A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent nodes are 18 cm` and `16 cm respectively. The minimum possible length of the string is:

Options

A. 144 cm
B. 204 cm
C. 288 cm
D. 72 cm

Answer

To solve the problem step by step, we will analyze the information given about the standing wave modes on a string fixed at both ends.

### Step 1: Understand the Concept of Nodes and Wavelength
In a string fixed at both ends, the ends are nodes. The distance between adjacent nodes corresponds to half the wavelength (\(\lambda/2\)). Therefore, if we know the distance between two adjacent nodes, we can find the wavelength.

### Step 2: Identify the Given Distances
We are given two distances between adjacent nodes:
- Distance for the first mode (node 1) = 18 cm
- Distance for the second mode (node 2) = 16 cm

### Step 3: Calculate the Wavelengths
Using the relationship that the distance between adjacent nodes is \(\lambda/2\):
- For the first mode:
\[ \frac{\lambda_1}{2} = 18 \text{ cm} \implies \lambda_1 = 36 \text{ cm} \]
- For the second mode:
\[ \frac{\lambda_2}{2} = 16 \text{ cm} \implies \lambda_2 = 32 \text{ cm} \]

### Step 4: Establish the Relationship Between Modes
The length of the string \(L\) can be expressed in terms of the wavelengths and the mode numbers \(n_1\) and \(n_2\):
\[ L = n_1 \frac{\lambda_1}{2} = n_2 \frac{\lambda_2}{2} \]
This means we can set up the equation:
\[ n_1 \cdot 18 = n_2 \cdot 16 \]

### Step 5: Solve for the Ratio of Mode Numbers
Rearranging the equation gives us:
\[ \frac{n_1}{n_2} = \frac{16}{18} = \frac{8}{9} \]
This indicates that for every 8 nodes in the first mode, there are 9 nodes in the second mode.

### Step 6: Assign Values to Mode Numbers
Let:
- \(n_1 = 8k\)
- \(n_2 = 9k\)

Where \(k\) is a common integer factor.

### Step 7: Calculate the Length of the String
Substituting \(n_1\) and \(n_2\) back into the length equation:
\[ L = n_1 \cdot 18 = 8k \cdot 18 = 144k \text{ cm} \]
\[ L = n_2 \cdot 16 = 9k \cdot 16 = 144k \text{ cm} \]
Both expressions for \(L\) confirm that they are equal.

### Step 8: Determine the Minimum Length
To find the minimum possible length of the string, we set \(k = 1\):
\[ L = 144 \text{ cm} \]

### Final Answer
The minimum possible length of the string is **144 cm**.