Question

Two identical straight wires are stretched so as to produce $6$ beats per second when vibrating simultaneously. On changing the tension slightly in one of them, the beat frequency remains unchanged. Denoting by ${T_1},{T_2}$ the higher and the lower initial tensions in the strings $\left( {{T_1} > {T_2}} \right)$ , then it could be said that while making the above changes in tension
A. ${T_1}$ was decreased or ${T_2}$ was increased
B. ${T_1}$ was increased or ${T_2}$ decreased
C. ${T_1}$ was decreased or ${T_2}$ decreased
D. ${T_1}$ was increased or ${T_2}$ increased

Answer 1

Hint: In such cases when the problem is based on wave phenomena, we know that frequency of the waves directly varies with tension in the string, hence analyze every option with the scientific approach and check which option seems to be more appropriate for the given situation. Then, present the answer with a proper explanation.

Complete step by step solution:
Beat Frequency $f' = {f_1} - {f_2} = 6Hz$........(given)
As we know, the frequency is directly proportional to the velocity and the velocity is directly proportional to the tension in the string.
Since, ${T_1} > {T_2}$...........(given)
$\Rightarrow {v_1} > {v_2}$
And, ${f_1} > {f_2}$

Now, let us consider the 4 cases as:

${T_1}$ was increased	${T_1}$ was decreased	${T_2}$ was increased	${T_2}$ was decreased
In this case, if ${T_1}$ was increased then ${f_1}$ would also be increased. As a result of which, the difference between ${f_1}$ and ${f_2}$ increased. ${f_1} - {f_2} > 6Hz$	In this case, if ${T_1}$ was decreased then ${f_1}$ would also be decreased. As a result of which, the difference between ${f_1}$ and ${f_2}$ may be the same. ${f_2} - {f_1} = 6Hz$(In a condition, when${f_1}$ will become less than ${f_2}$)	In this case, if ${T_2}$ was increased then ${f_2}$ would also be increased. As a result of which, the difference between${f_1}$ and ${f_2}$ may be the same. ${f_2} - {f_1} = 6Hz$(In a condition, when ${f_2}$ will become more than ${f_1}$)	In this case, if ${T_2}$ was increased then ${f_2}$ would also be decreased. As a result of which, the difference between ${f_1}$ and ${f_2}$ increased. ${f_1} - {f_2} > 6Hz$
NOT OK	OK	OK	NOT OK

Hence, the correct option is A.

Note: Since this is a problem of multiple-choice questions (theory-based) hence, it is essential that given conditions be analyzed very carefully to give a precise explanation. While writing an explanation of this kind of conceptual problem, always keep in mind to provide the exact reasons in support of your explanation.