# Potential energy of spring and gravity combined

• kyva1929
In summary, the individual potential energies of a spring and gravity can be expressed as a sum, with one term being dependent on y^2 and the other on y. To show that this combined potential energy can be written in the form of 1/2ky^2, the expression obtained by integrating -(-ky+mg) with respect to y must be manipulated. The attached image shows the original problem and equations.
kyva1929

## Homework Statement

Please see the attached image

## Homework Equations

I'm stucked in part B, asking me to show that the combined potential energy of spring and gravity can be written in the form of 1/2ky^2

## The Attempt at a Solution

integrating -(-ky+mg) with respect to y, from y_0 to y, (and setting the potential energy in y_0 = 0), gives me an expression which is just the algebraic sum of the potential energy in each separate case. One of them is dependent on y^2 while the other is dependent on only y. I have no idea how this could be rewritten in the form of 1/2ky^2, please help!

#### Attachments

• 4.9.jpg
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## 1. What is the formula for calculating the potential energy of a spring and gravity combined?

The formula for calculating the potential energy of a spring and gravity combined is: PE = 1/2kx^2 + mgh, where k is the spring constant, x is the displacement of the spring from equilibrium, m is the mass, g is the acceleration due to gravity, and h is the height of the object.

## 2. How does changing the spring constant affect the potential energy of a spring and gravity combined?

Changing the spring constant directly affects the potential energy of a spring and gravity combined. A higher spring constant means the spring is stiffer and will require more force to stretch or compress, resulting in a higher potential energy. Conversely, a lower spring constant will result in a lower potential energy.

## 3. Does the potential energy of a spring and gravity combined change with displacement?

Yes, the potential energy of a spring and gravity combined is directly proportional to the displacement of the spring from equilibrium. As the displacement increases, the potential energy will also increase, and vice versa.

## 4. Can potential energy of a spring and gravity combined be negative?

Yes, the potential energy of a spring and gravity combined can be negative. This occurs when the object is at a height below the reference point, such as when the spring is compressed or when the object is below the ground level.

## 5. How is potential energy of a spring and gravity combined used in real life?

Potential energy of a spring and gravity combined is used in various real-life applications, such as in trampolines, pogo sticks, and shock absorbers in cars. It is also used in engineering and construction to determine the stability and strength of structures, such as bridges and buildings.

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