You push back and forth on a 5 m high flagpole and make it sway back and forth. When you push it with a frequency of 2 Hz you get it into its fundamental standing wave.

a) How fast do the waves travel through the flagpole?
b) What frequency would you have to apply to get it into its fourth harmonic standing wave?

To find the speed of the waves traveling through the flagpole, we need to use the equation v = λf, where v is the velocity of the waves, λ is the wavelength, and f is the frequency.

a) The fundamental standing wave means the lowest frequency at which the flagpole sways back and forth. In this case, the frequency is given as 2 Hz. To determine the wavelength, we need to know the length of the flagpole or the distance between two consecutive nodes (points of no motion).

If we assume that the length of the flagpole is L, then the distance between two consecutive nodes is equal to half the wavelength (λ/2). Since the flagpole is 5 m high, there is one node at the bottom and another node at the top. Therefore, the distance between the nodes is L/2.

Applying this information to the formula v = λf, we can rewrite it as v = (L/2) * f. Plugging in the values, we get:

v = (5m/2) * 2Hz = 5m/s.

Therefore, the waves travel through the flagpole at a speed of 5 m/s.

b) To find the frequency required to get the flagpole into its fourth harmonic standing wave, we can use the equation f = nf1, where f is the desired frequency, n is the harmonic number (in this case, 4), and f1 is the fundamental frequency.

From part a), we know that the fundamental frequency (f1) is 2 Hz. Plugging in the values, we get:

f = 4 * 2Hz = 8 Hz.

Therefore, to get the flagpole into its fourth harmonic standing wave, you would have to apply a frequency of 8 Hz.