Question

When a mass of 5 kg hangs from a vertical wire of length 2 m, waves travel on this wire with a speed of 10 m/s. What will the speed of these waves be if the wire is replaced with one with the same length but three times as heavy

Answer 1

Answer:

New speed of the wave, v = 5.77 m/s                                                                                                                                    

Explanation:

It is given that,

Mass of the object, m = 5 kg

Length of the wire, L = 2 m

Speed of the wave, v = 10 m/s

The speed of wave in terms of tension is given by :

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

$10=\sqrt{\dfrac{T}{\mu}}$

If the wire is replaced with one with the same length but three times as heavy, the mass per unit length is given by :

$\mu'=\dfrac{3m}{L}$

New speed of the wave is given by :

$v'=\sqrt{\dfrac{T}{\mu'}}$

$v'=\dfrac{1}{\sqrt{3} }\sqrt{\dfrac{T}{\mu}}$

$v'=\dfrac{10}{\sqrt{3} }$

v' = 5.77 m/s

So, the speed of the new wave is 5.77 m/s. Hence, this is the required solution.

Answer 2

The speed of these waves will be 5.8 m/s

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Further explanation

Let's recall the speed of wave formula as follows:

$\boxed {v = \lambda \times f}$

where:

c = speed of wave ( m/s )

λ = wavelength ( m )

f = frequency of wave ( Hz )

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

where:

v = speed of wave on string ( m/s )

T = tension in the string ( N )

μ = linear density ( kg/m )

Let us now tackle the problem!

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Given:

mass of the object = M = 5 kg

length of vertical wire = L = 2 m

initial speed of wave = v₁ = 10 m/s

initial mass of the wire = m₁ = m

final mass of the wire = m₂ = 3m

Asked:

final speed of wave = v₂ = ?

Solution:

$v_1 : v_2 = \sqrt{\frac{T}{\mu_1}} : \sqrt{\frac{T}{\mu_2}}$

$v_1 : v_2 = \sqrt{\mu_2} : \sqrt{\mu_1}$

$v_1 : v_2 = \sqrt{\frac{m_2}{L}} : \sqrt{\frac{m_1}{L}}$

$v_1 : v_2 = \sqrt{m_2} : \sqrt{m_1}$

$v_1 : v_2 = \sqrt{3m} : \sqrt{m}$

$10 : v_2 = \sqrt{3}$

$v_2 = 10 \div \sqrt{3}$

$v_2 = \frac{10}{3} \sqrt{3} \texttt{ m/s}$

$v_2 \approx 5.8 \texttt{ m/s}$

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Learn more

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Answer details

Grade: High School

Subject: Physics

Chapter: Wave