<< Chapter < Page Chapter >> Page >
Calculate the energy in a wave on a string.

Energy transport

Lets calculate the energy in a wave of a string:

Consider a fragment of string so small it can be considered straight, as is shown in the figure

The the kinetic energy is K = 1 2 m v 2 for the string fragment m = μ x

Why is this x and not s ? Lets consider that the string is not perturbed, then it is horizontal and hasmass as given. When the string is perturbed it stretches a little bit - but the mass does not increase.

So we have K = 1 2 μ x ( y t ) 2 and using this we can define the energy per unit length, ie. the kinetic energy density: K x = 1 2 μ ( y t ) 2 When the string segment is stretched from the length x to the length s an amount of work = T ( s x ) is done. This is equal to the potential energy stored in the stretched string segment. So the potential energy in this case is: U = T ( s x ) Now s = ( x 2 + y 2 ) 1 / 2 = x [ 1 + ( y x ) 2 ] 1 / 2 Recall the binomial expansion ( 1 + A ) n = 1 + n A + n ( n 1 ) A 2 2 ! + n ( n 1 ) ( n 2 ) A 3 3 ! + so s x + 1 2 ( y x ) 2 x U = T ( s x ) 1 2 T ( y x ) 2 x or the potential energy density U x = 1 2 T ( y x ) 2 To get the kinetic energy in a wavelength, lets start with y = A sin ( 2 π x λ ω t ) y t = ω A cos ( 2 π x λ ω t ) Lets evaluate it at time 0. y t | t = 0 = ω A cos ( 2 π x λ ) so K x = 1 2 μ ω 2 A 2 cos 2 ( 2 π x λ ) now integrate K = 0 λ K x x = 1 2 μ ω 2 A 2 0 λ cos 2 ( 2 π x λ ) x In order to do this integral we use the following trig identity: cos 2 A = cos 2 A + 1 2 so we get K = 1 2 μ ω 2 A 2 [ x 2 + λ 8 π sin 4 π x λ ] | x = 0 λ = 1 4 μ λ ω 2 A 2

In similar fashion the potential energy can be found to be U = 1 4 μ λ ω 2 A 2 . Deriving this will be assigned as a homework problem

So E = K + U = 1 2 μ λ ω 2 A 2 Power P = Δ E Δ t = 1 2 μ λ ω 2 A 2 τ = 1 2 μ ω 2 A 2 v Where I have used τ = 1 / ν and λ ν = v thus τ = λ / v

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Pdf generation problem modules. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10514/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Pdf generation problem modules' conversation and receive update notifications?

Ask