Length of a string tied to two rigid supports is 40 cm. Maximum length (wavelength in cm) of a stati

Q: Why does maximum wavelength correspond to only one loop?

Wavelength is inversely proportional to the number of loops. More loops mean the wave is 'squeezed' into the same length, making the wavelength shorter. Thus, the fewest loops (one) gives the longest wavelength.

Q: What exactly is a node and an antinode?

A node is a point on a stationary wave where the amplitude is always zero (no movement). An antinode is a point where the amplitude of vibration is maximum.

Q: Can a string fixed at both ends vibrate in half a loop?

No. A string fixed at both ends must have nodes at those ends. The simplest shape that connects two nodes is a single, full loop (which represents half a wavelength).

Q: How does this relate to the frequency of the wave?

Frequency and wavelength are inversely related ($v = f \lambda$). Since this mode has the maximum wavelength, it corresponds to the minimum possible frequency, known as the fundamental frequency.

Q: What would the wavelength be if it vibrated in two loops?

If it vibrated in two loops, there would be three nodes (two at the ends, one in the middle). The length of the string would then equal one full wavelength ($\lambda = l = 40\text{ cm}$).

Question

Length of a string tied to two rigid supports is $40	ext{ cm}$. Maximum length (wavelength in cm) of a stationary wave produced on it, is

Options:

(A) $20$

(B) $80$

(C) $40$

(D) $120$

Sigma Percentile · Accepted Answer

The physics of stationary waves on a string is one of the most elegant and visually intuitive concepts in mechanics. When a string is plucked, waves travel back and forth, reflecting off the fixed boundaries. Under the right conditions, these interfering waves create a stable pattern known as a **stationary wave** or **standing wave**.

### Analyzing the Setup

Imagine you are holding a string that is tightly stretched between two rigid walls. The length of this string is given to us as $l = 40	ext{ cm}$. Because the ends of the string are firmly attached to the walls, they are physically restricted from moving. In the language of waves, these fixed points of zero amplitude are called **Nodes**.

The question asks us to find the maximum possible wavelength ($\lambda$) of a stationary wave that can be produced on this string. To maximize the wavelength, we need to minimize the number of oscillations or "loops" the string makes.

### The Master Equation

The simplest possible vibration pattern for a string fixed at both ends is a single, continuous loop. This is known as the **fundamental mode** of vibration. In this mode, the string bows out in the middle, creating a single point of maximum amplitude called an **Antinode**.

From the geometry of stationary waves, we know that the distance between two consecutive nodes is exactly half of a wavelength.

$$ 	ext{Distance between nodes} = \frac{\lambda}{2} $$

Since our string is vibrating in just one loop, the entire length of the string ($l$) is equal to the distance between these two end nodes. This gives us our master equation:

$$ l = \frac{\lambda}{2} $$

### Final Calculation

Now, we simply substitute the known length of the string into our equation. We know that $l = 40	ext{ cm}$.

$$ 40 = \frac{\lambda}{2} $$

To isolate the wavelength ($\lambda$), we multiply both sides of the equation by $2$:

$$ \lambda = 2 	imes 40 $$

$$ \lambda = 80	ext{ cm} $$

And there we have it! The maximum wavelength of the stationary wave that can be produced on this string is **$80	ext{ cm}$**.

I know this might seem almost too simple, but that is the beauty of fundamental physics. By visualizing the physical constraints—the fixed nodes at the ends—the mathematics naturally unfolds. Always remember to draw the loops!

Animated Solution for Physics - Waves: Length of a string tied to two rigid supports is $40 cm$ . Maximum length (wavelength in cm) of a stationary wave produced on it, is

Select Answer:

Visualized Solution

String Setup

Fundamental Mode

Nodes and Antinodes

Wavelength Relation

Calculation

Final Answer

The Way Forward

The Sigma Insight: Standing Waves in Strings and Organ Pipes

Analyzing the Setup

The Master Equation

Final Calculation

Similar Questions

A string is clamped at both the ends and it is vibrating in its 4th harmonic. The equation of the stationary wave is $Y = 0.3 sin (0.157 x) cos (200 π t)$ . The length of the string is (All quantities are in SI units)

A $20 cm$ long string, having a mass of $1.0 g$ , is fixed at both the ends. The tension in the string is $0.5 N$ . The string is set into vibration using an external vibrator of frequency $100 Hz$ . Find the separation (in cm) between the successive nodes on the string.

A string of length 1 m and mass 5 g is fixed at both ends. The tension in the string is 8.0 N. The string is set into vibration using an external vibrator of frequency 100 Hz. The separation between successive nodes on the string is close to

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, $y (x, t) = (0.01 m) [sin (62.8 m^{- 1}) x] cos [(628 s^{- 1}) t]$ . Assuming $π = 3.14$ , the correct statement(s) is (are)

A wire of length $L$ and mass per unit length $6.0 \times 1 0^{- 3} kgm^{- 1}$ is put under tension of $540 N$ . Two consecutive frequencies that it resonates at are : $420 Hz$ and $490 Hz$ . Then, $L$ in metres is

A string 2.0 m long and fixed at its ends is driven by a 240 Hz vibrator. The string vibrates in its third harmonic mode. The speed of the wave and its fundamental frequency is

Velocity of sound in air is $320 m/s$ . A pipe closed at one end has a length of $1 m$ . Neglecting end corrections, the air column in the pipe can resonate for sound of frequency

Animated Solution for Physics - Waves: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Length of a string tied to two rigid supports is 40 cm. Maximum length (wavelength in cm) of a stationary wave produced on it, is