Tidal/harbour water level SHM

5. A vertical ladder is fixed to a wall in a harbour. On a particular day the minimum depth of water in the harbour occurs at 0900 hours. The next time the water is at its minimum depth is 2115 hours on the same day. The bottom step of the ladder is 1 m above the lowest level of the water and 9 m below the highest level of the water. The rise and fall of the water level can be modelled as simple harmonic motion and the thickness of the step can be assumed to be negligible. Find

1. the speed, in metres per hour, at which the water level is moving when it reaches the bottom step of the ladder,
2. the length of time, on this day, between the water reaching the bottom step of the ladder and the ladder being totally out of the water once more.
   

1. In a harbour, the water level rises and falls with the tides with simple harmonic motion.

On a particular day, the depths of water in the harbour at low and high tide are 4 m and 10 m respectively. Low tide occurs at 12:00 and high tide occurs at 18:20

1. Find, in \(\mathrm { mh } ^ { - 1 }\), the speed at which the water level is rising on this particular day at 13:35 A ship can only safely enter the harbour when the depth of water is at least 8.5 m .
2. Find the earliest time after 12:00 on this particular day at which it is safe for the ship to enter the harbour, giving your answer to the nearest minute.

4. A small shellfish is attached to a wall in a harbour. The rise and fall of the water level is modelled as simple harmonic motion and the shellfish as a particle. On a particular day the minimum depth of water occurs at 1000 hours and the next time that this minimum depth occurs is at 2230 hours. The shellfish is fixed in a position 5 m above the level of the minimum depth of the water and 11 m below the level of the maximum depth of the water. Find

1. the speed, in metres per hour, at which the water level is rising when it reaches the shellfish,
2. the earliest time after 1000 hours on this day at which the water reaches the shellfish.

6. The rise and fall of the water level in a harbour is modelled as simple harmonic motion. On a particular day the maximum and minimum depths of water in the harbour are 10 m and 4 m and these occur at 1100 hours and 1700 hours respectively.

1. Find the speed, in \(\mathrm { m } \mathrm { h } ^ { - 1 }\), at which the water level in the harbour is falling at 1600 hours on this particular day.
2. Find the total time, between 1100 hours and 2300 hours on this particular day, for which the depth of water in the harbour is less than 5.5 m .
   (Total 14 marks)

4. On a particular day, high tide at the entrance to a harbour occurs at 11 a.m. and the water depth is 14 m . Low tide occurs \(6 \frac { 1 } { 4 }\) hours later at which time the water depth is 6 m . In a model of the situation, the water level is assumed to perform simple harmonic motion.
Using this model,

1. write down the amplitude and period of the motion. A ship needs a depth of 9 m before it can enter or leave the harbour.
2. Show that on this day a ship must enter the harbour by 2.38 p.m., correct to the nearest minute, or wait for low tide to pass.
   (6 marks)
   Given that a ship is not ready to enter the harbour until 5 p.m.,
3. find, to the nearest minute, how long the ship must wait before it can enter the harbour.

3. The vertical motion of a point on the surface of the water in a certain harbour may be modelled as Simple Harmonic Motion about a mean level. The diagram shows that, on a particular day, the depth of water in the harbour at low tide is 2 m and the depth of the water in the harbour at high tide is 10 m . The table below shows the times of high and low tides on this day.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-4_405_912_621_233}

1. Write down the period and amplitude of the motion.
2. Let \(x \mathrm {~m}\) denote the height of water above mean level \(t\) hours after 5a.m. Find an expression for \(x\) in terms of \(t\).
3. The depth of water must be at least 4 m for boats to safely use the harbour. Determine the earliest time, after low tide at 5 a.m., at which boats can safely leave the harbour and hence find the latest possible time of return before the next low tide.
4. Calculate the rate at which the level of water is falling at 2 p.m.