In the arrangement shown in Figure P14.40, an object of mass, m = 6.0 kg, hangs

Question

In the arrangement shown in Figure P14.40, an object of mass, m = 6.0 kg, hangs from a cord around a light pulley. The length of the cord between point P and the pulley is L = 2.0 m.

(a) When the vibrator is set to a frequency of 145 Hz, a standing wave with six loops is formed. What must be the linear mass density of the cord?
in kg/m

(b) How many loops (if any) will result if m is changed to 216 kg?
in loops

(c) How many loops (if any) will result if m is changed to 8.64 kg?
in  loops

Explanation / Answer

(A) for six loop,

L = 3 lambda

lambda = 2/3 m ...(wavelength)

v = lambda f = (2/3) (145 Hz) = 96.7 m/s

and v = sqrt[ T / mu] and T = m g

(96.7)^2 = (6 x 9.8) / mu

mu = 6.29 x 10^-3 kg / L .....Ans

(B) v = sqrt[ m g / mu ]

f = v / lambda = constant

so sqrt(m) / (2 L / n) = constant

sqrt( m ) n = constant

sqrt(6) x 6 = sqrt(216) n

n = 1 loop

(C) sqrt(6) x 6 = sqrt(8.64) n

n = 5 loops

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