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Simple Pendulum Manual

lab notes for chap 4
Course

Introductory Physics II (PHYS 1002H)

17 Documents
Students shared 17 documents in this course
Academic year: 2021/2022
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Trent University

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1. Simple Pendulum Experiment

Introduction

In this experiment, you will learn the physics of the simple pendulum. You will verify the relationship

between the period of a pendulum with the length of the string and the mass of the bob.

Theory

A simple pendulum consists of a bob that is suspended by a string and allowed to swing freely from a

pivot point. Neglecting air resistance, the bob is subject to the force due to gravity, 𝐹 𝑔

= 𝑚𝑔, and to the

force due to the tension in the string, 𝐹

𝑇

. See Figure 1.

When released from a horizontal displacement 𝑥, the pendulum oscillates about the equilibrium position

(𝑥 = 0 , 𝜃

0

= 0 ) as a result from gravity’s restoring force acting on it. The period of oscillation, 𝑇, is

defined as the time it takes the pendulum to complete a full cycle (back and forth).

Amplitude of oscillation

The amplitude of oscillation, 𝜃 0

, can be related to the length of

the pendulum, 𝐿, and to the horizontal displacement, 𝑥, of the

bob using simple trigonometry

𝜃

0

= sin

− 1

(

𝑥

𝐿

) ( 1. 1 )

When oscillating at a small angle, the pendulum follows a

harmonic motion and its angular displacement, 𝜃(𝑡), is given by

𝜃

(

𝑡

)

= 𝜃

0

cos (

2 𝜋𝑡

𝑇

)

( 1. 2 )

where 𝜃 0

is the amplitude of the oscillation, 𝑡 is the time, and 𝑇

is the period.

Figure 1. Simple pendulum diagram.

Period of oscillation

For small oscillations (𝜃

0

≪ 60°), the period of the pendulum can be approximated by

𝑇 ≈ 2 𝜋√

𝐿

𝑔

( 1. 3 )

where 𝐿 is the length of the string and 𝑔 is the gravitational acceleration. As you can see, the period is

independent of the mass of the bob.

Apparatus

As the bob swings freely from a pivot point, one could

measure its period of oscillation using a photogate. This

instrument records the time at which an object blocks and

unblocks an infrared beam propagating across the device.

If the diameter, 𝑑, of the bob is known, it is possible to

determine the pendulum’s velocity at the equilibrium point

from the time it takes the bob to cross the infrared beam

(𝑣 = 𝑑/𝑡).

The length, amplitude angle, and the vertical displacement

of the pendulum can be determined by measuring the

horizontal 𝑥 displacement with a meter stick.

Procedure

  1. Measure the mass (𝑚 ± 𝜎 𝑚

) of each of the bobs. Determine their material based on their color. Record

the values in your write-up. Don’t forget to give the table where you’re recording them in a caption.

A. Length effect on the period of oscillation

  1. Setup your apparatus as shown in Figure 2.

✓ Use the “wrist” of the clamp to hold the string.

✓ Adjust the height and position of the clamps to make sure that the holes of the photogates (inner

part) are at the same level as the bob.

Figure 2. Experimental setup for the

simple pendulum experiment.

B. Mass effect on the period of oscillation

  1. Change the pendulum’s bob and, this time, record the average period of oscillation, 𝑇, for only one

length: 30cm. Use x = 10cm as your amplitude. This time you will also need to record the standard

deviation, 𝜎

𝑆𝐷

, and the number of samples, 𝑁 from the STAT panel:

✓ Determine the uncertainty, 𝜎

𝑇

, on the average period by calculating using the following

expression:

𝜎

𝑇

=

𝜎

𝑆𝐷

√

𝑁

( 1. 4 )

  1. Go to Analyze/Interpolation calculator. Enter your measured length and record the

corresponding extrapolated period, 𝑇

𝑒

.

✓ Don’t forget to click OK, so that the panel with the extrapolation gets displayed on your graph.

  1. In order to determine the uncertainty, 𝜎

𝑇 𝑒

, on the extrapolated period value repeat step 9 for length

values of 𝐿 − 𝜎

𝐿

and 𝐿 + 𝜎

𝐿

.

✓ Record your findings in your write-up.

✓ Estimate the uncertainty, 𝜎

𝑇

𝑒

, by calculating the difference between the obtained values and

dividing it by 2. Show this calculation:

𝜎

𝑇

𝑒

=

|

𝑇

2

− 𝑇

1

|

2

( 1. 5 )

  1. Verify with a demonstrator (your TA or instructor) that your formatting of the Analysis file is done

correctly for printing.

✓ Don’t forget to scale the graph to take up most of the space on the page ( ) and make sure

that extrapolation and curve fit panels do not interfere with the graph.

  1. Print data table and graph (File/Print...). Select Print Footer. Enter your name and student

number in Name option. Enter your workstation number in Comment. Make sure you’re printing

using the landscape option.

✓ Save a copy of the file: File/Save as; you can keep in the Downloads folder to save to

the Desktop.

✓ Remember to give your graph and table a descriptive hand-written caption once you receive

the printed copy.

Data Analysis

A. Length effect on the period of oscillation

  1. We would like to verify whether Equation ( 1. 3 ) a good representation for your data trend. One way to

do that is to compare the value for acceleration due to gravity, 𝑔, that is generated in the information

panel of the curve fit on your graph with the theoretical one of

(

981 ± 1

)

cm/s

2

. This comparison is

done using a consistency test:

∆

𝜎

∆

=

|𝑔 − 981 |

√𝜎

𝑔

2

+ 1

≤ 2

( 1. 6 )

If

∆

𝜎

∆

⁄ ≤ 2 , the two values are consistent. If

∆

𝜎

∆

⁄ > 2 , then they are inconsistent.

B. Mass effect on the period of oscillation

  1. Compare the interpolated value of the oscillation period (𝑇 𝑒

± 𝜎

𝑇

𝑒

) with the measured one (𝑇 ± 𝜎

𝑇

)

via the consistency test.

∆

𝜎

∆

=

|𝑇

𝑒

− 𝑇|

√

𝜎

𝑇

𝑒

2

+ 𝜎

𝑇

2

≤ 2

( 1. 7 )

Discussion

Your Discussion should be in a form of a short essay that provides a critique of the experiment. As part

of that essay you should address the following questions:

  1. Based on your comparison test in Step 13, what can you conclude about the accuracy of Equation

( 1. 3 ) in representing the dependence of the pendulum period on its length?

  1. What conclusions can you draw about period dependence on the mass of the pendulum bob? Is this

consistent with the predictions based on the theory?

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Simple Pendulum Manual

Course: Introductory Physics II (PHYS 1002H)

17 Documents
Students shared 17 documents in this course

University: Trent University

Was this document helpful?
BIT 1007 Winter 2023
1
1. Simple Pendulum Experiment
Introduction
In this experiment, you will learn the physics of the simple pendulum. You will verify the relationship
between the period of a pendulum with the length of the string and the mass of the bob.
Theory
A simple pendulum consists of a bob that is suspended by a string and allowed to swing freely from a
pivot point. Neglecting air resistance, the bob is subject to the force due to gravity, 𝐹𝑔=𝑚𝑔, and to the
force due to the tension in the string, 𝐹𝑇. See Figure 1.
When released from a horizontal displacement 𝑥, the pendulum oscillates about the equilibrium position
(𝑥=0,𝜃0=0) as a result from gravity’s restoring force acting on it. The period of oscillation, 𝑇, is
defined as the time it takes the pendulum to complete a full cycle (back and forth).
Amplitude of oscillation
The amplitude of oscillation, 𝜃0, can be related to the length of
the pendulum, 𝐿, and to the horizontal displacement, 𝑥, of the
bob using simple trigonometry
𝜃0=sin−1(𝑥
𝐿)
(1.1)
When oscillating at a small angle, the pendulum follows a
harmonic motion and its angular displacement, 𝜃(𝑡), is given by
𝜃(𝑡)= 𝜃0cos(2𝜋𝑡
𝑇)
(1.2)
where 𝜃0 is the amplitude of the oscillation, 𝑡 is the time, and 𝑇
is the period.
Figure 1. Simple pendulum diagram.

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