# Math Help - Displacement of water

1. ## Displacement of water

A water-filled container is sitting still on a platform as shown.

Suddenly, the platform starts shaking vertically due to the action of a nearby machine.

An accelerometer placed on the contained wall measures the vertical acceleration [m/s2] as

A cos wt

What is the smallest h (the distance from water surface to the top of the container wall) when the water will NOT spill over the edge? Ignore the friction between water and the container walls. Give your answer in millimeters.

A = 416;

w = 147;

Could someone please show me how to do this?

2. Hello m_i_k_o
Originally Posted by m_i_k_o
A water-filled container is sitting still on a platform as shown.

Suddenly, the platform starts shaking vertically due to the action of a nearby machine.

An accelerometer placed on the contained wall measures the vertical acceleration [m/s2] as

A cos wt

What is the smallest h (the distance from water surface to the top of the container wall) when the water will NOT spill over the edge? Ignore the friction between water and the container walls. Give your answer in millimeters.

A = 416;

w = 147;

Could someone please show me how to do this?
This is not particularly easy, but here's the general method:

• Sketch a graph of acceleration against time - a standard cosine graph.

• While the acceleration of the container exceeds $g$ downwards, the water will be moving upwards relative to the container.

• Solve the equation for $t: 416\cos147t=9.8$

• Interpret this solution in conjunction with the sketch-graph to find the first two values of $t$ for which the acceleration is less than $-9.8$; i.e. numerically greater than $9.8$ downwards.

• Writing $a = \frac{dv}{dt}$ and then $v = \frac{ds}{dt}$, solve the differential equation to find $v,s$ in terms of $t$.

• Use the two values of $t$ to find the velocity and the displacement of the container during this phase of the downward acceleration.

• Hence find the displacement of the container relative to the water, giving the minimum height of the wall above the water-level.