The wave speed on a string under tension is 200 m/s. What is the speed if he tension is doubled?

The aim of this question is to understand the key concepts of speed, frequency, wavelength, and tension in a string.

Whenever energy is transferred from one place to another through the successive vibratory motion of particles, this form of energy transferring agent is called a wave. All type of waves have some common properties such as the speed, frequency, wavelength etc.

The speed of a wave traveling through a string depends upon its tension $F_T$, mass of the string $m$, and the length of the string $L$. Given these parameters, it can be calculated using the following formula:

$$v_{wave}=\sqrt{{\displaystyle \frac{F_T\times L}{m}}}$$

Expert Answer:

Lets say:

$$\text{ speed of wave at original tension }=v_{wave}=\sqrt{{\displaystyle \frac{F_T\times L}{m}}}$$

$$\text{ speed of wave at doubled tension }=v_{wave}^{\prime }=\sqrt{{\displaystyle \frac{2\times F_T\times L}{m}}}$$

Notice that both $L$ and $m$ remain the same because they are the property of the string, which is not changed. Dividing both of the above equations:

$${\displaystyle \frac{v_{wave}^{\prime }}{v_{wave}}}={\displaystyle \frac{\sqrt{{\displaystyle \frac{2\times F_T\times L}{m}}}}{\sqrt{{\displaystyle \frac{F_T\times L}{m}}}}}$$

$$\Rightarrow {\displaystyle \frac{v_{wave}^{\prime }}{v_{wave}}}=\sqrt{{\displaystyle \frac{2\times F_T\times L\times m}{F_T\times L\times m}}}$$

$$\Rightarrow {\displaystyle \frac{v_{wave}^{\prime }}{v_{wave}}}=\sqrt{2}$$

$$\Rightarrow v_{wave}^{\prime }=\sqrt{2}{v}_{wave}\dots \dots \dots \dots (1)$$

Substituting values:

$$\Rightarrow v_{wave}^{\prime }=\sqrt{2}(200m/s)$$

$$\Rightarrow v_{wave}^{\prime }=280m/s$$

Which is the required answer.

Numerical Result

$$\Rightarrow v_{wave}^{\prime }=280m/s$$

Example

What happens to the speed of the wave if the tension in the string is raised by four times instead of doubling?

Lets say:

$$\text{ speed of wave at original tension }=v_{wave}=\sqrt{{\displaystyle \frac{F_T\times L}{m}}}$$

$$\text{ speed of wave at four times the tension }=v_{wave}^{\prime }=\sqrt{{\displaystyle \frac{4\times F_T\times L}{m}}}$$

Dividing both of the above equations:

$${\displaystyle \frac{v_{wave}^{\prime }}{v_{wave}}}={\displaystyle \frac{\sqrt{{\displaystyle \frac{4\times F_T\times L}{m}}}}{\sqrt{{\displaystyle \frac{F_T\times L}{m}}}}}$$

$$\Rightarrow {\displaystyle \frac{v_{wave}^{\prime }}{v_{wave}}}=\sqrt{{\displaystyle \frac{4\times F_T\times L\times m}{F_T\times L\times m}}}$$

$$\Rightarrow {\displaystyle \frac{v_{wave}^{\prime }}{v_{wave}}}=\sqrt{4}$$

$$\Rightarrow {\displaystyle \frac{v_{wave}^{\prime }}{v_{wave}}}=2$$

$$\Rightarrow v_{wave}^{\prime }=2v_{wave}\dots \dots \dots \dots (2)$$

Substituting values:

$$\Rightarrow v_{wave}^{\prime }=2(200m/s)$$

$$\Rightarrow v_{wave}^{\prime }=400m/s$$

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