Wave Speed on a Stretched String MCQ Quiz in मराठी - Objective Question with Answer for Wave Speed on a Stretched String - मोफत PDF डाउनलोड करा

Concept:

The fundamental frequency (n) of a vibrating string is inversely proportional to its length (l):

n ∝ 1/l

When a string of length l is divided into three segments of lengths l₁, l₂, and l₃, the fundamental frequencies of the segments are n₁, n₂, and n₃, respectively:

n₁ ∝ 1/l₁, n₂ ∝ 1/l₂, n₃ ∝ 1/l₃

The original frequency (n) of the string is the harmonic mean of the frequencies of the three segments.

Calculation:

To find the original fundamental frequency:

1/n = 1/n₁ + 1/n₂ + 1/n₃

Substituting the relationship of frequencies and lengths:

1/n = l₁/l + l₂/l + l₃/l

Since the total length of the string is l:

1/n = (l₁ + l₂ + l₃)/l

∴ The relationship between the original fundamental frequency and the segment frequencies is:

1/n = 1/n₁ + 1/n₂ + 1/n₃.

Concept:

Length of the String Under Tension:

• The extension of a string under tension is directly proportional to the applied tension and the original length.
• The formula for the length of the string under tension is: L = L₀ (1 + ΔL/L₀) = L₀ + (T / k), where T is the tension and k is the constant of proportionality.
• From the two given tensions, we can use the proportionality to form equations for the lengths.

Calculation:

Let the original length of the string be L₀.

When the tension is 6 N, the length is P: P = L₀ + (6 / k)

When the tension is 8 N, the length is Q: Q = L₀ + (8 / k)

Subtract the two equations: Q - P = (L₀ + 8 / k) - (L₀ + 6 / k)

Q - P = (2 / k)

k = 2 / (Q - P)

Substitute this value of k into one of the equations, say P = L₀ + (6 / k):

P = L₀ + (6 / (2 / (Q - P))) = L₀ + 3(Q - P)

L₀ = P - 3(Q - P) = 4P - 3Q

∴ The original length of the string is 4P - 3Q. Option 4) is correct.

CONCEPT:

The wave velocity of a string under Tension T and mass per unit length μ 

⇒ v = 

EXPLANATION:

For the first string,

Length = l, radius = r, density = ρ 

The volume of the wire = πr²L

∴ density = mass/volume

⇒ volume = mass/density = m₁/ρ 

So, we have, πr2L = m₁/ρ 

⇒ m₁/L = πr2ρ = mass per unit length = μ 

So, velocity v =  ---- (1)

Similarly for the second material of mass m₂ 

⇒ m₂/L = πr2(3ρ) ​ 

So, velocity v' = nv =  

∴ n = 0.58

Hence the correct answer is option 1.

Concept:

Fundamental frequency:

The lowest frequency of any vibrating object is called a fundamental frequency.

The fundamental frequency of a string is given by

Where l = length of the string, T = tension on the string and m =  linear mass density

​Calculation:

Given:

The fundamental frequency of a string (l) is given by:

As T and m is constant

⇒ n1l1 = n2l2 = n3l3 = k        [Where k = constant]

The original length of the string is

The total length of the string is 

⇒ l = l1 + l2 + l3

Substitute the value of l, l1, l2, and l3 in the above equation, we get

CONCEPT:

The wave velocity of a string under Tension T and mass per unit length μ 

⇒ v = 

EXPLANATION:

For the first string,

Length = l, radius = r, density = ρ 

The volume of the wire = πr²L

∴ density = mass/volume

⇒ volume = mass/density = m₁/ρ 

So, we have, πr2L = m₁/ρ 

⇒ m₁/L = πr2ρ = mass per unit length = μ 

So, velocity v =  ---- (1)

Similarly for the second material of mass m₂ 

⇒ m₂/L = πr2(3ρ) ​ 

So, velocity v' = nv =  

∴ n = 0.58

Hence the correct answer is option 1.

CONCEPT:

For a transverse wave the velocity of the transverse wave 

Where, T = tension and μ = mass per unit length

∴ T = μ v²

EXPLANATION:

We know the relation between T and v

⇒ T ∝ v2

(Where μ is a constant)

⇒ T1/T2 = (v1/v2)2

∴ T2 = T1/(v1/v2)2

Given,

v₁ = v and v₂ = v/2

⇒v1/v2 = 2

⇒T1 = 2.06 × 104 N

∴ T2 = T1/(v1/v2)2

= (2.06 × 104)/(2)² = (2.06/4) × 104 = 5.15 × 103 N

Hence the correct answer is option 2.

Concept:

The linear velocity of a wave traveling in a stretched string is given by the equation:

Where T is the tension in the stretched string, m is the mass of the string and L is the length of the string.

The general equation of a wave is y = A sin(ωt + kx + ϕ)

Where y is the displacement of the wave at time t, A is the amplitude, ϕ is the phase and k is the angular wavenumber.

Calculation:

Given:

m/L = 1.2 × 10-4 kg/m

y = (0.02 m) sin[x + 30t]

Comparing the general wave equation

The general equation of a wave is y = A sin(ωt + kx + ϕ)

A = 0.02 m, ω = 30 rad/s, k = 1.

Linear velocity:

T = 30 × 30 × 1.2 × 10-4 = 0.108 N

Concept:

• The frequency of oscillation is defined as the number of oscillations in one second.
• The distance between two nodes in terms of the wavelength λ the string is 
• The relation between the speed of the wave v, frequency f and wavelength λ is given by, v=f λ 
• The speed of the wave in string in terms of adiabatic constant,  

Explanation:

Let's consider a gas with a molar mass M, f is the frequency of a sound and T is the temperature. 

The distance between two nodes is 

So,  λ =2L

The speed of the wave is, 

Additional Information

• Isothermal expansion: A process in which temperature remains constant. For such a process, PV = constant
• Isobaric process: A process in which pressure remains constant. For this process, ">VT=constant" id="MathJax-Element-13-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">">PT=constant" role="presentation" style="display: inline; position: relative;" tabindex="0">
• ">PT=constant" role="presentation" style="display: inline; position: relative;" tabindex="0">Isocoric process: ">PT=constant" role="presentation" style="display: inline; position: relative;" tabindex="0">A process in which volume remains constant. For this process,  ">PT=constant" role="presentation" style="display: inline; position: relative;" tabindex="0">

The correct answer is option 4) i.e. 1 : 4 

CONCEPT:

• The fundamental frequency of a stretched string is given by the equation:

Where L is the length of the vibrating part of the string, T is the tension and μ is the linear density of the string.

EXPLANATION:

Given that:

The ratio of frequencies is v1 : v2 = 1 : 2

Since the wires are identical, they will have the same L and μ.  

The fundamental frequency, ν ∝ √T

Ratio, ν1 : ν2 = √T1 : √T2

⇒ν1² : ν2² = T1 : T2

1² : 2² = 1 : 4

Concept:

• Transverse wave: The wave generated such that the particles oscillate in the direction perpendicular to the propagation of the wave is called the transverse wave. 
  • The transverse wave can be observed when we pull a tight string a bit. 

• The Speed of Such Transverse wave is given as:

 ">v=Tμ" id="MathJax-Element-1-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">v=Tμv=Tμ

Where T is Tension in the tight String and μ is mass per unit length of the string. 

Calculation:

Given length of string = 7m

Mass of string = 0.35 kg

Tension in string = 60.5 N

Mass per unit length 

⇒ μ = 0.05

Speed of wave