PS113L Pendulum worksheet

PENDULUM WORKSHEET PS 113L – Intro Physics I Lab / 50pts Name: Hunter Porter Section # : 1 Remember to save your worksheet as a PDF and upload it to the Canvas Assignment for submission. MEASUREMENT DATA FROM PENDLUM EXPERIMENT (10 PTS) Run # Avg Pendulu m Period (s) Period Uncertainty (s) Pendulum Length, L (m) Vertical Displacement (m) Release Angle, θ (deg) Pendulu m Mass, m (kg) Variabl e Adjuste d 1 11.5266 ± 6.16 1 m 0.086 m 5 100 g L, θ, m 2 10.2446 ± 5.58 1 m 0.1 m 5.7 100 g θ 3 10.9517 ± 5.58 1 m 0.147 m 8.6 100 g θ 4 11.8981 ± 6.17 1 m 0.175 m 10 100 g θ 5 11.5846 ± 6.17 1 m 0.20 m 11.5 100 g θ 6 11.1306 ± 5.94 0.92m 0.08 m 5 100 g L 7 10.5949 ± 5.53 0.82 m 0.071 m 5 100 g L 8 9.7719 ± 5.2 0.72 m 0.062 m 5 100 g L 9 9.0577 ± 4.74 0.62 m 0.054 m 5 100 g L 10 8.7419 ± 5.08 0.52 m 0.045 m 5 100 g L 11 7.6382 ± 5.1 0.42 m 0.036 m 5 100 g L 12 9.7365 ± 5.03 1 m 0.095 m 5 200 g m 13 10.7821 ± 5.68 1 m 0.092 m 5 500 g m 14 ± m Page 1

PLOTS OF PENDULUM PERIOD DEPENDANCE ON EACH VARIABLE (12 PTS) In the space below, insert copies of your graphs you made in either Microsoft Excel or Pasco Capstone. Make sure when inserted they are well formatted and clearly readable (axis labels with variables and units are present, data points and numbers are large enough to see and read, if a fit equation was asked for make sure the line is visible and the equation is displayed). Plot 1: Pendulum Period vs Pendulum Length Plot 2: ln(Period) vs ln(Length), natural log plot of pendulum period vs length Page 2

ANALYSIS OF RESULTS Review of Observations and Plots (5 pts) Review the plots. Do any show a significant change in period with respect to the variable, is any change linear or nonlinear? Do any plots show no significant change/slope, i.e. is it constant and unaffected by changing that variable? Make sure any variations are significant , that is it’s well above the noise/uncertainty in the data. The scale of your y-axis here is important. From your data and observations, what varying factors actually affect the period of a pendulum? The pendulum period vs release angle showed the most change of all of the graphs. In period vs in length shows the least amount of change. Model Equation for fitting ln(Period) to ln(Length) of a Pendulum (2 pts) Let T = KL n be an exponential relationship where K and n are constants that fit the data best. This exponential function can be manipulated to resemble the standard linear function y = mx + b . ln ( T ) = ln ( K L n ) = ln ( K ) + ln ( L n ) ln ( T ) = n ln ( L ) + ln ( K ) y = mx + b This is the equation of a straight line where the y-variable is ln(T), the x-variable is ln(L), ln(K) is the constant y-intercept and n is the constant slope. This model equation should fit the data in your Plot #2 and be synonymous with the linear equation fit to that data. Note: The y-intercept is ln(K) therefore K = e y − intercept What are your values for K and n ? Substitute your values for K and n into T = KL n .  Slope (constant rate): -0.079 seconds/m  Y-intercept: 2.418 seconds Comparing your Model to expected outcome from Theory (4 pts) Using some basic principles of physics and mechanics it would be possible to derive an equation to predict pendulum period directly from theory. This would reveal a more general model equation where the numerical constants are replaced with physical constants, thus giving rise to an even broader predictive equation that would be useful under an even broader range of conditions. The equation of interest is: T = 2 π √ L g Page 4

T = 2 π √ L g or T = 2 π √ g L 1 / 2 = K L n where 𝐿 is the pendulum length and 𝑔 is the gravitational acceleration. As a comparison between theory and experiment compute the percent difference 1 between 1. K and 2 π / √ g : % Difference=120.32% 2. n and ½ : Using Observational Model to Predict g , gravitational acceleration (3 pts) Using your experimental model and physics theory, it is possible to use data from timing the period of pendulums to estimate the value of g , acceleration due to gravity on the surface of the Earth. Use your K value to determine an experimental value of g , and find the percent difference between your value and the reference value in Daytona Beach, Florida of 9.79265(±2)m/s 2 . 2 DISCUSSION OF RESULTS (4 PTS) Looking at your results and how they compare to expectations, discuss if you believe your results are successful, inconclusive, or unsuccessful. Were you able to identify what variable(s) affect a pendulum’s period? Did your experimental model agree with what physics theory predicts? Did you demonstrate you can use a pendulum to estimate local gravity? 0.0546% DISCUSSION OF UNCERTAINTIES (6 PTS) Identify and describe at least 3 likely sources of uncertainty that you believe affected your results in a non-trivial way. Be specific in the source, what was affected, and how it was affected (+bias, -bias, or +-random, etc.). Discuss how significant you think each source of uncertainty is (does one have a greater effect than others). An uncertainty can be random (affecting individual measurements or outcomes randomly) or systematic (always having the same impact, such as consistently over estimating something). 1. 2. 3. 1 Note: %Difference = | x experimental − x theory | x theory ∙ 100% 2 Provided by the US National Geodetic Survey via their online Su r f ac e G r a vity P r e d i c tion T o o l . Page 5