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Damped Oscillations Essay

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Damped and Undamped Oscillations



Objective Section

Investigate Newton's Laws of motion for undamped and damped motion using pendulum systems of a tennis ball and Styrofoam ball.

Equipment
  * Styrofoam and tennis ball
  * String
  * Pendulum apparatus
  * DataStudio with motion sensor
  * Graphical Analysis

Procedure Section

For part I, a tennis ball’s dimensions were calculated then connected to a pendulum. The period of oscillation of 6 different rope lengths were graphed to determine the experimental value of gravity. Afterwards, using one common length, 5 different amplitudes were measured and the change in period was observed.

In part II, a Styrofoam ball pendulum system was used to study damped motion. The damping oscillation was obtained from a motion sensor and a DataStudio program. The period, damping constant, time constant, and quality factor were calculated both through Graphical Analysis and manually using the oscillation graph (Appendix A).

Results

Part 1
g ± Δg | 10.09 ± 0.370 m/s2 |

Part 2

Graph estimate | Experimental |
ω | 4.01 rad/s | ω± Δ ω | 3.99 ± 0.011 rad/s |
τ | 58.00 s | τ± Δ τ | 57.90 ± 0.154 s |
Q | 232.48 | Q± Δ Q | 230.85 ± 0.885 |
b | 0.278 | b ±   Δ b | 0.279 ± (0.982) 10-5 |

(For all the data recollected and operation see Appendix A)

Data Analysis
Formulas used:

Part 1
g=4π2slope
Part 3- Experimental
θ0=RS- R0L

θt=θ0e-tτsinω't+ϕ
(ω')2=(ω0)2-(1τ)2

Q= ωτ

b=2mτ
Eyeball-Part II
ω=#of oscillationstime
f=1time
To find decay time τ: find the intersection in on the graph in which given by the value found after:
Amplitude⋅1e

Q=2πfτ
b=2mτ

Error propagation formulas:

Part1:
Δg2=∂g∂SlopeΔslope2

Δg2=4π2slope2Δslope2
Part 3:
ΔQ2=∂Q∂τΔτ2+∂Q∂ωΔω2

ΔQ2=ω Δτ2+τ Δω2

Δb2=∂b∂mΔm2+∂b∂τΔτ2

Δb2=2τΔm2+-2bτ2Δτ2

Finding the gravitational values with error in Part I

Finding Angular Frequency for both experimental and estimated values...

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Citations

MLA Citation

"Damped Oscillations". Anti Essays. 15 Dec. 2018

<http://parimatch-stavka7.com/free-essays/Damped-Oscillations-393606.html>

APA Citation

Damped Oscillations. Anti Essays. Retrieved December 15, 2018, from the World Wide Web: http://parimatch-stavka7.com/free-essays/Damped-Oscillations-393606.html


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