MHTCET 26 Apr 2025 Shift 2 Paper

Concept:Beat frequency is the difference between two frequencies, and the frequency of an organ pipe depends on its length and whether it is open or closed.Explanation:For an open pipe, the first overtone (second harmonic) frequency is $\frac{v}{L}$.For a closed pipe, the first overtone (third harmonic) frequency is $\frac{3v}{4L}$.Given that they produce 4 beats simultaneously:$$\frac{v}{L} - \frac{3v}{4L} = 4$$Simplifying this:$$\frac{v}{4L} = 4$$$$\therefore \frac{v}{L} = 16$$Now, the open pipe length is halved to $\frac{L}{2}$, so its fundamental frequency becomes:$$f_o = \frac{v}{2\left(\frac{L}{2}\right)} = \frac{v}{L}$$The closed pipe length is doubled to $2L$, so its fundamental frequency becomes:$$f_c = \frac{v}{4(2L)} = \frac{v}{8L}$$The new beat frequency is:$$f_o - f_c = \frac{v}{L} - \frac{v}{8L} = \frac{7v}{8L}$$Using $\frac{v}{L} = 16$:$$\text{Beats} = \frac{7}{8} \times 16 = 14$$Answer:The number of beats produced is 14, which corresponds to option C.