In the arrangement shown in the figure below, an object of mass m = 4.0 kg hangs

Question

In the arrangement shown in the figure below, an object of mass m = 4.0 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. (Ignore the mass of the vertical section of the cord.)

(a) When the vibrator is set to a frequency of 130 Hz, a standing wave with six loops is formed. What must be the linear mass density of the cord?
kg/m

(b) How many loops (if any) will result if m is changed to 9.00 kg? (Enter 0 if no loops form.)
loops

(c) How many loops (if any) will result if m is changed to 14 kg? (Enter 0 if no loops form.)
loops

Explanation / Answer

for a standing wave 6 loops are formed,6(lambda/2) = l

lambda = l/3 = 2/3 m

the spped of the waves in the string

v = lambda*f = 2/3*130 = 86.7 m/sec

the tension in the string f = mg = 4*9.8 = 39.2 N

v = sqrt(f/mue)

mue = f/v^2 = 39.2/86.7^2 = 5.2*10^-3 kg/m

b) f = mg = 9*9.8 = 88.2 N

v = sqrt(88.2/ 5.2*10^-3) = 130.24 m/sec

lambda = v/f =130.24/130 = 1.002 m

no of loops n = l/lambda/2 = 2*2/1.002 = 3.99 = 4

c)m = 14 kg

f = mg = 14*9.8 = 137.2 N

v = sqrt(137.2/ 5.2*10^-3) = 162.4 m/sec

lambda = 162.4/130 = 1.25 m

no of loops = n = l/lambda/2 = 3.2

its not an integer

no standing wave is formed

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