Show that the time when the level of the water becomes h<|h max| is equal to?

To show that the time when the level of the water becomes h<|h max| is equal to, we can use the following equation:

h(t) = h max * sin(ωt)

where:
h(t) is the level of the water at time t
h max is the maximum level of the water
ω is the angular frequency

When h(t) becomes less than |h max|, we can set h(t) equal to |h max| and solve for t:

|h max| = h max * sin(ωt)

Dividing both sides by h max:

1 = sin(ωt)

To find the time t when sin(ωt) = 1, we know that sin(π/2) = 1. Therefore, we can set ωt = π/2 and solve for t:

ωt = π/2
t = π/(2ω)

Therefore, the time when the level of the water becomes h<|h max| is equal to t = π/(2ω).

The time when the level of the water becomes h<|h max| is equal to:

$$t=\frac{1}{\sqrt{2 g}}\left[\left(\frac{\pi}{2}-\theta\right)\sqrt{|h_{max}|-h}+\sqrt{|h_{max}|}\left(\frac{\pi}{2}+\theta\right)\right]$$

where:

- \(g\) is the acceleration due to gravity,
- \(\theta\) is the angle of the container with the vertical,
- \(h_{max}\) is the maximum height of the water in the container,
- \(h\) is the height of the water in the container at time \(t\).