Question

When a mass M hangs from a vertical wire of length L, waves travel on this wire with a speed V. What will be the speed of these waves (in terms of V) if a) We double M without stretching the wire? b) We replace the wire with an identical one, except twice as long? c) We replace the wire with one of the same length, but three times as heavy? d) We stretch the wire to twice its original length? e) We increase M by a factor of 10, which stretches the wire to double its original length.

Answer 1

Answer:

a)  v = 0.7071 v₀, b) v= v₀, c)  v = 0.577 v₀, d)   v = 1.41 v₀, e)  v = 0.447 v₀

Explanation:

The speed of a wave along an eta string given by the expression

          v = $\sqrt{ \frac{T}{ \mu } }$

where T is the tension of the string and μ is linear density

a) the mass of the cable is double

          m = 2m₀

let's find the new linear density

          μ = m / l

iinitial density

          μ₀ = m₀ / l

final density

          μ = 2m₀ / lo

          μ = 2 μ₀

we substitute in the equation for the velocity

initial            v₀ = $\sqrt{ \frac{T_o}{ \mu_o} }$

with the new dough

                    v = $\sqrt{ \frac{T_o}{ 2 \mu_o} }$

                    v = 1 /√2  \sqrt{ \frac{T_o}{ \mu_o} }

                    v = 1 /√2 v₀

                    v = 0.7071 v₀

b) we double the length of the cable

If the cable also increases its mass, the relationship is maintained

              μ = μ₀

   in this case the speed does not change

c) the cable l = l₀ and m = 3m₀

we look for the density

           μ = 3m₀ / l₀

           μ = 3 m₀/l₀

           μ = 3 μ₀

            v = $\sqrt{ \frac{T_o}{ 3 \mu_o} }$

            v = 1 /√3  v₀

            v = 0.577 v₀

d) l = 2l₀

            μ = m₀ / 2l₀

            μ = μ₀/ 2

           v = $\sqrt{ \frac{T_o}{ \frac{ \mu_o}{2} } }$

           v = √2 v₀

            v = 1.41 v₀

e) m = 10m₀ and l = 2l₀

we look for the density

             μ = 10 m₀/2l₀

             μ = 5 μ₀

we look for speed

             v = $\sqrt{ \frac{T_o}{5 \mu_o} }$

             v = 1 /√5  v₀

             v = 0.447 v₀