JEE Advanced2024PhysicsWaves and SoundActual
Two uniform strings of mass per unit length and 4 , and length L and 2 L , respectively, are joined at point O , and tied at two fixed ends P and Q , as shown in the figure. The strings are under a uniform tension T . If we define the frequency v₀= 1 2 L T , which of the following statement(s) is(are) correct?
Options
- AWith a node at O , the minimum frequency of vibration of the composite string is v₀ .
- BWith an antinode at O , the minimum frequency of vibration of the composite string is 2 v₀ .
- CWhen the composite string vibrates at the minimum frequency with a node at O , it has 6 nodes, including the e
- DNo vibrational mode with an antinode at O is possible for the composite string.
Correct answer
A. With a node at O , the minimum frequency of vibration of the composite string is v₀ .
Step-by-step solution
C ₁= T , C ₂= T 4 = C ₁ 2 For node at O : L = n ₁ 2 , 2 ~L = m ₂ 2 ( n , m are integers) aligned & ₁= 2 ~L n , ₂= 4 ~L ~m & C ₁ ₁ = C ₂ ₂ & C ₁ 2 ~L n = C ₁ 2 4 ~L m & 4 n = m aligned For minimum frequency, n =1, ~m =4 v_ = C ₁ 1 2 ~L = 1 2 ~L T =v₀ The string will look like Total no. of nodes =6 including the end nodes For antinode at O : aligned & L =(2 n +1) ₁ 4 ; 2 ~L =(2 n +1) ₂ 4 (n, m are integers) & ₁= 4 ~L (2 n +1) ; ₂= 8 ~L (2 ~m +1) & C ₁ ₁ = C ₂ ₂ & C ₁ C ₂ = ₁ ₂ & 2= 4 ~L (2 n +1) 8 ~L (2 ~m +1) & 4= (