The speed of a longitudinal wave in a solid bar is given by v = √(X/ρ), where 'ρ' is density of the medium. What is the unknown term 'X'?
Young's Modulus of the medium
Understanding Wave Speed in a Solid Bar
The question asks us to identify the unknown term 'X' in the formula for the speed of a longitudinal wave in a solid bar, given as $v = \sqrt{X/\rho}$, where $\rho$ is the density of the medium.
Longitudinal Waves in Solids
Longitudinal waves are waves where the particles of the medium oscillate parallel to the direction of wave propagation. In a solid material, these waves can propagate, causing compressions and expansions within the material.
Formula for Wave Speed
The speed of mechanical waves in a medium depends on the elastic properties of the medium and its density. For a longitudinal wave propagating along a slender solid rod or bar, the relevant elastic property that resists change in length under stress is the Young's Modulus.
The standard formula for the speed of a longitudinal wave in a solid rod is given by:
$$v = \sqrt{\frac{Y}{\rho}}$$
where:
- $v$ is the speed of the longitudinal wave
- $Y$ is the Young's Modulus of the solid material
- $\rho$ is the density of the solid material
Comparing Formulas
We are given the formula:
$$v = \sqrt{\frac{X}{\rho}}$$
Comparing this given formula with the standard formula for the speed of a longitudinal wave in a solid rod ($v = \sqrt{Y/\rho}$), we can clearly see that the unknown term 'X' corresponds to the Young's Modulus ($Y$) of the medium.
Analyzing the Options
- Option 1: Young's Modulus of the medium - This matches our derivation based on the standard formula for longitudinal waves in solid bars.
- Option 2: Bulk Modulus of the medium - Bulk Modulus ($K$) is the elastic modulus associated with volume changes. While related to elasticity, Young's Modulus is specifically used for longitudinal stress and strain in solid rods, governing propagation along the length. Bulk modulus is used for waves in fluids or for pressure waves in solids related to volume compression.
- Option 3: linear mass density - Linear mass density is mass per unit length, which is a property related to the geometry and mass distribution, not the elastic resistance of the material to deformation under stress.
- Option 4: Density of the gas - The question specifies a "solid bar," making density of a gas irrelevant.
Based on the comparison and the understanding of wave propagation in solids, the unknown term 'X' is the Young's Modulus of the medium.
| Property | Symbol | Description | Relevance to Wave Speed |
|---|---|---|---|
| Young's Modulus | $Y$ | Measures resistance to elastic deformation under longitudinal stress (stretching/compressing) | Used for longitudinal waves in solid rods |
| Bulk Modulus | $K$ | Measures resistance to uniform compression (volume change) | Used for longitudinal waves in fluids and pressure waves in solids |
| Density | $\rho$ | Mass per unit volume | Affects inertia, present in formulas for speed of mechanical waves |
| Linear Mass Density | $\lambda$ | Mass per unit length | Used for wave speed in strings or wires ($v = \sqrt{T/\lambda}$) |
Revision Table: Speed of Mechanical Waves
| Medium | Wave Type | Formula for Speed ($v$) | Relevant Modulus |
|---|---|---|---|
| Solid (Rod) | Longitudinal | $v = \sqrt{Y/\rho}$ | Young's Modulus ($Y$) |
| Solid (Bulk) | Longitudinal (Pressure/Compression) | $v = \sqrt{(K + \frac{4}{3}G)/\rho}$ (where G is Shear Modulus) or $\sqrt{K/\rho}$ if Shear is negligible or for pressure waves | Bulk Modulus ($K$), Shear Modulus ($G$) |
| Fluid (Liquid or Gas) | Longitudinal (Sound) | $v = \sqrt{K/\rho}$ (or $\sqrt{B/\rho}$ where B is adiabatic Bulk Modulus for gases) | Bulk Modulus ($K$ or $B$) |
| Solid | Transverse | $v = \sqrt{G/\rho}$ | Shear Modulus ($G$) |
| Stretched String | Transverse | $v = \sqrt{T/\lambda}$ | Tension ($T$), Linear Density ($\lambda$) |
Additional Information: Elastic Moduli and Wave Types
Elastic moduli are properties of a material that describe its resistance to deformation under stress. They represent the stiffness of the material.
- Young's Modulus (Y): Describes the tensile or compressive stiffness of a solid material. It relates stress (force per unit area) to strain (fractional change in length) in one dimension. A higher Young's Modulus means the material is stiffer and harder to stretch or compress along its length.
- Bulk Modulus (K): Describes the resistance of a material (solid, liquid, or gas) to uniform compression. It relates pressure (stress) to the fractional change in volume (volume strain). A higher Bulk Modulus means the material is harder to compress.
- Shear Modulus (G): Describes the resistance of a solid material to shearing deformation (like twisting or bending). It relates shear stress to shear strain.
Mechanical waves require a medium to travel through and depend on the elastic properties (stiffness) and inertial properties (density) of that medium. Longitudinal waves in a solid bar cause compressions and expansions along the direction of the bar, which is governed by the material's resistance to changes in length, i.e., Young's Modulus.
Important Questions from The Speed of Sound
At standard temperature and pressure, in which of the following media does sound propagate with the greatest speed?
The velocity of sound in air is affected by change in the
I. Moisture content of air
II. Temperature of air
III. Composition of air
IV. Atmospheric pressure
Choose the correct answer.
Velocity of sound is maximum in:
What is the relation between the frequency f, wavelength λ, and speed v of the sound?
An object is 10.64 km below the sea level. A research team sends down a sonar signal to confirm this depth. After how long can it expect to get the echo? Take speed of sound in sea water = 1520 m/s.
