A string on a musical instrument is held under tension T and extends from the point x = 0 to the point x = L. The string is overwound with wire in such a way that its mass per unit length μ(x) increases uniformly from μ0 at x = 0 to μLat x = L. (a) Find an expression for μ(x) as a function of x over the range 0 ≤ x ≤ L. (b) Find an expression for the time interval required for a transverse pulse to travel the length of the string.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
A string on a musical instrument is held under tension T and extends from the point x = 0 to the point x = L. The string is overwound with wire in such a way that its mass per unit length μ(x) increases uniformly from μ0 at x = 0 to μL
at x = L. (a) Find an expression for μ(x) as a function of x over the range 0 ≤ x ≤ L. (b) Find an expression for the time interval required for a transverse pulse to travel the length of the string.
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