Calculate the wave speed in each string

[nick85]

In Fig. 17-30a, string 1 has a linear density of 2.30 g/m, and string 2 has a linear density of 4.10 g/m. They are under tension owing to the hanging block of mass M = 500 g.



(a) Calculate the wave speed in each string.
m/s (speed in string 1)
m/s (speed in string 2)

(b) The block is now divided into two blocks (with M1 + M2 = M) and the apparatus rearranged as shown in Fig. 17-30b. Find M1 and M2 such that the wave speeds in the two strings are equal.
g (mass of M1)
g (mass of M2)

why wouldn't the speed in string 1 be given by the sqr root of (500*9.8)/(4.1)?

Thanks for any help.

 

[Hootenanny]

A mass of 500g is five hundred grams not five hundred times g. Remember mass is not a force.

-Hoot

 

[kyle8921]

I would first clarify the context of this question. Are we assuming that the strings are identical in material and tension? Are we assuming that the blocks are hanging at the same height in both scenarios? These details can affect the calculation of wave speed.

Assuming that the strings are identical in material and tension, and the blocks are hanging at the same height in both scenarios, we can use the formula for wave speed, v = √(T/μ), where T is the tension in the string and μ is the linear density.

(a) For string 1: v = √(T/μ) = √(0.5*9.8/0.0023) = 156.6 m/s
For string 2: v = √(T/μ) = √(0.5*9.8/0.0041) = 110.2 m/s

(b) In order for the wave speeds to be equal, we can set the two expressions for wave speed in the two strings equal to each other and solve for M1 and M2:
√(0.5*9.8/M1) = √(0.5*9.8/M2)
Squaring both sides and rearranging, we get:
M1/M2 = 0.5/0.5 = 1
Since M1 + M2 = M, we can substitute M1 = M - M2 into the equation above and solve for M2:
(M - M2)/M2 = 1
M - M2 = M2
M2 = M/2
Therefore, M1 = M - M2 = M - M/2 = M/2

Therefore, the masses of M1 and M2 should be equal in order for the wave speeds in the two strings to be equal. Both M1 and M2 should be equal to 250 g.

In response to your question about the calculation of wave speed in string 1, the formula you used is for the frequency of a wave, not the wave speed. The correct formula for wave speed is v = √(T/μ), as mentioned above.