An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but it's too dangerous to go inside. Recalling your recent physics class, you decide to try setting up standing-wave resonances inside the tunnel. Using your subsonic amplifier and loudspeaker, you find resonances at 5.0 Hz and 6.4 Hz , and at no frequencies between these. It's rather chilly inside the tunnel, so you estimate the sound speed to be 333 m/s .

 L = 116.6 m

Explanation:

For this exercise we approximate the tunnel as a tube with one end open and the other closed, at the open end there is a belly and at the closed end a node, therefore the resonances occur at

               λ = 4L         1st harmonic

               λ = 4L / 3    third harmonic

               λ = 4L / 5    fifth harmonic

General term

               λ = 4L / n      n = 1, 3, 5,...    odd

                                    n = (2n + 1)      n are all integers

They indicate that two consecutive resonant frequencies were found, the speed of the wave is related to the wavelength and its frequency

            v = λ f

            λ = v / f

we substitute

             L =

for the first resonance n = n

             L = (2n + 1)

for the second resonance n = n + 1

             L = (2n + 3)

we have two equations with two unknowns, let's solve by equating

             (2n + 1) \frac{v}{4f_1}= (2n + 3) \frac{v}{4f_2}

             (2n + 1) f₂ = (2n +3) f₁

              2n + 1 = (2n + 3)

              2n (1 - \frac{f_1}{f_2}) = 3 \frac{f_1}{f_2} -1

we substitute the values

              2n (1- ) = 3 -1

              2n 0.21875 = 1.34375

              n = 1.34375 / 2 0.21875

              n = 3

remember that n must be an integer.

We use one of the equations to find the length of the Tunal

                  L = (2n + 1) \frac{v}{4f_1}

                  L = (2 3 + 1)

                  L = 116.55 m

the answer is the first electromagnet (a). this is because it has more coils around the ferromagnetic material. increasing the number of coils increases the surface area in which the metal interacts with the current in the coil. this enables more atoms in the metal to become aligned hence becoming more magnetized.