User:Edblah

World0

Linear density, linear mass density or linear mass is a measure of mass per unit of length, and it is a characteristic of strings or other one-dimensional objects. The SI unit of linear density is the kilogram per metre (kg/m). It is defined as:


 * $$\mu = \frac{\partial m}{\partial x}$$

where μ is the linear density of the object, m is the mass, and x is a coordinate along the (one dimensional) object.

For the common case of a homogenous substance of length L and total mass m, this simplifies to:


 * $$\mu = \frac{m}{L}$$

Let $$L$$ be the length of the string, $$m$$ its mass and $$T$$ the tension.

When the string is deflected it bends as an approximate arc of circle. Let $$R$$ be the radius and $$\theta$$ the angle under the arc. Then $$L = \theta\,R$$.

The string is recalled to its natural position by a force $$F$$:


 * $$ F = \theta\,T$$

The force $$F$$ is also equal to the centripetal force
 * $$F = m\,\frac{v^2}{R}$$
 * where $$v$$ is the speed of propagation of the wave in the string.

Let $$\mu$$ be the linear mass of the string. Then


 * $$m = \mu\,L = \mu\,\theta\,R$$

and


 * $$F = \mu\,\theta\,R\,\frac{v^2}{R}    =   \mu\,\theta\,v^2   $$

Equating the two expressions for $$F$$ gives:
 * $$\theta\,T = \mu\,\theta\,v^2$$

Solving for velocity v, we find


 * $$v = \sqrt{T \over \mu}$$

Frequency of the wave
Once the speed of propagation is known, the frequency of the sound produced by the string can be calculated. The speed of propagation of a wave is equal to the wavelength $$\lambda$$ divided by the period $$\tau$$, or multiplied by the frequency $$f$$ :


 * $$v = \frac{\lambda}{\tau} = \lambda f$$

If the length of the string is $$L$$, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so $$L$$ is half of the wavelength of the fundamental harmonic. Hence:


 * $$f = \frac{v}{2L} = { 1 \over 2L }  \sqrt{T \over  \mu}   $$

where $$T$$ is the tension, $$\mu$$ is the linear mass, and $$L$$ is the length of the vibrating part of the string.