1. Forces

Driving Force: \( F_g = mg \sin \theta \) (constant).
Retarding Force: Friction \( f = \mu N \).

2. Normal Force

The normal force \(N\) has two components: 1. From the wall due to gravity component: \( N_g = mg \cos \theta \). 2. From water pressure: \( N_p \). As the plate slides down distance \(x\), the average depth increases, increasing the hydrostatic pressure force. Force due to pressure \( F_p \propto x \). Specifically, total normal force \( N(x) = mg \cos \theta + C \cdot x \).

3. Equation of Motion

\[ ma = mg \sin \theta – \mu (mg \cos \theta + C x) \] \[ a = g(\sin \theta – \mu \cos \theta) – \frac{\mu C}{m} x \] This is of the form \( a = A – B x \), which represents Simple Harmonic Motion (SHM) about a shifted equilibrium. The motion is a half-cycle of SHM because the plate starts at rest and stops when \(v=0\).

4. Time Calculation

The angular frequency \(\omega\) is the coefficient of the \(x\) term (square rooted): \[ \omega^2 = \frac{\mu C}{m} \] Detailed calculation of \(C\) (Pressure Force coefficient): Pressure force \( F = \int \rho g h dA \). For a plate of area \(S = l^2\), the force increases linearly with displacement \(x\). \( F_{pressure} = \rho g (x \sin \theta) S \). So \( C = \rho g S \sin \theta \). \[ \omega = \sqrt{\frac{\mu \rho g S \sin \theta}{m}} \] The time to stop is half the time period: \[ t = \frac{T}{2} = \frac{\pi}{\omega} = \pi \sqrt{\frac{m}{\mu \rho g l^2 \sin \theta}} \]

Answer: \(\pi \sqrt{\frac{m}{\mu \rho_0 g l^2 \sin \theta}}\)