Wave Speed on a Stretched String MCQ Quiz - Objective Question with Answer for Wave Speed on a Stretched String - Download Free PDF

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:

• 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:

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

CALCULATION:

Given,

Tension in String T = 60N

Mass of String m = 5.0 × 10–3 kg

Length of String l = 0.72 m

Mass Per Unit Length of String 

⇒μ = 6.67 × 10^-3 kg 

Speed 

⇒ 

Solving it, we will get the approx value of v = 93 m/s. 

So, Option 3 is the correct answer.

CONCEPT:

• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
  • Example: Motion of an undamped pendulum, undamped spring-mass system.

The speed of a transverse waves on a stretched string is given by:

Where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.

EXPLANATION:

The speed of transverse waves on a stretched string is given by v = √(T/X).

1. Here X is mass per unit length or linear density of string. So option 1 is correct.
2. Bulk modulus of elasticity (B): It is the ratio of Hydraulic (compressive) stress (p) to the volumetric strain (ΔV/V).
3. Young’s modulus: Young's modulus a modulus of elasticity, applicable to the stretching of wire, etc., equal to the ratio of the applied load per unit area of the cross-section to the increase in length per unit length.
4. Density: The mass per unit volume is called denisty.

CONCEPT:

• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
• Example: Motion of an undamped pendulum, undamped spring-mass system.

 , where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.

CALCULATION:

Given length l = 2 m, mass of 2 m string m = 10 g, wave velocity v = 40 m/s.

  5 g/m = 5 × 10^-3 kg/m. from formula,

 ;

 ⇒ 40² × 5 × 10^-3 = 8 N.

T = 8 N.

Concept:

• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
  • Example: Motion of an undamped pendulum, undamped spring-mass system.

The speed of a transverse wave on a stretched string is given by:

Where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.

Calculation:

Let μ is the mass per unit length of the roap.

The speed of transverse waves on a stretched string is given by v = √(T/μ).

At a distance x from the lower end, we will find tension which will be

T = μ g x

So, using

  (Speed of transverse wave on a string)

v =

CONCEPT:

• Fundamental frequency: It is the lowest frequency of a periodic waveform. It is also known as natural frequency.

The fundamental frequency of a sonometer wire:

where f is the fundamental frequency, l is the length of the wire, T is the tension in the wire, and μ is the mass per unit length.

EXPLANATION:

The frequency is given by:

Given that:

Tension is made four times,

So new tension (T') = 4T

The new frequency is given by:

• Thus the frequency becomes 2 times. So option 1 is correct.

EXPLANATION:

→Speed of the transverse wave in any string \(v = \sqrt{\frac{T}{μ}} \) 

Where, T = tension in the string, μ = mass per unit length

Given: 

mass, m = 2.5 kg, length, l = 20.0 m

∴ μ = m/l = 2.5/20 = 0.125 kg/m

Hence speed, \(v = \sqrt{\frac{200}{0.125}}\) 

v = √1600 = 40 m/s

∴ Time taken for the wave to travel from one end to the other end

= Time taken for the wave to travel 20 m

t = distance/velocity = 20/40 = 0.5 s

So, the correct answer is option (2).

Explanation:

Given:

String A and B are of the same material and produce a frequency which is 6 Hz.

Therefore, The difference between f_A and f_B is 6 Hz

i.e, fA – fB = 6 Hz ---(1)

Where f_A is the frequency of A and f_B is the frequency of B respectively.

If tension decreases, f_B decreases and becomes f_B.

Now, the difference between f_A and f_B = 7 Hz (increases), which means that the frequency of string A is greater than the frequency of string B.

So, f_A > f_B

As we know, f_A = 530 Hz

Putting the value of f_A in equation (1) we get;

530 – f_B = 6 

⇒ -f_B = 6-530

⇒ -f_B = -524

⇒ f_B = 524 Hz.

Hence, option 4) is the correct answer.

The correct answer is option 1) i.e. 7.

CONCEPT:

• Frequency in a stretched string:
  • A stretched string is always subjected to a tension force along the length of the string.
  • When a stretched string is vibrated, it produces transverse waves.

The frequency of the transverse wave (ν) is related to the tension in the string as follows:

Where L is the length of the string, T is the tension in the string and μ is the mass per unit length of the given string.

• Beats: Beats are the periodic fluctuations heard in the intensity of a sound when two sound waves of nearly identical frequencies interfere with one another.
  • ​The number of beats is found from the difference in frequencies of two interfering waves. 

​CALCULATION:

Frequency, 

 137 Hz

 144 Hz

Beats = difference in frequencies = ν₂ - ν₁ = 144 - 137 = 7

Concept:

Wave Equation and Tension in a String:

• The wave equation for a string is given by: y = A sin( 2π ( t / T - x / λ ) )
• Where:
  • A = Amplitude of the wave
  • T = Period of the wave
  • λ = Wavelength of the wave
  • t = Time
  • x = Position along the string
• For a string under tension, the tension (T) is related to the wave speed and linear mass density (μ) by the formula: T = μ v² = μ ( ω / k )²
• Where:
  • μ = Linear mass density (kg/m)
  • v = Wave speed (m/s)
  • ω = Angular frequency (rad/s)
  • k = Wave number (rad/m)

Calculation:

Given,

Amplitude, A = 0.02 m

Linear mass density, μ = 0.04 kg/m

The wave number (k) is given by: k = 2π / λ = 2π / 0.50 = 2π / 0.50 rad/m

The angular frequency (ω) is given by: ω = 2π / T = 2π / 0.004 rad/s

The tension in the string is:

T = μ ( ω / k )² = 0.04 × ( (2π / 0.004) / (2π / 0.50) )² = 6.25 N

∴ The tension in the string is 6.25 N.