Question

A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to vw=50.00m/s. What are the wavelength and frequency of the first three modes that resonate on the string?

Answer 1

Answer

given,

Length of the string, L = 2 m

speed of the wave , v = 50 m/s

string is stretched between two string

For the waves the nodes must be between the strings

the wavelength  is given by

           $\lambda = \dfrac{2L}{n}$

where n is the number of antinodes; n = 1,2,3,...

the frequency expression is given by

            $f = n\dfrac{v}{2L}$

now, wavelength calculation

      n = 1

           $\lambda_1 = \dfrac{2\times 2}{1}$

                    λ₁ = 4 m

      n = 2

           $\lambda_2 = \dfrac{2\times 2}{2}$

                   λ₂ = 2 m

      n =3

           $\lambda_3 = \dfrac{2\times 2}{3}$

                    λ₃ = 1.333 m

now, frequency calculation

      n = 1

            $f = n\dfrac{v}{2L}$

            $f_1 =1\times \dfrac{50}{2\times 2}$

                    f₁ = 12.5 Hz

      n = 2

            $f = n\dfrac{v}{2L}$

            $f_2 =2\times \dfrac{50}{2\times 2}$

                    f₂= 25 Hz

      n = 3

            $f = n\dfrac{v}{2L}$

            $f_3 =3\times \dfrac{50}{2\times 2}$

                    f₃ = 37.5 Hz