Question

A string of mass m is fixed at both ends. The fundamental tone oscillations are excited in the strong with angular frequency ω and maximum displacement amplitude A. Find the total energy contained in the string.

A. 1/2mω²A²
B. 1/4mω²A²
C. 1/6mω²A²
D. 1/8mω²A²

Answer 1

The total energy contained in a string undergoing fundamental tone oscillations with angular frequency ω and amplitude A is E = ½mω²A².

Step-by-step explanation:

The total energy contained in the string, when fundamental tone oscillations are excited, can be found by summing the kinetic and potential energy of the oscillations.

For a mass element undergoing simple harmonic motion, the potential energy U is given by ½ksx², where ks represents the spring constant and x is the displacement.

In the context of the oscillating string, ks is equivalent to mω², making the potential energy ½mω²x².

The kinetic energy is given by ½mv². Since at maximum displacement, the velocity is zero, and at zero displacement the velocity is maximum, the kinetic energy also averages ½mω²A² over a full cycle.

Thus, the total energy E for an oscillation is E = ½mω²A², which matches the potential energy at maximum displacement, because kinetic and potential energies are equal at any instant in time for simple harmonic motion.