If we want to vibrate a 100 meters long string that weighs 500 grams to generate

Question

If we want to vibrate a 100 meters long string that weighs 500 grams to generate waves to deliver some messages faster than the speed of sound (340 m/s), how strong is the tension we need at least?

(3) Steel ( 7700 kg/m3) and nylon ( 1200 kg/m3) are typical materials for a guitar’s string. How much thicker should a nylon string be to give the same pitch as a steel string under the same tension?

Regarding the standing wave in Figure 2, if we want to decrease the node number by 1 by changing only one of the four quantities—i) oscillation frequency, ii) string tension, iii) string mass density, and iv) string length—how should we do it? (answer “increase” or “decrease”)

Explanation / Answer

1)

1) given

L = 100 m

m = 500 g = 0.5 kg

v = 340 m/s

linear mass density, mue = m/L

= 0.5/100

= 0.005 kg/m

apply, v = sqrt(T/mue)

==> T = v^2*mue

= 340^2*0.005

= 578 N <<<<<<<<--------------Answer

let

rho1 = 7700 kg/m^3

rho2 = 1200 kg/m^3

2)
when both the strings have same linear mass density both will produce same pitch.

let d1 and d2 are diamemters(thickness) of the two wires.

let A1 and A2 are cross sectional areas.

so we need, mue1 = mue2

rho1*A1 = rho2*A2

rho1*pi*d1^2/4 = rho2*pi*d2^2/4

d2 = d1*sqrt(rho1/rho2)

= d1*sqrt(7700/1200)

= 2.53*d1

so, the thickness of nylon string should be equal to 2.53 times of te thickness of the steel string.

3)

i) decrease

ii) increase

iii) decrese

iv) decrease

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