Question

String 1 in the figure has linear density 2.60 g/m and string 2 has linear density 3.30 g/m. A student sends pulses in both directions by quickly pulling up on the knot, then releasing it. She wants both pulses to reach the ends of the strings simultaneously.

What should the string length L1 be?

What should the string length L2 be?

Answer 1

Here is your answer please change up some words to remain plagraism free.

Step-by-step explanation:

To determine the required lengths of strings 1 and 2 so that pulses sent in both directions reach the ends of the strings simultaneously, we need to apply the principle that the time it takes for a wave pulse to travel a distance on a string is equal to the distance divided by the wave speed.

The wave speed, in turn, is determined by the tension in the string and the linear density of the string according to the formula:

v = sqrt(T/μ),

where v is the wave speed, T is the tension, and μ is the linear density.

Let L1 be the length of string 1 and L2 be the length of string 2. Since the wave speed is the same for both strings, we can set up the following equations:

L1/v = L2/v

sqrt(T1/μ1)*L1 = sqrt(T2/μ2)*L2

where T1 and T2 are the tensions in strings 1 and 2, respectively.

We can solve for L1 and L2 by combining these two equations and solving for each variable. Substituting the given linear densities of strings 1 and 2, we get:

sqrt(T1/2.60)*L1 = sqrt(T2/3.30)*L2

Squaring both sides and simplifying, we get:

(T1/T2) = (3.30/2.60) * (L1/L2)^2

Substituting the condition that the pulses reach the ends of the strings simultaneously, we know that the total time for a pulse to travel down string 1 and back up to the knot is equal to the time for a pulse to travel down string 2 and back up to the knot. This condition implies that the total length of string 1 (2L1) is equal to the total length of string 2 (2L2):

2L1 = 2L2

Solving this equation for L2 and substituting it into the expression for T1/T2 derived above, we get:

T1/T2 = (3.30/2.60) * (L1/2L1)^2 = 1.25

Solving for L1, we obtain:

L1 = sqrt(T1/μ1) * (2L2/v) = sqrt((1.25)*(2.60/3.30)) * (2L2)

Simplifying this expression, we get:

L1 = (2/3) * sqrt(2.60/3.30) * L2

Therefore, the required length of string 1 is (2/3) * sqrt(2.60/3.30) times the length of string 2. We can substitute the given length of string 2, say L2 = 1 meter, into this expression to obtain the required length of string 1:

L1 = (2/3) * sqrt(2.60/3.30) * 1 meter ≈ 0.693 meter.

Therefore, the required length of string 1 is approximately 0.693 meter and the required length of string 2 is 1 meter.

POWERED BY OPEN AI ALL RIGHTS RESERVED