Question

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\) (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true Statement \(-1:\) Two waves moving in a uniform string having uniform tension cannot have different velocities. Statement \(-2:\) Elastic and inertial properties of string are same for all waves in same string. Moreover speed of wave in a string depends on its elastic and inertial properties only. (A) a (B) \(b\) (C) \(c\) (D) \(\mathrm{d}\)

Short answer

The correct answer is (A) a.

Step-by-step solution

Evaluate the truth value of Statement \(-1\)#

Statement \(-1\) states that "Two waves moving in a uniform string having uniform tension cannot have different velocities". This statement is true because the velocity of a wave in a string depends on the tension and mass density of the string. If the string is uniform and has uniform tension, the velocity of any wave traveling through that string will be the same.

Evaluate the truth value of Statement \(-2\) #

Statement \(-2\) states that "Elastic and inertial properties of string are same for all waves in the same string. Moreover, the speed of the wave in a string depends on its elastic and inertial properties only." This statement is true as well. The velocity of a wave in a string depends on the square root of the ratio of tension (elastic property) and mass density (inertial property) of the string. As a result, if these properties are the same for all waves in the same string, their velocities will be the same as well.

Determine if Statement \(-2\) is the correct explanation for Statement \(-1\) #

Statement \(-1\) claims that in a uniform string with uniform tension, two waves cannot have different velocities. Statement \(-2\) explains that the elastic and inertial properties of the string are the same for all waves, and the speed of the wave depends on these properties only. Thus, statement \(-2\) does provide a correct explanation for statement \(-1\). Based on the analysis, statement \(-1\) is true, statement \(-2\) is true, and statement \(-2\) is the correct explanation for statement \(-1\). Therefore, we can conclude the correct option is: (A) a

Key concepts

String Properties

When talking about wave velocity in strings, it's important to understand what string properties are. These properties include:

• Mass Density: This is the mass of the string per unit length. It represents the inertial property of the string.
• Elasticity: This refers to how the string can stretch and return to its original form. It determines how the string responds to forces.

The uniformity of the string means that these properties are consistent throughout its length.
This uniformity ensures that waves traveling through it are influenced universally by those same properties.
Therefore, any variations in these properties can change the behavior and speed of the waves traveling through the string.

Uniform Tension

Uniform tension in a string is crucial for wave behavior. Tension refers to the force applied along the string to make it tight. In a string with uniform tension:

• The tension is consistent across the entire length of the string.
• This consistent tension ensures uniform wave velocity.
• It stabilizes the wave patterns moving through the string.

When the tension is uniform, the forces affecting the wave speed remain constant throughout the string.
This leads to the wave having the same velocity no matter where it is on the string.
With non-uniform tension, wave speeds would vary, which can lead to unpredictable behaviors.

Elastic and Inertial Properties

The elastic and inertial properties of a string dictate how waves travel through it. These two properties are:

• Elastic Property: Defined by the tension in the string, determining how the string reacts to stretch or compression.
• Inertial Property: Determined by the mass density of the string, indicating how resistant the string is to motion.

The wave speed is influenced by these properties and is calculated using the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension and \( \mu \) is the mass density.
This formula shows that the wave speed is independent of the wave's frequency or amplitude.
It depends solely on the string's characteristics, making these properties crucial for understanding wave mechanics.

Wave Mechanics

Wave mechanics explores the motion of waves through various mediums. The velocity of a wave in a medium like a string stems from the medium's mechanical properties:

• In a string, elasticity and inertia play dominant roles in determining wave velocity.
• Mechanical waves transfer energy without transferring matter, using the medium's properties.
• Uniform properties lead to uniform wave propagation.

Understanding wave mechanics helps in appreciating how waves behave under various conditions.
The consistency in a string's properties, like uniform tension and mass density, ensures predictable wave behavior.
This predictability forms a fundamental basis in wave-related studies, offering insights into broader physical phenomena.