1. Home
  2. General physics ii phy2202ca
  3. Oscillatory motion and waves
  4. The simple pendulum

1.4 The simple pendulum  (Page 2/3)

<< Chapter < Page
  General physics ii phy2202ca     Page 2 / 3
Chapter >> Page >

Measuring acceleration due to gravity: the period of a pendulum

What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?

Strategy

We are asked to find g size 12{g} {} given the period T size 12{T} {} and the length L size 12{L} {} of a pendulum. We can solve T = 2π L g size 12{T=2π sqrt { { {L} over {g} } } } {} for g size 12{g} {} , assuming only that the angle of deflection is less than 15º size 12{"15º"} {} .

Solution

  1. Square T = 2π L g size 12{T=2π sqrt { { {L} over {g} } } } {} and solve for g size 12{g} {} :
    g = 4π 2 L T 2 . size 12{g=4π rSup { size 8{2} } { {L} over {T rSup { size 8{2} } } } } {}
  2. Substitute known values into the new equation:
    g = 4π 2 0 . 75000 m 1 . 7357 s 2 . size 12{g=4π rSup { size 8{2} } { {0 "." "75000"" m"} over { left (1 "." "7357 s" right ) rSup { size 8{2} } } } } {}
  3. Calculate to find g size 12{g} {} :
    g = 9 . 8281 m / s 2 . size 12{g=9 "." "828"m/s rSup { size 8{2} } } {}

Discussion

This method for determining g g can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation sin θ ≈ θ size 12{"sin"θ approx θ} {} to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about 0.5º size 12{0 "." 5°} {} .

Making career connections

Knowing g g can be important in geological exploration; for example, a map of g size 12{g} {} over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.

Take home experiment: determining g g

Use a simple pendulum to determine the acceleration due to gravity g g in your own locale. Cut a piece of a string or dental floss so that it is about 1 m long. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). Starting at an angle of less than 10º size 12{"10"°} {} , allow the pendulum to swing and measure the pendulum’s period for 10 oscillations using a stopwatch. Calculate g size 12{g} {} . How accurate is this measurement? How might it be improved?

An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10 kg size 12{"10"`"kg"} {} . Pendulum 2 has a bob with a mass of 100 kg size 12{"100"`"kg"} {} . Describe how the motion of the pendula will differ if the bobs are both displaced by 12º size 12{"12"°} {} .

The movement of the pendula will not differ at all because the mass of the bob has no effect on the motion of a simple pendulum. The pendula are only affected by the period (which is related to the pendulum’s length) and by the acceleration due to gravity.

Phet explorations: pendulum lab

Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It’s easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of g on planet X. Notice the anharmonic behavior at large amplitude.

Pendulum Lab

Section summary

  • A mass m size 12{m} {} suspended by a wire of length L size 12{L} {} is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about 15º size 12{"15"°} {} .

    The period of a simple pendulum is

    T = 2π L g , size 12{T=2π sqrt { { {L} over {g} } } } {}

    where L size 12{L} {} is the length of the string and g is the acceleration due to gravity.

Conceptual questions

Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.

Problems&Exercises

As usual, the acceleration due to gravity in these problems is taken to be g = 9.80 m / s 2 , unless otherwise specified.

What is the length of a pendulum that has a period of 0.500 s?

6.21 cm

Some people think a pendulum with a period of 1.00 s can be driven with “mental energy” or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?

What is the period of a 1.00-m-long pendulum?

2.01 s

How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?

The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?

2.23 Hz

Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?

(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is 9 . 79 m/s 2 size 12{9 "." "79"`"m/s" rSup { size 8{2} } } {} is moved to a location where it the acceleration due to gravity is 9 . 82 m/s 2 size 12{9 "." "82"`"m/s" rSup { size 8{2} } } {} . What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.

(a) 2.99541 s

(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by ( 0 . 01 ) 2 = 0 . 01% size 12{ \( 0 "." "01" \) rSup { size 8{2} } =0 "." "01"%} {} so it is necessary to have at least 4 digits after the decimal to see the changes.

A pendulum with a period of 2.00000 s in one location ( g = 9 . 80 m/s 2 size 12{g=9 "." "80"`"m/s" rSup { size 8{2} } } {} ) is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?

(a) What is the effect on the period of a pendulum if you double its length?

(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?

(a) Period increases by a factor of 1.41 ( 2 size 12{ sqrt {2} } {} )

(b) Period decreases to 97.5% of old period

Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is 1 . 63 m/s 2 size 12{1 "." "63"`"m/s" rSup { size 8{2} } } {} .

At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is 1 . 63 m/s 2 size 12{1 "." "63"`"m/s" rSup { size 8{2} } } {} , if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon.

Slow by a factor of 2.45

Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.

If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?

length must increase by 0.0116%.

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, General physics ii phy2202ca. OpenStax CNX. Jul 05, 2013 Download for free at http://legacy.cnx.org/content/col11538/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'General physics ii phy2202ca' conversation and receive update notifications?

Ask