In the arrangement shown below, an object can be hung from a string (with linear

Question

In the arrangement shown below, an object can be hung from a string (with linear mass density ? = 0.002 00 kg/m) that passes over a light pulley. The string is connected to a vibrator (of constant frequency f), and the length of the string between point P and the pulley is L = 1.60 m. When the mass m of the object is either 9.0 kg or 16.0 kg, standing waves are observed; no standing waves are observed with any mass between these values, however.

(a) What is the frequency of the vibrator? (Note: The greater the tension in the string, the smaller the number of nodes in the standing wave.)

(b) What is the largest object mass for which standing waves could be observed?

Explanation / Answer

let

m1 = 9 kg

m2 = 16 kg

u = 0.00200 kg / m

here no masses between m1 and m2 produce standing waves

so here by using

2 * L * f / n = sqrt( m1 * g / u)

                                            ( m 2 > m1)

2 * L * f / n - 1 = sqrt(m2 * g / u )

then taking both equation

n - 1 / n = sqrt ( m 1 / m2 )

then by putting the values

n - 1 / n = sqrt ( 9 / 16)

then,

n-1 = 3   and   n = 4

so then putting the value of n in the equation

2 * L *f / n = sqrt( m1 * g / u )

f = (4 / 2 * 1.60 ) * sqrt ( 9 * 9.8 / 0.002 )

f = 262.5 Hz

the frequency of the vibrator is 262.5 Hz

b)

here by using

2 * L * f / n = sqrt ( m * g / u )

then for the standing wave n = 1

2 * L * f = sqrt( m * g / u )

m = 4 * L^2 * f^2 * u / g

m = (4 * 1.6^2 * 262.5^2 * 0.002) / 9.8

m = 144 kg

so the largest object mass for which standing waves could be observed is 144 kg

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