# Change in Gravitational Potential Energy in Oscillating water mass

• Physics2013
In summary, the conversation discusses the motion of water in a lake and how it can be approximated as a simple pendulum. The increased gravitational potential energy of the water system is given by U=(1/6)bpgL(Yo2). To solve the problem, one needs to find the height of the center of mass and express it in terms of Yo and D, the height of the left edge of the water surface from the bottom.
Physics2013

## Homework Statement

Seiche in a lake. The simplest motion of water in a lake can be approximated as simply the water surface tilting but remaining flat.
Imagine a lake of rectangular cross section of length L and with depth h where (h<<L). The problem resembles that of the simple pendulum, the kinetic energy is due to horizontal
motion and potential energy due to small change in height.
So, imagine at some instant the water level at the increased end is +Yo with respect to normal level. Show that the increased gravitational potential energy of whole mass system is given by
U=(1/6)bpgL(Yo2)

## Homework Equations

U=mgh
Figure attatched!

## The Attempt at a Solution

I know the mass of the water will be defined as the volumeXdensity. the volume would be, if width of lake is defined as b, Lbh. Density is denoted p. Thus, m=Lbhp.
The gravitational constant g is applicable so g=g. Finally, as the height is rising as much as it is falling I can take the average height and thus, h = h+(1/2)Yo?
But, now plugging this into the equation U=mgh I have
U=(Lbhp)g(h+(1/2 Yo), and this is definitely not the 'answer' stated in the problem.
Could anyone help identify where is my logic going wrong?

#### Attachments

• Graph.jpg
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This one is deadly wrong: h = h+(1/2)Yo (h = h + something?!)
Note that h is the height from the bottom to the average level of the water surface: h = const! What is left is to find the height of the center of mass (COM). Hint: apply the formula of COM's position and express COM's height in term of Yo and D, where D is the height of the left edge of the water surface from the bottom (D is not h!). Then find the geometrical relation between D and Yo.

## 1. What is gravitational potential energy?

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is the energy that is required to move an object from one point to another against the force of gravity.

## 2. How does the oscillating water mass affect gravitational potential energy?

The oscillating water mass affects gravitational potential energy by changing its position in the gravitational field. As the water mass moves up and down, its distance from the center of gravity changes, causing a change in its gravitational potential energy.

## 3. What factors influence the change in gravitational potential energy in an oscillating water mass?

The factors that influence the change in gravitational potential energy in an oscillating water mass include the mass of the water, the amplitude of the oscillation, and the height at which the oscillation occurs. Additionally, the strength of the gravitational field and the direction of the oscillation also play a role.

## 4. How is the change in gravitational potential energy in an oscillating water mass calculated?

The change in gravitational potential energy in an oscillating water mass can be calculated using the formula: ΔU = mgh, where m is the mass of the water, g is the acceleration due to gravity, and h is the change in height of the water mass.

## 5. What are some real-life applications of understanding the change in gravitational potential energy in oscillating water masses?

Understanding the change in gravitational potential energy in oscillating water masses can help in the design and operation of wave energy converters and tidal power plants. It can also be used in the study of tides and ocean currents, as well as in predicting the behavior of floating structures such as ships and oil rigs.

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