Question

The speed of a wave on a violin A string is 288 m/s and on the G string is 128 m/s. The force exerted on the ends of the string G is 110N, on the ends of the string A is 350N.

Use this information to determine the ratio of mass per unit length of the strings (A/G).

Answer 1

The ratio of mass per unit length of the A and G strings on the violin is approximately 3.3.

Step-by-step explanation:

To determine the ratio of mass per unit length of the strings A and G, we can use the formula for wave speed on a string:

Wave speed (v) = sqrt(Tension (F) / linear mass density (μ))

For the A string, the wave speed (vA) is 288 m/s and the tension (FA) is 350 N. For the G string, the wave speed (vG) is 128 m/s and the tension (FG) is 110 N. We can set up equations to find the linear mass density of each string:

vA = sqrt(FA / μA)
vG = sqrt(FG / μG)

Simplifying the equations, we get:

μA = FA / (vA)2
μG = FG / (vG)2

Substituting the given values, we find:

μA = 350 N / (288 m/s)2

μG = 110 N / (128 m/s)2

Calculating these expressions, we find that the ratio of mass per unit length of the strings (A/G) is approximately 3.3.

Answer 2

`(\mu_A)/(\mu_G)=0.197`

Step-by-step explanation:

given,

Speed of a wave on violin A = 288 m/s

Speed on the G string = 128 m/s

Force at the end of string G = 110 N

Force at the end of string A = 350 N

the ratio of mass per unit length of the strings (A/G). = ?

speed for string A

`v_A = \sqrt{(F_A)/(\mu_A)}`.......(1)

speed for string G

`v_G = \sqrt{(F_G)/(\mu_G)}`........(2)

Assuming force is same in both the string

now,

dividing equation (2)/(1)

`(v_G)/(v_A)=\frac{\sqrt{(F_G)/(\mu_G)}}{\sqrt{(F_A)/(\mu_A)}}`

`(v_G)/(v_A)=(√(\mu_A))/(√(\mu_G))`

`(128)/(288)=(√(\mu_A))/(√(\mu_G))`

`(\mu_A)/(\mu_G)=0.197`