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...