Lab Activity Phy223 Rotational Mechanics, Part II: Swinging Stick
Names:
Swinging Stick
Introduction
In this activity you will find the minimum period of a swinging stick (a roughly 8-foot-long piece of 1x2 nominal lumber) both theoretically and experimentally and compare the results.
Procedure and Instructions for Report
A. Abstract
Give a brief description of what you did. What is the purpose of this lab?
B. Finding the experimental values for xmin and Tmin
Your goal is to find the period T of oscillation as a function of x, the distance between the axis of rotation (pin) and the center of mass (cm). You will then use your data to experimentally find the distance , for which the period T has its minimum value .
Experimental Setup:
Procedure and Data:
1. Use every hole, starting with the first hole from the center of mass, as the axis of rotation.
2. For each hole: take two or three trials to measure the period T. Make sure to keep the amplitude low (less than 10⁰). Keep friction to a minimum. Record all data in an Excel spreadsheet (i.e. all values for T, their average, and the distance x) and include it in your report.
Analysis:
1. Create a graph showing the period T as a function of x, the distance between the axis of rotation and the center of mass.
2. Use your graph to find your experimental result for (the distance with a minimum T) and the corresponding period .
C. Finding the theoretical values for xmin and Tmin
The swinging stick is an example for a physical pendulum. The physical pendulum is covered in section 14.6 of your textbook. You will probably need your textbook to work through the following steps.
1. Draw an extended FBD for the swinging stick at a point half-way through an oscillation (maximum amplitude). Important hint: you can draw the FBD with the stick at the left-of-equilibrium position shown in the figure above or in the right-of-equilibrium position as shown in figure 14.21 of your textbook. Note that the angle is measured counterclockwise from the equilibrium position. This means that is negative in the figure above or positive in figure 14.21.
2. Use the parallel axis theoremto create an equation for the moment of inertia for your stick as a function of x(the distance the axis of rotation is away from the center of mass).
3. Write out the equation for about the axis of rotation (not the center of mass) using information from #1 and #2 above. What force is exerting the torque to speed up and slow down the stick? Is I the moment of inertia about the center of mass or not?
4. Solve the differential equation you obtained in step 3. Use section 14.6 in your textbook for guidance. You will get an equation for the angular frequency as a function of x.
5. Rewrite this equation to determine a function for the period T (i.e. express in terms of T and solve for T).
6. Use calculus to predict the position for the minimum period and the value for the minimum period. In Excel, create a graph of T vs. x, and include it in your report.
D. Conclusions
1. Compare - using a graph overlay - your theoretical function for the period to your experimental function.
2. Compare your theoretical and empirical values for and . Determine the percent difference . Discuss your results.
3. Give possible sources of error and discuss their significance.
4. Discuss the physical meaning of the minimum of the period. That is, what’s happening physically that reduces the period and then increases it again? Hint: in your equation for T vs. x, what is the physical meaning of the numerator and denominator?