Pendulum Motion Lab: Determining Period and Amplitude Dependence | Lecture notes Reasoning

Pendulum Motion Lab 1 – Spring 2021

PENDULUM MOTION:

MOVING AWAY FROM SIMPLE MOTION

Objective: • Determine how the period of a pendulum depends on amplitude.

Prelab:

1. Read this handout.

2. Consider the two pairs of data points shown at right with uncertainty bars. Is

pair A or pair B more likely to come from a measurement of the same value?

Explain.

3. Students A and B are both trying to measure the period of a pendulum. Both

students agree that the uncertainty in measuring the period is 0.6 s using their

chosen measurement apparatus. Student A measures one swing of the

pendulum and records a time of 3.4 ± 0.6 s. Student B measures twenty

swings of the pendulum and records a time of 74.2 ± 0.6 s. Which student

has measured the more precise period? Describe your reasoning.

4. How do you use the LINEST function in Excel? Write the command that

you would enter and what keyboard combination you need to press. Know

how to find this information each week.

Apparatus: Stopwatch, string, mass, clamp, meter sticks, protractor

Introduction: A simple pendulum is an idealization consisting of a point mass, swinging at

the end of a massless string. In class you have learned that for small angles of swing, the period

of a simple pendulum is approximately given by the formula

, (1)

where

ℓ

is the length of the string and g is the acceleration of gravity. As the amplitude of

oscillation approaches 0, the period of a simple pendulum approaches the value given by Eq.

1. In this lab we will investigate what happens to the period when the small angle

approximation is not valid.

Part I: Measuring the period of oscillation

1. Suspend the pendulum from the apparatus. Displace the mass from equilibrium and let it

swing. From your observations, select a pendulum length and displacement angle that

would allow you to reasonably measure the period of one oscillation.

2. Use the stopwatch to time one complete oscillation. Repeat for a total of 10 trials.

Calculate the average period, the standard deviation, and the standard error. Write your

result in standard form.

3. Start the mass oscillating again, but now time 10 complete oscillations. Estimate the

uncertainty in the time. Determine the period with uncertainty and write the result in

standard form.

4. Compare the two methods used to determine the period. Is one better than the other?

Explain. Discuss your thoughts with your instructor.

T=2



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Pendulum Motion Lab 1 – Spring 2021 PENDULUM MOTION: MOVING AWAY FROM SIMPLE MOTION Objective: • Determine how the period of a pendulum depends on amplitude. Prelab:

Read this handout.
Consider the two pairs of data points shown at right with uncertainty bars. Is pair A or pair B more likely to come from a measurement of the same value? Explain.
Students A and B are both trying to measure the period of a pendulum. Both students agree that the uncertainty in measuring the period is 0.6 s using their chosen measurement apparatus. Student A measures one swing of the pendulum and records a time of 3.4 ± 0.6 s. Student B measures twenty swings of the pendulum and records a time of 74.2 ± 0.6 s. Which student has measured the more precise period? Describe your reasoning.
How do you use the LINEST function in Excel? Write the command that you would enter and what keyboard combination you need to press. Know how to find this information each week. Apparatus: Stopwatch, string, mass, clamp, meter sticks, protractor Introduction: A simple pendulum is an idealization consisting of a point mass, swinging at the end of a massless string. In class you have learned that for small angles of swing, the period of a simple pendulum is approximately given by the formula , (1) where ℓ is the length of the string and g is the acceleration of gravity. As the amplitude of oscillation approaches 0, the period of a simple pendulum approaches the value given by Eq.
In this lab we will investigate what happens to the period when the small angle approximation is not valid. Part I: Measuring the period of oscillation
Suspend the pendulum from the apparatus. Displace the mass from equilibrium and let it swing. From your observations, select a pendulum length and displacement angle that would allow you to reasonably measure the period of one oscillation.
Use the stopwatch to time one complete oscillation. Repeat for a total of 10 trials. Calculate the average period, the standard deviation, and the standard error. Write your result in standard form.
Start the mass oscillating again, but now time 10 complete oscillations. Estimate the uncertainty in the time. Determine the period with uncertainty and write the result in standard form.
Compare the two methods used to determine the period. Is one better than the other? Explain. Discuss your thoughts with your instructor.

T = 2 π

Pendulum Motion Lab 1 – Spring 2021 Part II: Period as a function of Amplitude Procedure: Given the tools available, design an experiment that will allow you to determine the period T

as a function of angular amplitude q m , with minimal uncertainty in each quantity. To determine

q m , you may use the protractor or trigonometry. Measure the period of the pendulum for as

broad a range of angular amplitudes as practical. Plot T vs. q m including error bars for both T

and q m. Be sure to explain your methods, including estimates of error bars, in your notebook.

Later in your physics education you may be lucky enough to derive the relation where is the value given by eq. 1 and is the amplitude correction. The function

can be written as a power series in q m ,

Note that some of the a i may be zero. If q m is not too large ( ), the higher order terms

in the power series become progressively smaller very quickly and only the first non-zero term after the 1 is important. Determine the power of the first non-zero term after the 1 by plotting your data in such a way

that it will give a straight line if the first term is of order q m , q m 2 , or q m 3. See if you can

convince your instructor that the correction term is linear, quadratic, or cubic using your data. Determine the value of the leading non-zero coefficient, ( ) and its uncertainty. T = T 0 ⋅ Q ( θ m ) T 0 Q ( θ m ) Q ( θ m ) Q ( θ m ) = 1 + a 1 θ m + a 2 θ m^2 + a 3 θ m^3 +.... θ radians ≤ 1 a 1 , a 2 , or a 3

Pendulum Motion Lab: Determining Period and Amplitude Dependence, Lecture notes of Reasoning

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