This question involves analyzing a standing wave on a string fixed at both ends. The given wave equation is y(x, t) = (0.01 m) sin[(62.8 m⁻¹)x] cos[(628 s⁻¹)t].

Let's break down the standard form of a standing wave equation for a string fixed at both ends:

$y(x,t)=2A\sin \left(kx\right)\cos \left(\omega t\right)$

By comparing this with the given equation, we can identify the wave parameters.

Step 1: Identify the wave number (k) and angular frequency (ω)

From the equation: $k=62.8m^{-1}$ and $\omega =628s^{-1}$.

Step 2: Relate wave number to wavelength (λ)

The relationship is: $k=\frac{2\pi }{\lambda }$

Given π = 3.14, we can solve for λ:

$62.8=\frac{2\times 3.14}{\lambda }=\frac{6.28}{\lambda }$

$\lambda =\frac{6.28}{62.8}=0.1m$

Step 3: Determine the harmonic number and find the length (L) of the string

For a string fixed at both ends, the wavelength for the nth harmonic is given by:

$\lambda =\frac{2L}{n}$

The problem states it is the fifth harmonic, so n = 5.

$0.1=\frac{2L}{5}$

Solving for L:

$L=\frac{0.1\times 5}{2}=\frac{0.5}{2}=0.25m$

Therefore, the statement "The length of the string is 0.25 m" is correct.

Step 4: Find the fundamental frequency (f₁)

First, find the frequency (f) of the fifth harmonic from the angular frequency.

$\omega =2\pi f$

$628=2\times 3.14\times f$

$f=\frac{628}{6.28}=100\mathrm{Hz}$

This is the frequency of the fifth harmonic (f₅). The fundamental frequency (f₁) is related to the harmonic frequency by:

$f_n=n{f}_1$

For n=5:

$100=5f_1$

$f_1=\frac{100}{5}=20\mathrm{Hz}$

Therefore, the statement "The fundamental frequency is 100 Hz" is incorrect; it is 20 Hz.

Step 5: Count the number of nodes

For a string fixed at both ends vibrating in its nth harmonic, the number of nodes (including the two fixed ends) is n + 1.

For the fifth harmonic (n=5), the number of nodes is 5 + 1 = 6.

Therefore, the statement "The number of nodes is 5" is incorrect.

Step 6: Find the maximum displacement of the midpoint

The amplitude of the standing wave at any point x is given by $2A\left|\sin \right(kx\left)\right|$. The maximum possible value of this amplitude is 2A, which occurs at an antinode where sin(kx) = ±1.

From the given equation, 2A = 0.01 m. Therefore, A = 0.005 m.

The midpoint of the string is at x = L/2 = 0.25/2 = 0.125 m.

Let's find the amplitude at the midpoint:

$Amplitude=2A\left|\sin \right(kx\left)\right|=0.01\times \left|\sin \right(62.8\times 0.125\left)\right|$

First, calculate kx:

$kx=62.8\times 0.125=7.85$

Now, we need to see if this position is a node or an antinode. For the fifth harmonic, the midpoint is a node. Let's verify this by checking if sin(kx) = 0.

We know k = 2π/λ. For the midpoint x = L/2:

$kx=\left(\frac{2\pi }{\lambda }\right)\left(\frac{L}{2}\right)$

But from the harmonic relationship, λ = 2L/n. Substituting:

$kx=\left(\frac{2\pi }{\frac{2L}{n}}\right)\left(\frac{L}{2}\right)=\left(\frac{2\pi n}{2L}\right)\left(\frac{L}{2}\right)=\frac{\pi n}{2}$

For n=5 (an odd harmonic), the midpoint is an antinode. For n=5:

$kx=\frac{\pi \times 5}{2}=\frac{5\pi }{2}$

$\sin \left(\frac{5\pi }{2}\right)=\sin (2\pi +\frac{\pi }{2})=\sin \left(\frac{\pi }{2}\right)=1$

Therefore, the amplitude at the midpoint is |0.01 × 1| = 0.01 m. This is the maximum possible displacement for any point on the string.

Therefore, the statement "The maximum displacement of the midpoint of the string, from its equilibrium position is 0.01 m" is correct.

Final Answer: The correct statements are the first one (length is 0.25 m) and the fourth one (midpoint displacement is 0.01 m).

Related Topics & Formulae

Standing Waves on a String: Waves that remain in a constant position, resulting from the interference of two waves travelling in opposite directions.

Key Formulae:

• Standard Equation: $y(x,t)=2A\sin \left(kx\right)\cos \left(\omega t\right)$
• Wave Number: $k=\frac{2\pi }{\lambda }$
• Angular Frequency: $\omega =2\pi f$
• Wavelength for nth Harmonic: $\lambda =\frac{2L}{n}$
• Frequency for nth Harmonic: $f_n=n{f}_1$
• Number of Nodes: n + 1
• Number of Antinodes: n