Question

-Mary and Pippen hold a rope tightly between them. The rope has a set length of 2.3 m and a mass of 0.55 kg, and initially there is a tension in the jumprope of 690 N. What is the mass per unit length of the rope (aka. the little "mu" symbol in the equation for wave speed)? (Give your answer to 3 sig figs. and in units of kg/m)  

-If Mary sends a wave towards Pippen under these conditions, what is the speed of the wave in the rope? (Give your answer to 3 sig figs.)

-Now, Pippen holds his end of the rope stationary while Mary shakes the rope up and down at a constant frequency that produces a standing wave on the rope with only one "bump". This means that each end of the rope acts as a node and there is a single peak right in the middle of the rope. If Mary's frequency of rope-shaking represents the fundamental frequency of the rope, how many wavelengths can be observed on the rope? (Give your answer to 1 sig fig. and with no units)

-Assuming Mary's frequency of rope-shaking represents the fundamental frequency of the rope, what is the wavelength of the wave? (Give your answer to 3 sig figs. and in units of m)

-Assuming the string has the same tension as in part (1), find the fundamental frequency of Mary's rope-shaking. (Give your answer to 3 sig figs. and in units of Hz)

-Now, Mary wants to shake the rope at a higher rate so that he creates a standing wave with 4 "bumps". Assuming he does this, how many wavelengths can be observed on the rope? (Give your answer to 3 sig figs. with no units)

-Assuming Mary shakes the rope at such a rate that he creates a standing wave with 4 "bumps", what is the frequency of this standing wave? (Give your answer to 3 sig figs. and in units of Hz)

Answer 1

(a) The mass per unit length is,

(b) The speed of the wave on the string is given by, T is the tension.

(c) For the one bump which is the fundamental frequency, only (1/2) of the wavelength can be observed on the rope. Since the length of the string is L, the first harmonic will be such that,

ie. only half of the wavelength can be observed.

(d) We will use the above formula to find the wavelength,

(e) we know that the product of frequency and wavelength is velocity of the wave on the string.

(We know that at same tension the velocity is calculated in part (b))