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

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