Andrew TealAP Physics 1

26 March 2015Pendulum Design Lab: Designing a Clock Pendulum

Purpose

For this lab, the goal is to first design experiments to test the various characteristics of a pendulum. Next, these factors of mass, amplitude and pendulum length must be tested using the experiments created. With the results, a pendulum must then be designed that will keep a period of 1.875 seconds for the grandfather clock to operate and keep time.

Materials

Notebook and pencil for data recording Laptop computer LabPro Quest Vernier Photogate and adapter cable 2 support clamps and stands 1 table clamp 100g, 200g and 300g masses with hooks String of Meter stick Protractor Graphing software (Excel 2013)

Procedure

For the experimental procedures, three separate tests will be run varying the amplitude, mass and length of the pendulum to measure any changes in period for each separate characteristic. From the data collected, relationships can be derived for the separate variables and it will be apparent as to which characteristic truly affects period. With this, a pendulum can be designed to fit the grandfather clock. Because the pendulum for the clock requires a specific period, the data will show which part of the pendulum’s design must be manipulated to fit the requirements.

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Andrew TealAP Physics 1

26 March 2015Process:

1. Place a support stand on the edge of a table and secure it to the table with a clamp or by placing a heavy mass on the base. Tie two strings of equal length to the 200 g mass. Attach the strings about 15 cm apart to a horizontal rod extended from the support stand. The pendulum should hang over the edge of the table, as shown on the previous page. This arrangement will let the mass swing only along a line, and will prevent the mass from striking the photogate. The length of the pendulum is the distance from the point on the rod halfway between the strings to the center of the mass. The pendulum length should be at least 1 m (you do not need an exact measurement now).

2. Attach the photogate to a second support stand. Place the support stand on the floor and position the photogate so that the mass blocks the photogate while hanging straight down. Connect the photogate to the LabPro and turn it on.

3. Observe the reading on the screen. Temporarily hold the mass out of the center of the photogate. Block the photogate with your hand; note that the photogate is shown as blocked. When you remove your hand, the display should change to unblocked. Now, set the time mode. Select PENDULUM from the TIMING MODES menu.

4. Now you can perform a trial measurement of the period of your pendulum.Hold the mass displaced about 10° from vertical and release. (For a pendulum that is 100 cm long, that corresponds to pulling the mass about 15 cm to the side.) After the mass has passed through the photogate, the average period begins to display on the calculator screen.

Part 1: Amplitude

5. Measure the period for four different amplitudes. Choose a wide range of amplitudes, from just barely enough to unblock the photogate, to about 40°. For each trial, use the protractor to measure the amplitude before releasing the mass. Collect the data and record it in a data table like the one shown for Part I in the Data Tables section.

Part 2: Length

6. Now you will measure the effect of changing pendulum length on the period. Use a constant mass and consistent amplitude of 10° for each trial. Vary the pendulum length in steps of 10 cm, from 100 cm down to 50 cm. Be sure to measure the pendulum length from the rod to the center of the mass. You may have to raise the photogate as the length changes. Record your data in a data table like the one shown for Part II in the Data Tables section.

Part 3: Mass

7. Use three different masses to determine if the period is affected by changing the mass. Be sure to attach the masses so that the center of mass remains unchanged; otherwise, you are also

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Andrew TealAP Physics 1

26 March 2015affecting the length. For each mass trial, keep the length of the pendulum the same and use constant amplitude of 10°. Record your data in a data table for later use in Excel.

Data

Below are the three data tables recording the relationships between period and variations in amplitude, mass and length of the pendulum.

Data Analysis

Graphs

Using the tables from data collection, values were plotted in a scatter plot and then overlaid with a line of best fit and the equation for that line. This gives an accurate representation of the relationships between the period and the variables of amplitude, mass and length for the pendulum.

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Amplitude (degrees)

Period (s)

10 2.02520 2.02930 2.03640 2.054

Mass (kg) Period

(s)0.1 2.01670.2 2.040.3 2.036

Length (m)

Period (s)

1 2.0850.9 1.97850.8 1.84450.7 1.77850.6 1.6760.5 1.543

Andrew TealAP Physics 1

26 March 2015

To further explore possible relationships between the variables and their resulting period, the length data is then graphed again, but with the length values being both squared and square rooted to check for any possible relationships in the data.

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Andrew TealAP Physics 1

26 March 2015

Calculations

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Andrew TealAP Physics 1

26 March 2015In analyzing all data collected, the graphs are their lines of best fit must be used, as a line of best fit minimizes chance for errors and small variations in measurements.

Using reason and the five graphs under data analysis, one must be selected that reveals the realistic relationship with a variable and its effect on the period of the pendulum. Reasoning through the different variables, one assumption can be made in that any pendulum with varying length should have a period of 0 when the length is 0, as a pendulum cannot function with a length of 0. Therefore, the three length graphs must be observed to find the most appropriate relationship. Using the lines of best fit and the included y intercepts, it is apparent that the Period vs. sqrt(Length) graph is closest to having a y intercept of 0, as is expected. Therefore, this is the graph and line of best fit that must be analyzed, at the data shows a realistic and accurate relationship with length and period.

For the Period vs. sqrt(Length) graph, the line of best fit is: y = 1.7978x + 0.2711

Using the above line of best fit, the y intercept must be replaced with 0, as the small difference is likely due to experimental error and it should be assumed that a length of 0 results in a period of 0.

Therefore, the new equation is simply: y = 1.7978x

Because period is on the y axis and the square root of length is on the x axis, variables y and x in the above equation can be replaced to better represent the relationship that is being observed.

Therefore, it can be re-written as: T (s.) = 1.7978 * sqrt(L) (m.)

The slope of the equation above therefore defines the relationship between the square root of length and the period of a pendulum. The slope can be assigned the variable K, so that:

K = 1.7978

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Andrew TealAP Physics 1

26 March 2015With knowledge of the period equation for a pendulum, it is known that this K value is theoretically

equivalent to 2 π√ g . However, this is not necessary to know, as this value will not change considering

gravity is constant in this case. Therefore, the value of K can be used to calculate the relationship between any period or length of a pendulum. Next, this equation must be used to solve for the length of a pendulum that has a period of 1.875, as this is required for the grandfather clock. Setting up the equation by plugging in this period, it’s now written as:

1.875 (s.) = 1.7978 * sqrt(L) (m.)

Rearranging, length can be solved for through algebra.

L (m.) = (1.875 (s.) / 1.7978)²

so

L = 1.088 m

Conclusion

Using calculations from data analysis and the graph of Period vs. sqrt(Length), the desired length of a pendulum for a period of 1.875 seconds is 1.088 meters.

Because the clock might need adjusting over time for any reason, the pendulum can include an adjustable mass that can be raised or lowered to shift the pendulum’s center of mass. The length is derived from the pendulum’s center of mass, and a mass on the pendulum that can be raised or lowered would shift this center of mass up and down to essentially vary the length of the pendulum and therefore change its period.

As mentioned previously, a pendulum’s length is specifically the distance from the pivot point to the center of mass, and not the vertical distance from top to bottom. This is due to the moment of inertia of the pendulum. As observed in rotational motion, an object rotating with mass dispersed towards the outer edges will slow down the speed of the rotations. This is the same concept with a pendulum. When the center of mass is closer to the pivot point, the length is therefore less and the period also decreases. If this distinction is not made, the overall length of a pendulum from top to bottom could vary in periods

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Andrew TealAP Physics 1

26 March 2015for different pendulums, as the center of masses determine the true length, and this can vary depending on materials and design of the pendulum. In essence, length of a pendulum is measured from the pivot point to the center of mass, as the moment of inertia is critical to this calculation and the way that mass is dispersed can have a great effect on the speed and period of the pendulum when in motion.

Three different variables were measured when testing the period of a pendulum. The most direct relationship found for the pendulum’s period is the square root of the pendulum’s length, and mass and amplitude were found to be inconsequential in determining period. Mass and amplitude both do not have any real effects on the period, and this is proven through the data analysis graphs. For both of these variables, the line of best fit reveals almost no slope at all, and the periods only varied slightly with changes in the variables as due to experimental error. Had either of these variables been important to period, their graphs would show a relationship to prove so.

Pendulum Design

The antique store owner needs a pendulum of length 1.088 m, as this provides a period of 1.875 seconds.

Because amplitude and mass do not matter for the period, the materials that are used for the pendulum do not truly matter, but a stainless steel is recommended for its price and longevity. Titanium should be used for the shaft, as it is much lighter and allows for easier adjustments of the center of mass. This can have any mass necessary, as it doesn’t really matter. For the amplitude, or angle at which the pendulum swings, it is recommended to use an angle of 10 degrees, as this is a moderate level of swinging that is noticeable but safe and within the data tested for amplitudes. Again, as with mass, the amplitude doesn’t actually affect the period, so any angle is acceptable, but the length and mass tests have both been run at 10 degrees, and this is therefore the recommended angle of use.

Below is a sketch of the designed pendulum for the antique dealer. It includes necessary measurements and a screw system for adjustments. The stylish screws on each side of the pendulum can be loosened to shift the pendulum up or down its shaft, thus shifting its center of mass and consequentially varying the length and period of the pendulum’s movement.

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Andrew TealAP Physics 1

26 March 2015

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