# Relationship between linear density and tension

### The Speed of a Wave on a String Objective: To find the relationship between the velocity and wave length of standing square root of the tension (T) over the linear density (µ). The string can vary in length, its tension and its linear density (mass / length). Fig. 2. . For a standing wave, the distance between adjacent nodes or adjacent. A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced {T \over \mu }}. This relationship was discovered by Vincenzo Galilei in the late s. its linear density.

Wave Speed on a String under Tension To see how the speed of a wave on a string depends on the tension and the linear density, consider a pulse sent down a taut string Figure When the taut string is at rest at the equilibrium position, the tension in the string FT is constant. The mass element is at rest and in equilibrium and the force of tension of either side of the mass element is equal and opposite. Mass element of a string kept taut with a tension FT. The mass element is in static equilibrium, and the force of tension acting on either side of the mass element is equal in magnitude and opposite in direction. If you pluck a string under tension, a transverse wave moves in the positive x-direction, as shown in Figure The mass element is small but is enlarged in the figure to make it visible. The small mass element oscillates perpendicular to the wave motion as a result of the restoring force provided by the string and does not move in the x-direction.

The tension FT in the string, which acts in the positive and negative x-direction, is approximately constant and is independent of position and time. A string under tension is plucked, causing a pulse to move along the string in the positive x-direction.

Assume that the inclination of the displaced string with respect to the horizontal axis is small.

## Physics Study Guide/Standing waves

The net force on the element of the string, acting parallel to the string, is the sum of the tension in the string and the restoring force. The x-components of the force of tension cancel, so the net force is equal to the sum of the y-components of the force. The magnitude of the x-component of the force is equal to the horizontal force of tension of the string FT as shown in Figure For waves to travel through the low E string at the same wave speed as the high E, would the tension need to be larger or smaller than the high E string?

What would be the approximate tension?

### Physics Study Guide/Standing waves - Wikibooks, open books for an open world

Solution Use the velocity equation to find the speed: The tension would be slightly less than N. Use the velocity equation to find the actual tension: The ability of one particle to pull on its neighbors depends on how tightly the string is stretched—that is, on the tension see Section 4.

The greater the tension, the greater the pulling force the particles exert on each other, and the faster the wave travels, other things being equal. Along with the tension, a second factor influences the wave speed.

For a given net pulling force, a smaller mass has a greater acceleration than a larger mass. Therefore, other things being equal, a wave travels faster on a string whose particles have a small mass, or, as it turns out, on a string that has a small mass per unit length.

The mass per unit length is called the linear density of the string. The effects of the tension F and the mass per unit length are evident in the following expression for the speed v of a small-amplitude wave on a string: In these instruments, the strings are either plucked, bowed, or struck to produce transverse waves.

Example 2 discusses the speed of the waves on the strings of a guitar. The effect of your adjustments on the wavelength of the wave is displayed, as is the time for the wave to travel the length of the string. From the travel time you can assess how the speed of the wave depends on the mass of the string. The length of each string between its two fixed ends is 0. Each string is under a tension of N. Find the speeds of the waves on the two strings. Since the tension is the same for both strings, and smaller linear densities give rise to greater speeds, we expect the wave speed to be greatest on the string with the smallest linear density. Solution The speeds of the waves are given by Equation Conceptual Example 3 offers additional insight into the nature of a wave as a traveling disturbance.

• Homework Help: Wave speed, tension, linear density
• 16.3: Wave Speed on a Stretched String
• String vibration

A string particle moves up and down in simple harmonic motion about the undisturbed position of the string.