Simple Harmonic Motion Essay, Research Paper

OBJECT:

The objective of this lab is to obtain the spring constant by using the simple harmonic motion of the spring-mass system. Once the spring constant is obtained it is to be compared to the spring constant obtained by Hooke?s Law.

PROCEDURE:

1) Using a meter stick measure the distance from the attached point of the spring to the end of the spring, at this time there is to be no mass on the spring. Once this measurement is obtained the elongations can be calculated by subtracting the new measurements from this first measurement.

2)Add a weight to the spring and record the distance. The new distance is to be subtracted from the first distance.

3)Using the same weight pull the mass down an additional 20cm. Once the spring is elongated it is to be let go. When the spring is released from it?s elongated position the stopwatch is started. Once the spring has returned to it?s original starting position 25 times the timer is to be stopped and the time is recorded. Once two times are taken for every weight increment they are to be averaged together.

4)Steps 2 and 3 are to be repeated eight times using a new weight each time.

5) When all eight trials are done the spring is to be weighed and recorded.

SAMPLE CALCULATIONS

Mass used in each trial, in kilograms:

- 50 g / 1000 g = 0.05 kg

Elongation of the loaded spring, in meters:

- 18.5 / 100 cm = 0.185 m

Calculation of x:

- 22.6 cm / 100 cm = 0.226 m

- x = 0.226 m ? 0.185 m

- x = 0.041 m

Calculation for the theoretical value of spring constant ?k:

- k = m g / x

- k = (0.05 kg) (9.8 m/s) / 0.041 m

- k = 12.0 N/m

Calculation for the average value of the theoretical values of ?k?:

- kavg = k1 + k2 + k3 + k4 + k5 + k6 + k7 + k8

8

- kavg = 12.0 + 10.9 + 10.2 + 9.95 + 10.2 + 9.95 + 10.2 + 9.90 + 9.89 + 9.80

8

- kavg = 10.4 N/m

Calculation for the average time ?t?:

- tavg = t1 + t2

2

- tavg = 16.5 + 17.2

2

- tavg = 16.9 s

Calculation of the period ?T? for one revolution:

- T = t / 25

- T = 16.9 / 25

- T = 0.676 s

Calculation of ?T2?:

- T2 = (0.676)2

- T2 = 0.457 s2

Calculation for the experimental value of the spring constant ?k? when slope of graph = 4.05:

- k = 4P2 / slope

- k = 4P2 / 4.05

- k = 9.75 N/m

Calculation of ?f? when: – y-intercept (b) of graph = 0.261

– mass of spring, ms = 0.173 kg

- f = k b / 4P2 ms

- f = (9.75) (0.261) / 4P2 (0.173 kg)

- f = 0.373

RESULTS:

Average theoretical value of ?k?: k = 10.4 N/m

Experimental value for ?k?: k = 9.75 N/m

ERROR:

Average theoretical value of ?k?: k = 10.4 N/m

Experimental value for ?k?: k = 9.75 N/m

- Actual Error = (Theoretical Value ? Experimental Value) x 100%

Theoretical Value

- Actual Error = (10.4) ?( 9.75) x 100%

10.4

-Actual Error = 6.25%

REASONS FOR ERROR

There are many possible reasons for error in this lab. Once such reason is that the instrument used are not precision instruments. For instance if the meter stick wasn?t perfectly level a measurement which is not accurate would be used in calculating the Theoretical Value of ?k?. If the stopwatch wasn?t used properly or stopped or started according to the procedures the time used will directly affect the result of the Experimental Value for ?k? thus creating an inaccurate value for ?k?. The previous mentioned sources of error can to a certain extent be controlled. However, those sources of error that cannot be controlled are having no friction and having zero air resistance.

CONCLUSION

Considering all the possible sources of error, this lab proved to be rather successful. It is apparent with reference to the relatively low Actual Error of 6.25%.

In having such a low error this concludes that one can find the spring constant either dynamically or with Hooke?s Law.