Consider the phase speed of transverse waves on a wire, with the changes listed

Question

Consider the phase speed of transverse waves on a wire, with the changes listed below. Answer `true' (T), `false' (F), or `cannot tell' (C) to each of the five statements.

A) If the tension is increased and the length of the wire is increased (assuming the mass remains the same), then the phase speed increases.

B) If the tension in the wire is doubled, the phase speed of the transverse wave decreases by a factor of square root of 2.

C) If the tension is quadrupled, the mass of the wire is doubled, and the length is halved, then the phase speed remains the same.

D) If the mass of the wire is doubled and the tension and length remain the same, the phase speed of the transverse wave decreases by a factor of 2.

E) If the length of the wire increases by a factor of 4 and the tension and mass remain the same, the phase speed of the transverse wave decreases by a factor of 2.

Explanation / Answer

Use eqn,

v=sqrt[T/µ] =sqrt[T/(m/L)] = sqrt[TL/m] ----------(1)

A) True

From eqn (1),

v is directly proportional to T and L hence when T and L increases v icreases.

B) False

When T’=2T

T’/T=2

v' sqrtT’

v’/v=sqrt(T’/T)

v’=v*sqrt(T’/T)

v’=v*sqrt(2

This v’ is increased by sqrt2

C) True

T’=4T

T’/T=4

m’=2m

m’/m=2

L’/L=1/2

Putting these values in eqn (1),

v'=sqrt[T’L’/m]

v’=sqrt[(4T*L/2)/(2m)]

v’=sqrt[1*TL/m]

v’=v

Thus no change in velocity after changing T, L or m

D) True

T’=T

m’=2m

m’/m=2

L’=L

Putting these values in eqn (1),

v'=sqrt[T’L’/m]

v’=sqrt[TL/(2m)]

v’=sqrt(TL/m)*sqrt(1/2)

v=v*1/sqrt2

Thus v’ is decreased by sqrt2.

E) b

T’=T

m’=m

L’=4L

Putting these values in eqn (1),

v'=sqrt[T’L’/m]

v’=sqrt[T*4L/(m)]

v’=sqrt(TL/m)*sqrt(4)

v=v*2

Thus v’ is increased by 2.

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