Question

A string of mass $100\,gm$ is clamped between two rigid supports. A wave of amplitude $2\,mm$ is generated in string. If the angular frequency of a wave is $5000\text{ }rad/sec$ then the total energy of the wave in the string is
A. $25\,J$
B. $5\,J$
C. $0.5\,J$
D. $2.5\,J$

Answer 1

Energy is the quantitative property that must be transferred to a body or physical system to perform work on the object or heat it. Energy is a conserved quantity; according to the law of conservation of energy, energy can be converted in form but cannot be created or destroyed.

Complete step by step answer:
Waves of various types have distinct characteristics. These characteristics aid in the differentiation of wave types. One way traveling waves are distinguished is by the orientation of particle motion relative to the direction of wave propagation.

The following are the various types of waves classified according to particle motion:
Pulse Waves: A pulse wave is a wave that travels through a transmission medium with only one disturbance or crest.
Continuous-wave: A continuous-wave is a waveform with a constant amplitude and frequency.
Transverse Waves: In a transverse wave, the particle's motion is perpendicular to the wave's propagation direction.
Longitudinal Waves: Longitudinal waves are the waves in which the motion of the particle is in the same direction as the propagation of the wave.

They have one thing in common, despite their differences, and that is the ability to transport energy. $E=\dfrac{1}{2}k{{A}^{2}}$ is the energy of an object in simple harmonic motion. We can conclude from this equation that the energy of a wave is directly proportional to the square of its amplitude, i.e. $E\propto {{A}^{2}}$.

This relationship between energy and amplitude is critical in determining the damage that can be caused by earthquake shock waves. Because the waves are spread out over a larger area, they become weaker as they spread out. So, by using the above relation,
$E=\dfrac{1}{2}k{{A}^{2}}$
$\Rightarrow E=\dfrac{1}{4}m{{\omega }^{2}}{{A}^{2}}$
$\Rightarrow E=\dfrac{1}{4}\times \left( \frac{1}{10} \right){{\left( 5000 \right)}^{2}}\times 4\times {{10}^{-6}}$

\therefore E=2.5\,J$$ **Thus, the answer is option D.** **Note:** Interference is a phenomenon in which two waves superimpose to form a resultant wave of lower, greater, or equal amplitude. Constructive and destructive interference occurs as a result of the interaction of waves that are correlated with each other, either because they have the same frequency or because they are emitted from the same source. Interference effects can be seen in all types of waves, including gravity waves and light waves.