Single and Compound Harmonic Motion

7/21/2019 Single and Compound Harmonic Motion

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Single and Compound Harmonic MotionJonathan Healy

Introduction

The purpose of this lab was to determine the normal mode frequencies of two different systems. The first system was a mass with

two springs connected to it, one on either side (Figure 1). The second (Figure 2) is similar but more complex consisting of two

masses with one spring between the two masses and a spring on the other side of each mass for a total of three springs.


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Theory

1. Simple Harmonic Motion One Cart Oscillator

The equation of motion if x is the displacement of mass from equilibrium is:ma = -2kx or a + !2x = 0 where !2= 2k/m (1)

This is the differential equation for simple harmonic motion (SHM) whose general solution is:

x = Acos(!t + ") = Acos(2#ft + ")

where f = !/2#is the frequency and !is the angular frequency.

2. Compound Harmonic Motion Two Cart Oscillator

If x1and x2are the displacements of two equal masses m from equilibrium, then the equations of motion are:max1= -kx1+ k(x2 x1) (2)max2= -kx2 - k(x2 x1) (3)

Letting x1 = A1cos(!t) and x2 = A2cos(!t) and substituting:

A1(2k/m - !2) A2(k/m) = 0 (4)

- A1(k/m) + A2 (2k/m - !2) = 0 (5)

In matrix form:

| 2k/m - !2 -k/m | | A1 | | 0 |

| -k/m 2k/m - !2 | | A2 | = | 0 |

The determinant of the matrix must vanish therefore:

(2k/m - !2)2 (k/m)2= 0

Which gives two normal mode frequencies:

!s2= k/m (symmetric)

and:

!A2= 3k/m (anti-symmetric)

For != !s , equations (4) and (5) imply that A1 = A2 and both masses move in the same direction with equal amplitude.


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With != !A they imply that A1 = -A2and that they move in opposite directions with equal amplitudes.

Apparatus

Data / Results

"#$$%&'( "#$$%)&'( *+, %)&(

"#$$ ,#-./ 01/'2 3'01/2 3'3334,#-. 0 014'5 3'0145 3'3334

#6&' ,#-.$ 010'75 3'01075 3'3334

89:&;9< "#$$9$ %&'( *+,%&'( =>?

03 /@'@0 3'/

53 A@'@2 3'/

/33 @@'@4 3'/

033 /@@'@5 3'/


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$B-:+&$ ? B#-./ / ? ,#-. C$,:DD#.C-

/ "#$$%&'( *+,%&'( =>?

"&

%)&E">$F0( *+,'%)&(=>? G%,"( G%"( *+,%"( =>? ) /'51//03 3'/ 3'/@7 3'333/ /3'5 3'/35 3'335

A3 3'0 3'4@0 3'3330 0/'2 3'0/2 3'35

/33 3'/ 3'@2 3'333/ 51'4 3'514 3'35

/53 3'0 /'A1 3'3330 2@'4 3'2@4 3'35

/13 3'4 /'777 3'3334 /3A'2 /'3A2 3'35

0 "#$$%&'( *+,%&'( =>? "& %)&E">$F0( G%,"( G%"( *+,%"( =>? ) /'735@

53 3'/ 3'A@ 3'333/ 02'0 3'020 3'35

13 3'0 3'727 3'3330 A3'0 3'A30 3'35

/33 3'/ 3'@2 3'333/ 51'A 3'51A 3'35

/03 3'0 /'/17 3'3330 13'5 3'135 3'35

/53 3'0 /'A1 3'3330 2@'4 3'2@4 3'35

#6&' ) /'5225

/3 ,H,D9$ .:"9 / %$'( .:"9 0 %$'( #6&' I ? / ,H, J-9K*9+,H L />I

$DCB9

9--C-$ /'755A35A35 "#G

/2'A5 /2'A7 /'2A55 3'5A/252515 /'A10@A52@0 ":+

J/ L />0B: E $K-.%0)>"( 3'5A40202A J/%"#G )( 3'55A735@@@ J/%":+ M( 3'504/A@A54

)

=>? 3'0


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H L /'735@G = 3'3A4/

3

3'0

3'A

3'7

3'2

/

/'0

/'A

/'7

3 3'/ 3'0 3'4 3'A 3'5 3'7 3'1 3'2 3'@ /

NB-:+& 0 ? "& 6$' G%"'(


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H L /'51//G = 3'3A@A

3

3'0

3'A

3'7

3'2

/

/'0

/'A

/'7

/'2

3 3'0 3'A 3'7 3'2 / /'0

!"#$%& ( ) *& +,- ./*-0


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Part Two

H L 4'434/G ? 3'3/57

3

3'0

3'A

3'7

3'2

/

/'0

3 3'35 3'/ 3'/5 3'0 3'05 3'4 3'45 3'A

,"#$%& 12 ) *& +,- ./*-0N9-:9$/ O:+9#- %N9-:9$/(


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$B-:+&$ ? B#-.0 0 ? ,#-. C$,:DD#.C-

4P "#$$%&'( *+,%&'( =>?

"&

%)&E">$F0( *+,'%)&(=>? G%,"( G%"( *+,%"( =>? ) 4'434/

03 3'/ 3'/@7 3'333/ 7'1 3'371 3'30

53 3'/ 3'A@ 3'333/ /5'0 3'/50 3'34

13 3'0 3'727 3'3330 03'1 3'031 3'34

/33 3'/ 3'@2 3'333/ 43'5 3'435 3'34

4Q "#$$%&'( *+,%&'( =>?

"&

%)&E">$F0( *+,'%)&(=>? G%,"( G%"( *+,%"( =>? ) 4'4407

03 3'/ 3'/@7 3'333/ 1'4 3'314 3'30

53 3'/ 3'A@ 3'333/ /5'4 3'/54 3'34

13 3'0 3'727 3'3330 0/ 3'0/ 3'34

/33 3'/ 3'@2 3'333/ 43'2 3'432 3'34

4R "#$$%&'( *+,%&'( =>?

"&

%)&E">$F0( *+,'%)&(=>? G%,"( G%"( *+,%"( =>? ) 4'A03@03 3'/ 3'/@7 3'333/ 2'4 3'324 3'30

53 3'/ 3'A@ 3'333/ /7'0 3'/70 3'34

13 3'0 3'727 3'3330 03'1 3'031 3'34

/33 3'/ 3'@2 3'333/ 4/'4 3'4/4 3'34

#6&' ) 4'4500

,#-.$ $#"9 I $DCB9 9--C-$ A'/130/0177 "#G

47'15 47'42 /'20205 3'5A7@1//A1 0'100000000 ":+,#-.$ CBBC$:.9 ? 3'1

03 ,H,D9$ .:"9 / %$'( .:"9 0 %$'( #6&' I ? / ,H, J-9K*9+,H L />I

03'/2 03'55 /'3/205 3'@203113@4

J0 L />0B: E $K-.%)>"( 3'55237/25@ J0 %"#G )( 3'700A42551 J0%":+ )( 3'5302@1/A4

J4 L />0B: E $K-.%4)>"( 3'@775@/A@A J4 %"#G )( /'3123@5037 J4%":+ )( 3'21/3A4A30


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H L 4'A03@G ? 3'3770

3

3'0

3'A

3'7

3'2

/

/'0

3 3'35 3'/ 3'/5 3'0 3'05 3'4 3'45

!"#$%& 13 ) *& +,- ./*-0


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H L 4'4407G ? 3'34/@

3

3'0

3'A

3'7

3'2

/

/'0

3 3'35 3'/ 3'/5 3'0 3'05 3'4 3'45

,"#$%& 14 ) *& +,- ./*-0

N9-:9$/ O:+9#- %N9-:9$/(


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Conclusion

For the one cart oscillator a calculated reference value of (0.54 +/- 0.02) HZ. compares favorably with the frequency calculated thru

timing cycles with a stop watch which is calculated at 0.54 HZ. Similarily for the two cart oscillator with the two carts moving in thesame direction a reference value of (0.56 +/- 0.06) HZ compares well with 0.55 HZ. For the two cart oscillator system with the 2 carts

moving in opposite directions a reference value of (0.97 +/- 0.10) HZ also compares well to a value of 0.98 HZ.

Discussion

The large amount of error in this experiment comes from trying to calculate the spring constant of the springs. Notice that even with a

lot less error our results are still very close. Trying to line up a measurement on the ruler with the amount a spring stretched under a

specific amount of weight is not an exact science considering one is only using their eye to try to match numbers. In retrospect theuncertainty involved in doing this might have been slightly but not greatly exaggerated. However like was stated above the results

compare well even without this so heavily considered. There is very minimal error associated with the frequencies obtained via the

stopwatch as the data is gathered over multiple periods. Also the uncertainties in masses of the carts do not weigh heavily.

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Single and Compound Harmonic Motion