When mass M is tied to the bottom of a long, thin wire...

When mass $M$ is tied to the bottom of a long, thin wire suspended from the ceiling, the wire's second-harmonic frequency is $200 \mathrm{Hz}$. Adding an additional $1.0 \mathrm{kg}$ to the hanging mass increases the second-harmonic frequency to $245 \mathrm{Hz}$. What is $M ?$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Harmonics

Harmonics are the integer multiples of the fundamental frequency at which a system can naturally oscillate. The second harmonic, for example, is twice the fundamental frequency and corresponds to the formation of an additional node in the vibrating system, altering the energy distribution within the medium.

Tension-Frequency Relationship

The frequency at which a string or wire vibrates in its harmonic modes is directly related to the square root of the tension in the string. When a hanging mass is attached to a wire, the tension is approximately equal to the gravitational force on that mass, making the frequency proportional to the square root of the sum of the masses. This relationship allows changes in mass (and thus tension) to significantly alter the vibrational frequency.

Standing Waves

Standing waves are patterns that result from the interference of two waves of the same frequency traveling in opposite directions along a medium, such as a wire or string. They are characterized by nodes and antinodes, and they form the basis for understanding the vibration modes in musical instruments and other physical systems.