Superposition: Question 1

Worked solution

Step 1: Recall the boundary condition for a string fixed at both ends

A string fixed at both ends must have a node at each fixed end. If the string vibrates with $n$ equal loops, each loop is exactly half a wavelength, so the string length $L$ contains $n$ half-wavelengths: $L = n\frac{\lambda}{2} \quad\Rightarrow\quad \lambda = \frac{2L}{n}$

Step 2: Substitute the given values

Here $L = 1.50\text{ m}$ and the string shows $n = 5$ loops: $\lambda = \frac{2 \times 1.50}{5} = \frac{3.00}{5} = 0.60\text{ m}$

Check (independent method): the distance between adjacent nodes is always $\lambda/2$. With $5$ loops there are $6$ nodes spaced evenly over $1.50\text{ m}$, so there are $5$ node-to-node gaps of length $1.50/5 = 0.30\text{ m}$ each. Since a node-to-node gap is $\lambda/2$, this gives $\lambda = 2 \times 0.30 = 0.60\text{ m}$, the same result.

Step 3: Why the other options are wrong

• A ($0.30$ m): this is the loop length itself (half a wavelength), mistaken for the whole wavelength.
• C ($1.50$ m): this treats the entire string as a single wavelength, ignoring that $5$ separate loops are shown.
• D ($3.00$ m): this is the wavelength of the fundamental ($n=1$) mode, $\lambda = 2L$, applied incorrectly to a pattern that actually shows the $5$th harmonic.

Final answer

• Wavelength $= \boxed{0.60\text{ m}}$, option B.