1. Position of the $n$th bright fringe in YDSE

For slit separation $d$, screen distance $D$, and wavelength $\lambda$, the position of the $n$th bright fringe is

$y_n = \frac{n\lambda D}{d}.$

Here, for the $8^{\text{th}}$ bright fringe,

$y_8 = \frac{8\lambda D}{d(t)}.$

Since $d$ changes with time, $y_8$ also changes with time.

2. Given time-dependent slit separation

$d(t) = (0.8 + 0.04\sin \omega t)\,\text{mm}$

with

$\omega = 0.08\,\text{rad s}^{-1}.$

Convert to SI units:

$d(t) = \left(0.8 + 0.04\sin \omega t\right)\times 10^{-3}\,\text{m}.$

Also,

$\lambda = 6000\,\text{\AA} = 6\times 10^{-7}\,\text{m}, \qquad D=1\,\text{m}.$

3. Differentiate to get fringe speed

$y_8 = \frac{8\lambda D}{d}$

So,

$\frac{dy_8}{dt} = -\frac{8\lambda D}{d^2}\frac{dd}{dt}.$

Hence the speed is

$v = \left|\frac{dy_8}{dt}\right| = \frac{8\lambda D}{d^2}\left|\frac{dd}{dt}\right|.$

Now,

$d(t) = (0.8+0.04\sin\omega t)\times 10^{-3}$

Therefore,

$\frac{dd}{dt} = 0.04\omega \cos\omega t \times 10^{-3}\,\text{m/s}.$

Its maximum value is when $|\cos\omega t|=1$:

$\left|\frac{dd}{dt}\right|_{\max} = 0.04\times 0.08\times 10^{-3} = 3.2\times 10^{-6}\,\text{m/s}.$

4. For maximum fringe speed

Since

$v \propto \frac{|dd/dt|}{d^2},$

the maximum occurs when $|dd/dt|$ is maximum and $d$ is approximately at its mean value. Using the standard small-oscillation treatment here,

$d \approx 0.8\times 10^{-3}\,\text{m}.$

Thus,

$v_{\max} = \frac{8\times 6\times 10^{-7}\times 1}{(0.8\times 10^{-3})^2}\times 3.2\times 10^{-6}.$

Now compute:

$(0.8\times 10^{-3})^2 = 0.64\times 10^{-6} = 6.4\times 10^{-7}.$

So,

$\frac{8\times 6\times 10^{-7}}{6.4\times 10^{-7}} = \frac{48}{6.4} = 7.5.$

Hence,

$v_{\max} = 7.5\times 3.2\times 10^{-6} = 24\times 10^{-6}\,\text{m/s}.$

Convert to $\mu$m/s:

$24\times 10^{-6}\,\text{m/s} = 24\,\mu\text{m/s}.$

5. Final answer

The maximum speed of the $8^{\text{th}}$ bright fringe is

$\boxed{24}$

in $\mu\text{m/s}$.

6. Comparison with stored answer

Stored correct answer = $24$.

This matches the derived answer.