Question

Two uniform strings of mass per unit length \(\mu\) and \(4\mu\), and length \(L\) and \(2L\), 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 \(v_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}\), which of the following statements is(are) correct?

• A. With a node at \(O\), the minimum frequency of vibration of the composite string is \(v_0\).
• B. With an antinode at \(O\), the minimum frequency of vibration of the composite string is \(2v_0\).
• C. When the composite string vibrates at the minimum frequency with a node at \(O\), it has 6 nodes, including the end nodes
• D. No vibrational mode with an antinode at \(O\) is possible for the composite string.

Answer 1

The Correct Option is A, C, D

The correct option is (A): With a node at \(O\), the minimum frequency of vibration of the composite string is \(v_0\).,(C): When the composite string vibrates at the minimum frequency with a node at \(O\), it has 6 nodes, including the end node and (D): No vibrational mode with an antinode at \(O\) is possible for the composite string.