The transverse displacement of a stretched string from equilibrium as a function

Question

The transverse displacement of a stretched string from equilibrium as a function of time and position is given by:
y=0.13 cos(3 x + 81 t). x and y are in m; t is in s.
True False Greater than Less than Equal to  The wave moves in the positive x direction
True False Greater than Less than Equal to  The period is ..... 0.1 seconds
True False Greater than Less than Equal to  The wavelength is ..... 1 m
True False Greater than Less than Equal to  The speed of the wave is ..... 27 m/s

Calculate the average power transmitted by the string. Data: mass of a 163 m long piece of the string is 2.35 kg

Explanation / Answer

from general equation for wave on a string

y(x,t) = A cos (kx-wt) for wave moving from +ve value to origin

A is amplitude

k is waveno. = 2pi/L

W is ang frequency = 2pi f

t is time period

so here from this y=0.13 cos(3 x + 81 t).

The wave moves in the positive x direction -----True

-------------------------------------------------

Wt = 2pif t = 81

f = 81/2pi

t = 2pi/81

t = 0.038 secs
The period is ..... 0.1 seconds is FALSE

--------------------------------------

as k = 2pi/L = 3

wavelength L = 2pi/3 = 6.28/3 = 2.09 m

The wavelength is ..... 1 m is FALSE

-----------------------------------------------------------------

Velocity V^2 = T/u

where T s tension = mg

u is mass per unit length

so

V^2 = 2.35* 9.81/(2.35/163)

V = 40 m/s

Power P = F v

P = 2.35* 9.8 * 40

P = 921.2 Watts

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