular Motion and Centripetal Force

Measure the centripetal force with the radius, mass, and speed of a particle in uniform circular motion.

When a particle moves in a circle with constant speed, the

motion is called uniform circular motion. There is no com-

ponent of acceleration parallel to the path. The acceleration

vector is perpendicular to the path and hence directed inward

toward the center of the circular path. This causes the direc-

tion of the velocity to change without changing the speed.

Fig. 1(a) shows a particle moving with constant speed in a

circular path of radius with center at . The particle

moves from to in a time Δ . The vector change in

velocity during this time is shown in Fig. 1(b). The an-

gles Δ in Fig. 1(a) and 1(b) are the same because is

perpendicular to the line and is perpendicular to the

line , so

| | Δ

or| | Δ (1)

The magnitude of the average acceleration during Δ is

| |

ΔΔΔ

(2)

The magnitude of the instantaneous acceleration at

point is the limit of this expression as we take point

closer and closer to point

lim→

ΔΔ

lim→

ΔΔ

(3)

Fig. 1 Finding the velocity change , average accelerat-

ing , and instantaneous accelerating for a particle moving in a circle with constant speed

Objective

Theory

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

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

3.4 Motion in a Circle (p.106~109)

4.3 Newton’s Second Law (p.132~136)

5.4 Dynamics of Circular Motion (p.174~175)

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


Lab Manual




If the time interval Δ is short, Δ is the distance the parti-

cle moves along its curved path. So the limit of Δ Δ⁄ is the

speed at point . Also, can be any point on the path,

so we can drop the subscript and let represent the speed

at any point. Then

(4)

In uniform circular motion, the direction of the instantaneous

acceleration is perpendicular to and inward along the radi-

us, i.e. toward the center of the circle as in Fig. 1(c). Because

of this reason, the acceleration in uniform circular motion is

sometimes called centripetal acceleration. The word “cen-

tripetal” is derived from two Greek words meaning “ seeking

the center.” Fig. 2 shows the direction of the velocity and ac-

celeration vector at several points for a particle moving uni-

form circular motion.

We can also express the magnitude of the acceleration in

uniform circular motion in terms of the period (the time for

one revolution) of the motion. In a time the particle travels

a distance equal to the circumference 2π of the circle, so

its speed is

2

(5)

Substituting Eq. (5) into Eq. (4) yields

4

(6)

Fig. 2 Acceleration and velocity for a particle at uniform

circular motion

Uniform circular motion, like all other motion of a particle, is

governed by Newton’s second law, ∑ . To make the

particle accelerate toward the center of the circle, the net

force ∑ on the particle must always be directed toward the

center as in Fig. 3. The magnitude of the acceleration is con-

stant, so the magnitude of the net force must also con-

stant. If the inward net force stops acting, the particle flies off

in a straight line tangent to the circle, as in Fig. 4.

The magnitude of the net force on a particle with mass

in uniform circular motion is then

(7)

Fig. 3 Net force, acceleration and velocity in uniform circular motion

Fig. 4 What happens if the inward radial force suddenly ceases to act on a body in circular motion?


Lab Manual




1. List

Item(s) Qty. Description

PC / Software

Data Analysis: Capstone

1 Records, displays and analyzes the data measured by

various sensors.

Interface

1

Data acquisition interface designed for use with various

sensors, including power supplies which provide up to

15 watts of power.

Force Sensor

1

Measures the magnitude of force.

Range: 50N~50N

Resolution: 0.03N

Photogate

(Cable included)

1 Measures high-speed or short-duration events.

Circular Motion Apparatus 1 Allows objects to undergo uniform circular motion using

electric motor.

Swivel

1 Allows connected objects to rotate.

Wire

1 Connects the Mass to the Force Sensor

Mass Set 1 set

Includes 8 masses

- Free mass: Hook embedded

- Fixed mass: Perforated

- Plastic, Aluminum, Al. with Brass, and Stainless Steel

Fixed Mass Holder

(Nut included)

1 Secures a fixed mass.

Equipment


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Item(s) Qty. Description

Bubble Level

1 Checks the level of a surface.

Patch Cords

(with banana plugs)

2 Carry electric current.

A-shaped Base

Multi-clamp

Support Rod 600mm

Support Rod 300mm

1

1

1

1

Provide stable support for experiment set-ups.

Electronic Balance

Measure mass.

2. Details

(1) Photogate

The Photogate sensor is an optical timing device used for

very precise measurements of high-speed or short-duration

events. It consists of a light source (infrared LED) and a light

detector (photodiode). When an object moves through and

blocks the infrared beam between the source and the detec-

tor, a signal is produced which can be detected by the inter-

face.

When the infrared beam is blocked, the output signal of the

photogate becomes ‘0’ and the LED lamp on the photogate

goes on. When the beam is not blocked, the output signal

becomes ‘1’ and the LED goes off. This transition of signal

can be used to calculate quantities such as the period of a

pendulum, the velocity of an object, etc.


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(2) Force Sensor

The Force Sensor measures both pulling and pushing forc-

es in the range of 50N to 50N.

The sensor uses a strain gauge attached to an aluminum

beam. The gauge consists of an insulating flexible backing

which supports a metallic foil pattern. As the aluminum beam

is deformed, the foil is deformed, causing its electrical re-

sistance to change. The gauge is wired to form a full-bridge

circuit that is driven by a constant voltage source. The volt-

age across the bridge circuit is proportional to the applied

force.

(3) Mass Set

Free Mass Fixed Mass

Use Measuring Force Balancing

Shape

with Hook

with Center Hole

How

to useHook it to the wire.

Secure it to the rotating

arm using a mass holder.

MassMeasure the mass using

an electronic balance.

Do not have to measure

the mass.

Material Color Mass range g

Plastic White 11.50~12.50

Aluminum Light Gray 23.50~24.50

Al + Brass Gray / Yellow 43.00~44.00

Stainless Steel Dark Gray 66.50~67.50


Lab Manual




Setup 1. Equipment Setup

Setup

CAUTION

To avoid possible injury from the rotating arm hitting the

body, keep at least 1m distance from the rotating arm

when running the motor.

To avoid damaging the apparatus, keep all cords away

from the rotating arm.

To avoid damaging any other equipment or facilities near

the apparatus, especially your LCD monitor, keep the

apparatus away from them.

CAUTION

Do not tighten the swivel too hard. It could cause the

sensor to fail.

NOTE

① Thread the wire from the swivel hook through the pul-

ley as shown below.

② Hook the free mass to the end of the wire.

③ Hold the fixed mass firmly using the mass holder.

NOTE

To adjust the rotational radius of a free mass, lower or

raise the height of the upper (horizontal) support rod.


Lab Manual




Setup 2. Software Setup

(1) Run Capstone software.

(2) Add a Photogate.

Click the input port which you plugged the Photogate into

and select [Photogate] from the list.

(3) Create and configure a timer.

To measure the period of the circular motion, click [Timer

Setup] in the [Tools] palette and follow the steps below.

① Create a timer.

Choose [Build your own timer].

② Select the timing device for the timer.

Check [Photogate, Ch1] and then click [Next].

③ Arrange the sequence of timing events.

When the rod of the fixed mass holder passes the Photo-

gate, the Photogate beam is blocked and a “Blocked” signal

is generated. The time interval between the signals equals

the period of the motion, the time for one revolution. (You

will use this measurement to calculate the speed of the mass

in step (5).)

Click [▼] and then select [Blocked].

Click [▼] again and select [Blocked] one more time. Now

you have two [Blocked] events as below. Click [Next] for the

next step.


Lab Manual




④ Specify the timer’s name.

Enter the name of the timer (“period” is recommended) and

click [Finish].

(5) Configure calculator.

In a period , the mass travels a distance equal to the cir-

cumference 2π of the circle, so its speed is

2

(5)

The period will be measured through the process defined

in step (4). Now you define and using [Calculator].

Define the speed of the mass. Enter the equation of in

the first row of the [Calculator].

To define the equation 2 / , type “v=2*pi*R/” and “[“.

Make sure that you enter a left-bracket “[“ instead of “T” be-

cause you will insert “measured” data for . (Unlike numerals

or math functions, “measured” data are given in brackets

“[“ and “]”.)

When you type “[“, a pop-up list appears as below.

Select [period(s)], which you named in step (4). When you

select it, the equation is completed as shown below.

When you press the enter key, two cells are automatically

typed. One is for the unit of the parameter of the first equation,

and the other is for the variable R which you do not define yet.


Lab Manual




Enter the unit of the speed .

Enter the value and unit of R. You have to enter a correct

value for R whenever you vary the radius of the circular

path of the mass.

(6) Configure the Force Sensor.

The interface automatically recognizes the Force Sensor.

If the sensor is not in the panel, click the input port which

you plugged the sensor into. A drop down menu of sensors

will appear. Select [Force Sensor, High Resolution] from the

list and the sensor’s icon will be added to the panel.

Click the Force Sensor icon in the [Hardware Setup] panel

and then click the properties button (☼) in the lower right cor-

ner.

In the [Properties] window, check [Change Sign].

The sign of the outputs of the Force sensor is initially posi-

tive for the pressing force and negative for the pulling force.

In this experiment, you will measure the pulling force so you

have to change the sign of the outputs.


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(7) Configure the signal generator.

Do not add any sensor for [OUTPUTS 1].

Click [Signal Generator] in the [Tools] palette, and select

[850 Output 1].

Change the [Waveform] of the generator to [DC].

Set the [DC Voltage] to 4V.

Click [Auto] button to automatically activate the generator.

NOTE

Output options of [Signal Generator]:

[On] : The signal generator always output the signal.

[Off] : The signal generator is not activated.

[Auto] : The signal generator automatically outputs the

signal when you start recording, and automatically stops

when you stop recording.


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(8) Create displays.

You need 4 displays.

Data Table - -

Graph 1 -

Graph 2 -

Digital Meter

Click and drag [Table], [Graph], or [Digits] icon from the

[Displays] palette into the workbook page.

① Configure [Table] display.

[Table] display initially has two columns. To insert one more

column, click [Insert empty column] icon.

Click <Select Measurement> of each column and select

Time(s), Force(N), and v(m/s). Time(s) is a fundamental

measurement which shows the elapsed time from when the

measurement starts.

You can see the table as below.

② Configure [Graph] displays.


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Click <Select Measurement> to define the variables of each

axis.

Make a - graph. Select Time(s) for the -axis and

Force(N) for the -axis.

In the same way, make a - graph. Select v(m/s) for the

-axis and Force(N) for the -axis

③ Configure [Digits] display.

Drag [Digits] from the [Displays] palette into the workbook

page, and select v(m/s).

Move and resize the individual displays for easy view.

(9) Adjust the sample rate of the measurement.

Select [10.00Hz ] (collecting data every 0.1 second) for the

sensor in the [Controls] palette.

(10) Start/Stop recording data.

Click the [Record] button at the left end of the [Controls] pal-

ette to begin recording data. It will toggle to [Stop]. Click [Stop]

to stop data collection.

You can delete any data run using [Delete Last Run] or

drop-down menu in the [Controls] palette.


Lab Manual




Experiment 1.

In this experiment, you will vary the speed by changing

the voltage to the electric motor as the centripetal force is

continuously measured by the Force Sensor. The radius

and mass are held contant as the speed is increased.

(1) Follow the equipment setup instruction.

[Setup 1. Equipment Setup] on page 6

(2) Follow the software setup instruction.

[Setup 2. Software Setup] on pages 7-12

(3) Set the radius and the mass.

① Mount a set of masses (heavier set recommended).

② Adjust the radius (9cm is recommended) of the free mass.

To vary the radius, adjust the height of the Force Sensor,

keeping the wire tight.

③ Place the fixed mass at the same radius as the free mass

to ensure balancing the arm as it rotates.

④ Be sure to enter the correct value for in the calculator.

If the radius of the circular path of the mass is 9cm, enter

0.09 m .

Procedure

NOTE

Mass set Information

Free Masses Fixed Masses

Use Measuring Force Balancing

Shape

with Hook

with Center Hole

How

to use Hook it to the wire.

Secure it to the rotating

arm using a mass holder.

Mass Measure the mass using

an electronic balance.

Do not have to measure

the mass.

Material Color Mass range g

Plastic White 11.50~12.50

Aluminum Light Gray 23.50~24.50

Al + Brass Gray / Yellow 43.00~44.00

Stainless Steel Dark Gray 66.50~67.50

Measure the mass of free masses using the electronic

balance on the lecture table at the front of the lab.

Free Masses Mass g

Plastic

Aluminum

Aluminum + Brass

Stainless Steel

NOTE

Prior to recording each data run, press the [Zero] button

on the Force Sensor to zero the sensor.


Lab Manual




(4) Set the speed.

Speed is varied by changing the voltage supplied to the

electric motor. This will be displayed on [Digits] display. Apply

4, 5, 6, 7, and 8VDC to the motor.

Set [DC Voltage] of [Signal Generator] at 4V.

(5) Measure the force.

Click the [Record] button at the left end of the [Controls]

palette to begin recording data.

As the rotating arm rotates, the displays show the speed of

the mass and the force acting on the sensor. Wait a second

for stable rotation.

(6) Repeat measurement.

Repeat measurement for 4, 5, 6, 7, and 8VDC.

(7) Stop data collection.

Click the [Stop] button to stop data collection.

(8) Analyze the data.

Click [Show coordinates…] to read off data points.

(9) Repeat experiments.

Repeat measurement more than 3 times for each voltage

and calculate the average value to reduce the possibility of

experimental error.

NOTE

You can use [Pin] tool for easy view of the displays

.

NOTE

Collected data are stored in memory and appear in all

displays. The data runs are listed in the legend for each

display. (You can delete any run by clicking [Delete Last

Run] or drop-down menu in the [Controls] panel.)


Lab Manual




(10) Record your results.

Mass Radius Speed Force

(Theo.)

Force

(Result)

kg m m/s kg ⋅ m⁄ N

(11) Plot - graph.

Plot - graph and verify the centripetal force is propor-

tional to the square of the speed. Use the method of least

squares if required. (Refer to the appendix of [Free Fall] ex-

perimental manual.)

CAUTION

It is strongly recommended that you save the data file

with a different file name if you need to change the value

in the calculator. Changing makes all pre-measured

speed recalculated with the new value .

NOTE

If - graph vibrates as below,

① Make sure all the rotating parts are aligned.

② It could occur due to mechanical problems of the

swivel ball bearing. In this case, search the data for the

time interval of interest and calculate the average value of

the force to find the experimental value.


Lab Manual




Experiment 2.

In this experiment, the radius and the speed are held

contant as the mass is varied. By replacing the mass set,

the mass of the system becomes increased.





(3) Set the radius and the speed.

① Set [DC Voltage] of [Signal Generator] at 7V.

② Adjust the radius (9cm is recommended) of the free mass.

③ Place the fixed mass at the same radius as the free mass.

④ Enter 0.09 m for the radius in the calculator.

(4) Set the mass.

Use the lightest mass set.

(5) Measure the centripetal force.

Click the [Record] button at the left end of the [Controls] pal-

ette to begin recording data.

As the rotating arm rotates, the displays show the speed of

the mass and the force acting on the sensor. Wait a second

for stable rotation.

Record the speed. This value will be used to fine-tune the

voltage in the step (7).

Repeat measurement more than 3 times for each mass and

calculate the average value to reduce the possibility of exper-

imental error.



(7) Replace the mass set.

For all mass sets, repeat the steps (5) and (6).

Make sure the speed is equal to that of step (5). The speed

could be slightly different for each mass set even if the volt-

age remains constant, i.e. you need to fine-tune the voltage

for the each mass set.


Lab Manual




(8) Analyze the result.

Click the icon below to select the data run of interest.

Click [Show coordinates…] to read off data points.



(Theo.)

Force

(Result)


(10) Plot - graph.

Plot - graph and verify the centripetal force is propor-

tional to the mass. Use the method of least squares if re-

quired. (Refer to the appendix of [Free Fall] experimental

manual.)

NOTE

Recorded data run has a default name Run#%1,

where %1 is an automatically generated run number. You

can change the name of each data run if required.

① Click [Data Summary] in the [Tools] palette

② Select [Show Sensor Data] tap.

③ Right-click on the run name of interest.

④ Select [Rename] from the pop-up list.

CAUTION

It is strongly recommended that you save the data file

with a different file name if you need to change the value

in the calculator. Changing makes all pre-measured

speed recalculated with the new value .


Lab Manual




Experiment 3.

In this experiment, the mass and the speed are held

constant as the radius is varied. By lowering the Force

Sensor, the radius increases. As the radius increases, the

fixed mass must be moved to a matching radius to balance

the rotating arm.





(3) Set the mass.

Choose any mass set.

(4) Set the speed. (You will finally set the speed in step (6).)

The (linear) speed is dependent to the radius of the circular

path of the mass, i.e. whenever you change the radius, you

have to adjust the voltage so the speed remains constant.

(5) Set the radius.

① Adjust the radius of the free mass to 9cm.

② Place the fixed mass at the same radius as the free mass

③ Enter 0.09 m for the radius in the calculator.

(6) Find the appropriate voltage output.

Click the [Record] button to monitor the speed of the mass.

You can monitor the speed of the mass in [Digits] display.

Fine-tune the [DC voltage] near 4V so you can find any volt-

age at which the speed shows easy value.

This speed is the target speed of this experiment.

In the later steps, you have to adjust the voltage to rotate

the mass under the same linear speed. (As the radius de-

creases, the voltage must be increased.)

In the pre-experiment, the linear speed was 2.0m/s in the

conditions of 9cm and 4.0V output. And then 7.2V was

supplied to rotate the mass with 2.0m/s in the condition of

5cm. (Required voltages depend on your motor status.)

You should choose the voltage between 3V and 9V. The

motor may not work in a lower voltage than 3V, and could

make a noisy vibration in a greater voltage than 9V.


Lab Manual




(7) Measure the centripetal force.

If you have any useless data during monitoring the speed in

the previous step, delete it using [Delete Last Run] or drop-

down menu in the [Controls] palette.

In the condition of the voltage as you set in the step (6),

begin recording data.



(9) Change the radius.

Adjust the radii of the masses to 8cm.

Change in the [Calculator] to 0.08 m .

(10) Repeat measurement.

In the condition of 8cm, you must adjust the output

voltage so the speed shows the predetermined value in the

step (6).

Repeat measurement for R 9, 8, 7, 6, and 5cm.



(Theo.)

Force

(Result)


(12) Plot - 1⁄ graph.

Plot - 1⁄ graph and verify the centripetal force is propor-

tional to the inverse of the radius. Use the method of least

squares if required. (Refer to the appendix of [Free Fall] ex-

perimental manual.)

CAUTION

You MUST save the data file with a different file name

for each data run in this experiment. Changing makes

all pre-measured speed in the prior step recalculated

with the new value .


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.

□ Do NOT disassemble the equipment.

□ Keep the Mass Set, Mass Holder, Wire, and Bubble Level in the compartment box.

Result & Discussion

End of LAB Checklist

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