Question

A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent nodes are $ 18cm $ and $ 16cm $ respectively. The minimum possible length of the string is:
(A) $ 72cm $
(B) $ 144cm $
(C) $ 108cm $
(D) $ 216cm $

Answer 1

Hint :To solve this question, we have to know about the distance. We know that, Distance is a mathematical estimation of how far separated articles or focuses are. In physical science or ordinary utilization, distance may allude to an actual length or an assessment dependent on different measures (for example "two districts over").

Complete Step By Step Answer:
Let us consider $ {n_1} $ and $ {n_2} $ to be no. of loops when $ 18cm $ and $ 16cm $ are distances between nodes respectively.
We can write, $ {n_2} - {n_1} = 1 $
Or, $ {n_2} = {n_1} + 1 $
Let L be the length of the string. So,
We can write,
 $ L = 18 \times {n_1} \\
  L = 16 \times {n_2} \\ $
After solving the above three equations we will get,
 $ 18 \times {n_1} = 16 \times {n_ 2 } $
Or, $ 18 \times {n_1} = 16(1 + {n_1}) $
Or, $ {n_1} = 8 $
Similarly we can write, after putting the value of $ {n_1} $ on the above equation,
 $ {n_2} = 9 $
Now, using these two values we can say,
 $ L = 18 \times 8 = 144 $
So, the right option will be option number B.

Note :
We also have to say that, SI unit of distance is a meter according to the International System of Units. Interestingly, using this as the base unit and some equations, many other derived units or quantities are formed like volume, area, acceleration, and speed. We can say, the unit of distance is a centimetre in the CGS framework. In material science, distance is characterized as the actual estimation of how far an article has moved. Distance is a scalar amount. We have to keep these in our mind.