REPORT FOR LAB B3: THE PHYSICAL PENDULUM

Nathaniel Kan4/01/02

ABSTRACT:

In the Physical Pendulum lab we calculated the local acceleration of gravity by using a Kater pendulum and measuring the period at different amplitudes. Our best value for g was 9.78128 m/s2 which is within 0.04% of the accepted value of 9.81 m/s2.

PURPOSE:

The purpose of this lab was to analyze the oscillations of a physical pendulum, in this case a Kater pendulum, which is equipped with weights at both ends of the shaft. We were to calculate the local acceleration of gravity based on our data taken of the period of the pendulum’s swing. We also looked for a relationship between the variation of the period and the amplitude of the swing.

PRINCIPLES:

A simple pendulum consists of a weight m attached to a massless string of fixed length L oscillating at a small enough angle so sin(theta) = theta, will possess a period:

In reality, however, there are no massless strings or point masses, which adds complications. For our lab, we used a Kater pendulum, which possesses counterweights on each end of the pendulum’s length.

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Kater Pendulum

The Kater pendulum uses two fixed knife edges where the periods about each knife edge are the same within 1 part per 105. This means that the distance between the two knife edges can be used as L to determine g. While a normal small amplitude pendulum gives its period as:

By substituting in T1, which corresponds to the period about the knife edge distance h1 from the center of mass, and T2, which corresponds to the period about the edge h2 from the COM, along with values h1 and h2, it is possible to determine g:

In our lab, we will use this equation to calculate our value of g.

PROCEDURE:

We began this lab by setting up the pendulum and the photocell timer, making a few trial measurements of to familiarize ourselves with the timer’s use. We then connected the timer to an external BK counter, which is able to display the period of the pendulum for the most recent oscillation.

We then took the pendulum off the wall and used a balance to measure and mark the center of mass. We also used microscopes to measure the exact length of the distance between the two knife edges, taking painstaking care to determine the length compared to an actual one meter metal rod. This proved a difficult task, due to the lack of intense light and the instability of the microscope lenses. We did finally get an accurate measurement

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for the length, however (see lab book page 66 for values of L). We then measured the lengths between the COM and the knife edges, h1 and h2.

Next we rehung the pendulum on the first knife edge and took measurements of the period for varying amplitudes, including d0 = 5 cm, 10 cm, 15 cm and 20 cm. Then we turned the pendulum around and took measurements for the pendulum oscillating about the other knife edge.

CALCULATIONS:

To calculate g (with unadjusted T), we averaged our T1 values, and averaged our T2 values and then inputted them and our h1 and h2 values into the formula:

This yielded results as shown below:

g (with unadjusted T)

T @ d0 = 5 9.78129T @ d0 = 10 9.77199T @ d0 = 15 9.76747T @ d0 = 20 9.75206

To calculate g (with adjusted T), we used the following series expansion to adjust our T, and then inputted the new values into the formula used above:

This yielded the results shown below:

g (with adjusted T)

T @ d0 = 5 9.77898T @ d0 = 10 9.77209T @ d0 = 15 9.76075T @ d0 = 20 9.74517

RESULTS:

In this lab we successfully calculated a close estimate of g from measurements of the Kater pendulum. Our T adjustment approximation had, however, a larger margin of error than our non-adjusted T. This could possibly be because of the small number of degrees used in the approximation. Our best final value of g was 9.78128 m/s2, close to the accepted value of 9.81 m/s2 within 0.04%. We also observed a linear connection between amplitude and period, which is shown in the following graph.

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