Question

127. If λ1, λ2 and λ3 are wave lengths of the waves giving resonance with fundamental ,first and second over tones of a closed organ pipe the ratio of wave lengths λ1 : λ2 : λ3 is
(a) 1:2:3
(b) 1:1/3:1/5
(c) 1:3:5
(d) 4:5:3:1
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Answer 1

AnSwer :-

If λ$_1$, λ$_2$ and λ$_3$ are wave lengths of the waves giving resonance with fundamental ,first and second over tones of a closed organ pipe the ratio of wave lengths λ$_1$ : λ$_2$ : λ$_3$ is

(a) 1 : 2 : 3

☑ (b) 1 : ⅓ : ⅕

(c) 1 : 3 : 5

(d) 4 : 5: 3 : 1

Option (b) 1 : ⅓ : ⅕ is right.

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Explaination :-

In fundamental frequency

$\sf\ell = \dfrac{ \lambda_{1}}{4} \Longrightarrow \lambda _{1} = 4 \ell$

In first overtone

$\: \ell = \dfrac{3 \lambda _{2}}{4} \Longrightarrow \lambda _{2} = \dfrac{4 \ell}{3}$

In second overtone

$\: \ell = \dfrac{5\lambda _{2}}{4} \Longrightarrow \lambda _{2} = \dfrac{4\ell}{5}$

hence,$\: \lambda _{1} : \lambda _{2} : \lambda _{3} = 1 : \dfrac{1}{3} : \dfrac{1}{5}$

Thus , Option (b) is correct.

Answer 2

Answer

Answer:

If λ_1

1

, λ_2

2

and λ_3

3

: λ_2

2

: λ_3

3

is

(a) 1 : 2 : 3

☑ (b) 1 : ⅓ : ⅕

(c) 1 : 3 : 5

(d) 4 : 5: 3 : 1

Option (b) 1 : ⅓ : ⅕ is right.

Explanation:

In fundamental frequency

\sf\ell = \dfrac{ \lambda_{1}}{4} \Longrightarrow \lambda _{1} = 4 \ellℓ=

4

λ

1

⟹λ

1

=4ℓ

In first overtone

\: \ell = \dfrac{3 \lambda _{2}}{4} \Longrightarrow \lambda _{2} = \dfrac{4 \ell}{3}ℓ=

4

3λ

2

⟹λ

2

=

3

4ℓ

In second overtone

5λ

2

⟹λ

2

=

5

4ℓ

:λ

2

:λ

3

=1:

3

1

:

5

1

Thus , Option (b) is correct.

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