Question

One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at 120 Hz. The other end passes over a pulley and supports a 1.50-kg mass. The linear mass density of the rope is 0.0480 kg/m.(a) What is the speed of a transverse wave on the rope?(b) What is the wavelength?(c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 kg?

Answer 1

Answer:

(A) 14.7 N

(B) 0.15 m

(C) the speed will become 24.8 m/s while wavelength becomes 0.21 m

Explanation:

frequency (f) = 120 Hz

mass (m) = 1.5 kg

linear mass density (μ) = 0.0480 kg/m

acceleration due to gravity (g) = 9.8 m/s^{2}

(A) tension on the rope (T) = mg = 1.5 x 9.8 = 14.7 N

     speed = $\sqrt{\frac{T}{μ} }$

     speed =  $\sqrt{\frac{14.7}{0.0480} }$

     speed = 17.5 m/s

(B)  wavelength = velocity / frequency

     wavelength = 17.5 / 120 = 0.15 m

(C) when the mass are increased to 3.00 kg

    tension now becomes = mg = 3 x 9.8 = 29.4 N

   therefore speed =  $\sqrt{\frac{T}{μ} }$ =  $\sqrt{\frac{29.4}{0.048} }$ = 24.8 m/s

 wavelength now becomes = velocity / frequency = 24.8 / 120 = 0.21 m

hence the speed will become 24.8 m/s while wavelength becomes 0.21 m

Answer 2

This question involves the concepts of tension, frequency, and wavelength.

(a) The speed of the transverse wave on the rope is "17.5 m/s".

(b) The wavelength of the transverse wave on the rope is "0.1458 m".

(c) The changed speed and wavelengths are "24.76 m/s" and "0.2063 m", respectively.

(a)

The tension in the rope will be equal to the weight of the hanging mass:

T = mg = (1.5 kg)(9.81 m/s)

T = 14.72 N

Now, the speed of the transverse wave on the rope is given by the following formula:

$v=\sqrt{\frac{T}{\mu}}$

where,

v = speed = ?

μ = linear mass density = 0.048 kg/m

Therefore,

$v=\sqrt{\frac{14.72\ N}{0.048\ kg/m}}$

v = 17.5 m/s

(b)

Now, the wavelength is given by the following formula:

$v=f\lambda\\\\\lambda=\frac{v}{f}$

where,

λ = wavelength = ?

f = frequency of tuning fork = 120 Hz

Therefore,

$\lambda = \frac{17.5\ m/s}{120\ Hz}$

λ = 0.1458 m

(c)

Now, the mass changes. Hence, the values will change as follows:

T' = (3 kg)(9.81 m/s²) = 29.43 N

Speed will become:

$v'=\sqrt{\frac{T'}{\mu}}=\sqrt{\frac{29.43\ N}{0.048\ kg/m}}$

v' = 24.76 m/s

The wavelength will then become:

$\lambda'=\frac{v'}{f}=\frac{24.76\ m/s}{120\ Hz}$

λ' = 0.2063 m

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