Option 1 : Linear mass density

__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:

\({\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}}\)

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).

- Here
**X is mass per unit length or linear density of string.**So option 1 is correct. - Bulk modulus of elasticity (B): It is the ratio of Hydraulic (compressive) stress (p) to the volumetric strain (ΔV/V).
- 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.
**Density**: The mass per unit volume is called denisty.

Option 3 : 93 m/s

__ 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 transverse wave can be observed

- The Speed of Such Transverse wave is given as:

\(v = \sqrt{\frac{T}{\mu}}\)

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 \(\mu = \frac{5\times 10^{-3}kg}{0.72m}\)

⇒μ = 6.67 × 10^{-3} kg

Speed \(v = \sqrt{\frac{T}{\mu }}\)

⇒ \(v = \sqrt{\frac{60}{6.67\times10 ^{-3} }} m/s \)

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

So, Option 3 is the correct answer.

Option 3 : 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**.

\({\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}}\) , 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.

\({\rm{\mu }} = \frac{{\rm{m}}}{{\rm{l}}} = \frac{{10}}{2} = \) 5 g/m = 5 × 10^{-3} kg/m. from formula,

\({\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}} \) ;

\(40 = \sqrt {\frac{T}{{5 \times {{10}^{ - 3}}}}} \) ⇒ 40^{2} × 5 × 10^{-3 }= 8 N.

**T = 8 N**.

Option 1 : 2

__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**:

\(f =\frac{1}{2l} . \sqrt{T\over μ}\)

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:

\(f =\frac{1}{2l} . \sqrt{T\over μ}\)

Given that:

Tension is made four times,

So new tension (T') = 4T

The new frequency is given by:

\(f' =\frac{1}{2l} . \sqrt{4T\over μ} =2 \times \frac{1}{2l} . \sqrt{T\over μ} = 2f\)

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

Option 1 : Linear mass density

__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:

\({\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}}\)

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).

- Here
**X is mass per unit length or linear density of string.**So option 1 is correct. - Bulk modulus of elasticity (B): It is the ratio of Hydraulic (compressive) stress (p) to the volumetric strain (ΔV/V).
- 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.
**Density**: The mass per unit volume is called denisty.

Option 1 : 7

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**.

- A stretched string is always subjected to a

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

\(ν = \frac{1}{2L} \sqrt{\frac{T}{μ}}\)

Where **L **is the l**ength **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**.

- The number of beats is found from the

__ CALCULATION__:

Given that: | String 1 |
String 2 |

Length (l) |
l_{1} = 51.6 cm = 0.516 m |
l_{2} = 41.9 cm = 0.419 m |

Tension (T) |
T_{1} =20 N |
T_{2} = 20 N |

Mass per unit length (μ) |
μ_{1} = 1 g/m = 10^{-3} kg/m |
μ_{2} = 1 g/m = 10^{-3} kg/m |

Frequency, \(ν = \frac{1}{2L} \sqrt{\frac{T}{μ}}\)

\(ν_1 = \frac {1}{2 \times 0.516} \sqrt{\frac{20}{10^{-3}}} = \) **137 Hz**

\(ν_2 = \frac {1}{2 \times 0.491} \sqrt{\frac{20}{10^{-3}}} = \) **144 Hz**

**Beats = **difference in frequencies = ν_{2} - ν_{1} = 144 - 137 = 7

Option 4 : \(\sqrt {gx}\,\)

__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:

\({\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{μ }}}}\,\)

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

\({\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{μ }}}}\,\) (Speed of transverse wave on a string)

\({\rm{v}} = \sqrt {\frac{{\rm{\mu \,g\,x}}}{{\rm{μ }}}}\,\)

v =\(\sqrt {gx}\,\)

Option 1 : \(\frac{v}{L}\)

__CONCEPT__:

- The stationary wave is also known as the standing wave.
- It is a combination of two waves, with the same amplitude and the same frequency, moving in the opposite direction.
- It is a result of interference.

- It is a combination of two waves, with the same amplitude and the same frequency, moving in the opposite direction.
- Harmonics of an instrument: A musical instrument has a set of natural frequencies at which it vibrates when a disturbance is introduced into it.
- This set of natural frequencies are known as the harmonics of the instrument.

The Frequency of nth harmonic in the standing wave is given by:

\(ν_n=\frac{nv}{2L}\)

where n is nth harmonic, v is the speed of sound, L is the length string.

__CALCULATION__:

For second harmonic n = 2

Frequency of 2^{nd} harmonic in the standing wave

\(ν=\frac{nv}{2L}\)

\(ν=\frac{2v}{2L}\)

\(ν=\frac{v}{L}\)

So the correct answer is option 1.

Option 4 : 1 : 4

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

__CONCEPT__:

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

\(\nu =\frac{1}{2L} \sqrt {\frac{T}{μ}}\)

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

**⇒ν1 ^{2} : ν2^{2} = T1 : T2**

1^{2} : 2^{2} = 1 : 4

1A string of 7 metre length has a mass of 0.035 kg. If tension in the string is 60.5 N, then speed of a wave on the string is

Option 4 : 110 metre/sec

__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:

\(v = \sqrt{\frac{T}{\mu}}\)$v=\frac{T}{\mu}$

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

\(μ = \frac{0.35kg}{7m}\)

⇒ μ = 0.05

Speed of wave

\(v = \sqrt{\frac{T}{\mu}}\)

\(v = \sqrt{\frac{60.5}{0.05}}\)