A string of mass m_string and length L is attached to a vibrating box on one end

Question

A string of mass m_string and length L is attached to a vibrating box on one end and a mass m_end via a pulley on the other. The mass pulls the string taut and the box vibrates at a frequency f, creating the standing wave as shown. The waves setup on the string can be described by the wave function: Psi(x, t) = A sin (kx - omega t). A = 0.060 m f = 60.0 Hz m_string = 0.06 kg L = 4.0 m Y = 20 times 10^10 Pa a. How many nodes are set up on the string? How many antinodes are set up on the string? How many wavelengths are set up on the string? Which normal mode (which harmonic) is set up on the string? b. Find the end mass (m_end) necessary to setup the standing wave shown. c. Calculate the average power carried by the wave. d. Find an expression for the transverse speed |v_Psi| of a particle in the string. Use this to find the transverse speed of a particle in the string when t = 3.0 s and x = 0.26 m.

Explanation / Answer

a. 5 nodes

4 antinodes

2 wavelengths

3rd harmonic

(b) wavelength = L / 2 = 2 m

f= 60 Hz

v = 2 x 60 = 120 m/s

and v = sqrt[ T / (m / L)]

120^2 = T / (0.06 / 4)

T = 216 N

and T = m g

m = 216 / 9.8 = 22.04 kg ........Ans

(c) P = mu w^2 A^2 v / 2

= (0.06 / 4) (2 x pi x 60)^2 (0.06^2) (120) / 2

= 460.5 W

(d) V = A w cos(kx - wt)

v = 22.62 m/s cos(pi x - 120pi t) ......Ans

putting x = 0.26 m and t = 3 s

v =22.62 cos(-1130.16 rad) = 15.5 m/s ......Ans

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