A transverse wave travelling along a string given by: where the units within the

Question

A transverse wave travelling along a string given by:

where the units within the parentheses are meters and seconds

a) Verify that this expression is a solution for the wave equation

b) Determine the amplitude, wavelength and period of the wave.

c) Determine the wave phase velocity.

d) Find the transverse displacement and transverse velocity of the string at z=5m and t=0.1sec.

e) Given that tension on the string is supplied by a pulley with a weight attached to a pulley. The weight has a mass 10 times the mass of the string, calculate the length of the string

Explanation / Answer

a) Wave equation :

y(z,t) = A cos(wt - kx)

where A = 0.5m

w = 50 pi

and k = 4pi

b)Amplitude, A = 0.5m   .....Ans

k = 2pi / lambda

Wavelength = 2pi / (4pi) = 0.5 m .....Ans

time period= 2pi / w = 2pi / 50pi

T = 0.04 s

c) phase velocity = w / k = 50pi/ 4pi = 12.5 m/s

d) displacement y (5,0.1) = 0.5 cos[ (50 x pi x 0.1) - (4pi x 5) ]

y = 0.5 cos(-15pi) = - 0.5 m

v = dy/dt = - 0.5 x 50pi sin[ 50pi t - 4 pi z ]

at t = 0.1s and z = 5 m

v = - 25pi sin[ (50 pi 0.1) - (4 pi 5 )]

v = - 25 pi sin(-15pi) = 25 pi m/s Or 78.54 m/s

e) phase velocity = sqrt [ tension / linear mass density ]

suppose mass of string is m and length is L.

then Tension = 10m

linear mass density = m / L

12.5 = sqrt[ 10m / (m/L)]

156.25 = 10 L

L = 15.625 meter

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