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Data Analysis Assignment: Pendulum

You will measure the period of a pendulum and value of gravity, switching out pieces of the setup for new data runs (e.g., swap a metal ball). Record your data, compute means and errors for each data set, and compare your results to the theory for each setup you test. Equipment’s:

1. Pendulum PHET Simulation: http://phet.colorado.edu/sims/html/pendulum- lab/latest/pendulum-lab_en.html

2. Meter Stick: within the Simulation 3. Stopwatch: Cell phone 4. Protractor: within the Simulation 5. Mass: within the Simulation

Theory: A simple pendulum consists of a mass m suspended on a massless string of length L. The string is connected to a fixed point above the mass. The only forces on the mass are gravity and the string tension. The period of this simple pendulum for small oscillations is

(1) In real pendula, the string always has a mass, but as long as the mass of the string is much smaller than the mass hanging, it will be very close to the idealistic case of the simple pendulum model. Setup: Pendulum for 2 different masses and at two angles (100 and 450 ).

Procedure and Data Collection 1. Record the instruments. Before doing anything else, record the precision errors of your

measurement instruments:

TOOL PRECISION ERROR METER STICK DIGITAL STOPWATCH

PROTRACTOR

2. Weigh objects: A single measurement for each will do.

OBJECT MASS ± PRECISION ERROR MASS 1 MASS 2

Precision error here will be the 0.1 kg(from the simulation) 3. Experimental procedure:

a) Let the string come to rest. This is the equilibrium position. b) Lift the mass to the appropriate angle (10° or 45°) from equilibrium, then release.

http://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html

c) Start timing with your stopwatch. Keep timing for the required number of periods, then stop your stopwatch at the end of the last period.

4. Measure pendulum length. Measure and record the length of the pendulum from the pivot to the center of the mass.

Pendulum Length Data and Results (String, Mass 1)

Trial #

Pendulum Length,

(units: ______) 1 2 3 4 5

Mean of length = Standard deviation = Standard error of length = (random error) Result with standard error:

5. Measure and record single and multiple periods. For this part of only, you’ll first

investigate whether measuring a single pendulum period multiple times or multiple periods fewer times produces the best results.

Individual periods (Mass 1, 10° Amplitude)

Trial #

Single Period Time

(units: ____)

1 2 3 4 5

Mean of single period time = Standard deviation = Standard error of single period time = (random error) Result with standard error:

Now use the stopwatch to record the time it takes the pendulum to swing 10 full periods once. Calculate the time it takes for a single period of oscillation, dividing the results by 10 (value and error) as appropriate to obtain your measured value for a single period.

10 full periods (Mass 1, 10° Amplitude)

Trial # Ten Period

Times (units: _____)

10- period time result with error: Single period time result with standard error:

Comparing the different ways you’ve just measured 10 periods, answer which one:

• is the most time efficient? • gives the best sample population statistics?

6. Analysis. For each 10 full swing trial in the table, divide the results by 10 to obtain your

measured value for a single period. Be careful to calculate the proper error on these single periods, remembering that the total number of significant figures stays the same. (Remember you increase the precision in your period measurement when use multiple periods. You have the same total number of significant figures. So if the time recorded for 10 swings of a trial is 7.84s, then the period for a single swing will be 0.784s).

Pendulum Period: Case 1 (Mass 1, 10° Amplitude)

Tri al #

10-Period Time (units: ______)

Single Period Time (T) (units: ______)

1 2 3 4 5

Mean of single period time = Standard deviation = Standard error of single period time =

Result with standard error:

Compare results to theory. Use the measured pendulum length and m/s2 to calculate the theoretically expected pendulum period, then compare to your experimental results. Ignore any error in the theoretical expectation at this point. Theoretical Expectation ( ) = Experimental Result: Case 1 (with Error)= Do they agree?

Analyze sources of error:

Pendulum Period: Case 2 (Mass 1, 45° Amplitude)

Tri al #

10-Period Time (units: ______)

Single Period Time (T) (units: ______)

1 2 3 4 5

Mean of single period time = Standard deviation = Standard error of single period time =

Result with standard error:

Theoretical Expectation ( ) = Experimental Result: Case 2 (with Error)= Do they agree? Analyze sources of error: Measure Length: (Use the ruler in the simulation to measure the length of the pendulum)

Pendulum Length Data and Results (Mass 2)

Trial #

Pendulum Length, L2

(units: ______) 1 2 3 4 5

Mean of length = Standard deviation = Standard error of length = (random error) Result with standard error: L2 =

Pendulum Period: Case 3 (String, Mass 2, 10° Amplitude) Tri al #

10-Period Time (units: ______)

Single Period Time (T) (units: ______)

1 2 3 4 5

Mean of single period time = Standard deviation = Standard error of single period time =

Result with standard error:

Theoretical Expectation ( ) = Experimental Result: Case 4 (with Error)= Do they agree? Analyze sources of error:

Pendulum Period: Case 4 (Mass 2, 45° Amplitude) Tri al #

10-Period Time (units: ______)

Single Period Time (T) (units: ______)

1 2 3 4 5

Mean of single period time = Standard deviation = Standard error of single period time =

Result with standard error:

Theoretical Expectation ( ) =

Experimental Result: Case 4 (with Error)= Do they agree? Analyze sources of error: 7. Analysis: Find g?

Considering the period of a simple pendulum for small oscillations:

Uncertainty in g is given by: Where is error in L and is error in T Compute g with error

Case 1 Case 2 Case 3 Case 4

Mass1 Amplitude:

10°

Mass1 Amplitude:

45°

Mass 2 Amplitude:

10°

Mass 2 Amplitude:

45° Measured Length (m)

Measured Period (s)

Experimental (m/s2)

Result: (m/s2)

Compare experimental quantity to accepted value:

“Agreement Test” Two values and , which are expected to be equal, are said to reasonably “agree” if

Accepted value, m/s2.

Do and agree? (Yes/No) Case 1: Case 2: Case 3: Case 4: 8. Determine what factors the frequency of oscillation depend on. The cyclic frequency is found with the formula f = N/t, where f is the cyclic frequency, N is the number of oscillations, and t is the time interval. Given a measured number-of-oscillations-with- error N ± δN (where δN is the counting error that will be ) and a fixed time-interval t, using

relative frequency , find the error in the cyclic frequency δf. 1) Fill in the table below (keep the length and amplitude fixed).

Trial 1 2 3 4 Time Interval (s) (t)

2 10 20 30

Number of Oscillations

(N)

Cyclic Frequency (Hz)

(f = N/t)

Relative Error

Error in Cyclic Frequency (δf) (Hz)

From the table what do you think, which time interval gives you better results and why?

Dependence of Frequency on Amplitude b) Fill in the data table below (keep the length and mass fixed).

Desired Amplitude (degrees)

Error (degrees)

Number of Oscillations

(N)

Time

Interval (s) (t)

Cyclic Frequency (Hz)

(f = N/t)

c) Based on your data, try to describe how frequency depends on amplitude. Dependence of Frequency on Mass a) Fill in the data table below (keep the length and amplitude fixed).

Mass (m) (kg)

Error (δm)(kg)

Number of Oscillations

(N)

Time

Interval (s) (t)

Cyclic Frequency (Hz)( f =

N/t)

b) Based on your data, try to describe how frequency depends on mass. Dependence of Frequency on Length Finally, we investigate how the frequency depends on the length of the string.

Desired String Length

(l) (m)

Error (δl) (m)

Number of Oscillations

(N)

Time

Interval (s) (t)

Cyclic Frequency

(Hz) ( f = N/t)

gexp

(g= 4π2lf2)

c) Based on your data, try to describe how frequency depends on length. 9. Find g using F

Frequency is inverse of Time period therefore frequency can be written as: Find:

a) The average value of g from table above: b) Standard deviation in average g: c) Standard error in average g:

Result g ± Standard error in g: Compare experimental quantity to accepted value ( m/s2):

Use agreement test ( )

Procedure and Data Collection