phys 275 experiment 8 standing waves on a string · • today we will study oscillations in a...

PHYS 275 – Experiment 8 Standing Waves on a String

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PHYS 275 – Experiment 8

Standing Waves on a String

Experiment Summary

• Today we will study oscillations in a string under tension

– The physics is a bit more complicate than last weeks: we will study standing waves

– The experimental apparatus consists of a string under tension (a weight is applied to it); we will measure the frequency of its oscillations

• Technical topic of the week: weighed averages

– We will combine measurements using weights, which will allow us to improve the precision of our estimate of the quantity of interest

PHYS 275 - Experiment 8 2

Standing-waves Theory

• Standing (or

stationary) waves are

distinguished using

their mode number n

– Node = location

where the amplitude

is zero

– Mode number n =

number of nodes + 1

– Wavelength l = 2L/n

PHYS 275 - Experiment 8 3

Page 4: PHYS 275 Experiment 8 Standing Waves on a String · • Today we will study oscillations in a string under tension ... Standing-waves Theory • Standing (or stationary) waves are

More Theory

• The speed and frequency of a standing wave are related:

– 𝑣𝑠 = 𝜆𝑓 =𝑇

𝜇

• T is the tension of the string, measured in Newton

• 𝜇 is the linear mass density of the string (mass/length), measured in kg/m

• 𝜆 is the wavelength, 𝜆 =2𝐿

𝑛• 𝑓 is the wave frequency,

measured in …

• 𝑣𝑠 is the wave speed

PHYS 275 - Experiment 8 4

𝑓𝑛𝑛=

2𝐿 𝜇𝑇1/2

This is a form of

dispersion relation

Page 5: PHYS 275 Experiment 8 Standing Waves on a String · • Today we will study oscillations in a string under tension ... Standing-waves Theory • Standing (or stationary) waves are

Measurements

• We will measure the frequency corresponding to different modes, and combine these measurements to obtain the best estimate of the linear mass density– The idea is that when you combine measurements with

different precision, you want to give more weight to the more precise ones

• Here are the main formulae– Let us assume that we have the measurements 𝑥1, 𝜎1 , 𝑥2, 𝜎2 , …, 𝑥𝑛, 𝜎𝑛

– We take the weight of each measurement to be 𝑤𝑖 =1

𝜎𝑖2

– Finally: 𝑥 = 𝑖=1𝑛 𝑤𝑖𝑥𝑖

𝑖=1𝑛 𝑤𝑖

= 𝑖=1𝑛

𝑥𝑖𝜎𝑖2

𝑖=1𝑛

1𝜎𝑖2; 𝜎 𝑥 =

𝑖=1𝑛 𝑤𝑖

𝑖=1𝑛

1𝜎𝑖2

PHYS 275 - Experiment 8 5

Page 6: PHYS 275 Experiment 8 Standing Waves on a String · • Today we will study oscillations in a string under tension ... Standing-waves Theory • Standing (or stationary) waves are

Experimental Apparatus

PHYS 275 - Experiment 8 6

Page 7: PHYS 275 Experiment 8 Standing Waves on a String · • Today we will study oscillations in a string under tension ... Standing-waves Theory • Standing (or stationary) waves are

Some Suggestions

PHYS 275 - Experiment 8 7

Page 8: PHYS 275 Experiment 8 Standing Waves on a String · • Today we will study oscillations in a string under tension ... Standing-waves Theory • Standing (or stationary) waves are

More Suggestions

PHYS 275 - Experiment 8 8

Page 9: PHYS 275 Experiment 8 Standing Waves on a String · • Today we will study oscillations in a string under tension ... Standing-waves Theory • Standing (or stationary) waves are

(Yet) More Suggestions

PHYS 275 - Experiment 8 9

Page 10: PHYS 275 Experiment 8 Standing Waves on a String · • Today we will study oscillations in a string under tension ... Standing-waves Theory • Standing (or stationary) waves are

Notes and Reminders

• Submit your Excel spreadsheet on ELMS and

turn in your check sheet before leaving the lab

• Complete the final version of your report by

1pm next week

• Finish the homework set in Expert-TA by 2pm

next week

• Turn off your equipment and clean up your

bench area before leaving the classroom

• Save your data on the local disk frequently!

PHYS 275 - Experiment 8 10

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