Question

A wave pulse travels along a string at a speed of 160cm/s . Note that parts a - d are independent and refer to changes made to the original string.
a)What will be the speed if the string's tension is doubled?

b)What will be the speed if the string's mass is quadrupled (but its length is unchanged)?

c)What will be the speed if the string's length is quadrupled (but its mass is unchanged)?

d)What will be the speed if the string's mass and length are both quadrupled?

Answer 1

Answer:

a) $v'=\sqrt{2} \times v$

   Speed will become √(2) times the initial speed.

b) $v'=\frac{1}{2}\times v$

   i.e Speed of the pulse becomes half of the initial speed.

c) $v'=2 \times v$

   i.e Speed of the pulse becomes twice of the initial speed.

d) $v'= v$

   i.e Speed of the pulse remains the same as initial.

Explanation:

Given:

• speed of wave pulse along a string, $v=160\ cm.s^{-1}$

We know the relation between the velocity of wave pulse in a string, tension in the string and its linear mass density as:

$v=\sqrt{\frac{T}{\mu} }$  initially

where:

$T=$ tension in the string

$\mu=\frac{m}{l}$ is the linear mass density as mass per unit length of the string.

a)

Speed of the pulse if the tension is doubled:

$v'=\sqrt{\frac{2T}{\mu} }$

$v'=\sqrt{2} \times v$

Speed will become √(2) times the initial speed.

b)

Speed of the pulse if the mass of the string is quadrupled:

We firstly find μ for this string.

$\mu'=\frac{4m}{l}$

i.e

$\mu'=4\mu$

Now,

$v'=\sqrt{\frac{T}{4\mu} }$

$v'=\frac{1}{2}\times v$

i.e Speed of the pulse becomes half of the initial speed.

c)

Speed of the pulse if the length of the string is quadrupled:

We firstly find μ for this string.

$\mu'=\frac{m}{4l}$

i.e

$\mu'=\frac{ \mu}{4}$

Now,

$v'=\sqrt{\frac{T}{\frac{m}{4l} }$

$v'=2 \times v$

i.e Speed of the pulse becomes twice of the initial speed.

d)

Speed of the pulse if both the length & the mass of the string is quadrupled:

We firstly find μ for this string.

$\mu'=\frac{4m}{4l}$

i.e

$\mu'=\mu$

Now,

$v'=\sqrt{\frac{T}{\mu} }$

$v'= v$

i.e Speed of the pulse remains the same as initial.