Vertical spring energy transformation

In summary: Kelly explains the transformation of energy in a mass-spring system. In horizontal motion, there is maximum elastic potential energy at the two ends and maximum kinetic energy at the equilibrium position. However, in vertical motion, gravitational potential energy is also present. At the bottom, there is maximum elastic potential energy and at the top, there is maximum gravitational potential energy. Throughout the motion, there is a balance of energy between gravitational potential energy, elastic potential energy, and kinetic energy. When the spring is vertical, there is an interesting concept of whether to measure the stretching distance from the horizontal or vertical equilibrium position, which affects the application of conservation of energy. When measured from the vertical equilibrium position, the potential energy stored in the spring and the kinetic energy do not
  • #1
plutonium
15
1
a mass is attached to a spring and released. it then oscillates in simple harmonic motion. what is the transformation of energy?

i understand how it works horizontally (max Ee at the 2 ends, max Ek at the equilibrium position), but how does it work vertically now that Eg is also present?

this is how i see it: setting the maximum stretch as h = 0, upon release of the mass, Eg = max (let's set max as 1 unit), Ee = 0, Ek = 0. at the bottom, Ee = 1, Eg = 0, Ek = 0. but at the middle, which is the equilibrium point, Ek should be max, and Eg and Ee should be 0, but it is half way of the unstretched position, so Ee = 1/2, and Eg is also 1/2 since it is half way between max height and min height (which i set to 0), so how can there be Ek? this is what's confusing me.
 
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  • #2
Ee=1/2kx^2. At the halfway point, the Ee is not half of that at the max strecth.
 
  • #3
OKay okay,let's try and clear up things once and for all. So in the case of the vertical mass and spring, there are three distinct kinds of energy transformations that are occurring. They are Kinetic energy, gravitational potential energy and elastic potential energy. In the simple case, their formuals are (in order): 1/2mv^2, mgh and 1/2kx^2,where m is the mass of the BLOCK, v is the velocityof the block, h is the change in height of the block, k is the spring constant, and x is the stretching distance of the spring.

OKay, that was a mouthful, now for the big picture. So when the mass is released. it falls...simple. This is due to the gravitaional potenial energy! As it falls, that energy is being converted to both kinetic AND elastic potential enegry, which should make sense; the block is picking up speed while the spring is stretching. Throughout the process, its a sort of balance scale of energy, where at the VERY top of the oscillation you have GRAVITATIONAL potential energy, and ELASTIC potential energy at the bottom. Kinetic is just "there" so to speak in the midst of transition.

Mkay, I know that was a lot, but hopefully it put the whole picture together.
 
  • #4
plutonium said:
a mass is attached to a spring and released. it then oscillates in simple harmonic motion. what is the transformation of energy?

i understand how it works horizontally (max Ee at the 2 ends, max Ek at the equilibrium position), but how does it work vertically now that Eg is also present?

this is how i see it: setting the maximum stretch as h = 0, upon release of the mass, Eg = max (let's set max as 1 unit), Ee = 0, Ek = 0. at the bottom, Ee = 1, Eg = 0, Ek = 0. but at the middle, which is the equilibrium point, Ek should be max, and Eg and Ee should be 0, but it is half way of the unstretched position, so Ee = 1/2, and Eg is also 1/2 since it is half way between max height and min height (which i set to 0), so how can there be Ek? this is what's confusing me.

There is actually something interesting and nontrivial in the case of a vertical spring. In calculating [itex] {1 \over 2 } k x^2 [/itex], where is the x measured from? One can measure it from the equilibrium position corresponding to the length when the spring is horizontal (let's call this the "unstretched" equilibrium position) OR one can measure it from the new equilibrium position when the spring is vertical (the "stretched eq. pos.). This affects the way one applies conservation of energy.

Usually, most people start working with respect to the new, "stretched" equilibrium position (so that x goes from +A to -A, where A si the amplitude of the motion). In that case, it turns out that when applying conservation of energy, one does not need to include mgh! One simply uses the potential energy stored in the spring and the kinetic energy.
It's easy to rpove this explicitly and is ultimately due to the fact that the force of the spring is a linear force. Shifting the origin in [itex] {1 \over 2 } k x^2 [/itex] adds a linear term in the shift and this term is exactly canceled by mgh. Quite neat to see at work, actually.

Patrick
 

1. What is vertical spring energy transformation?

Vertical spring energy transformation is the process by which potential energy is converted into kinetic energy and vice versa through the use of a spring that is oriented vertically. This can occur when an object is dropped onto a spring, causing it to compress and store potential energy, which is then released as kinetic energy when the object is launched upward.

2. How does the spring store and release energy?

The spring stores energy by compressing or stretching when a force is applied to it. This causes potential energy to be stored in the spring, which is then released as kinetic energy when the spring returns to its original shape. This process can be repeated as long as there is enough energy to compress or stretch the spring.

3. What factors affect the amount of energy transformed by a vertical spring?

The amount of energy transformed by a vertical spring is affected by several factors, including the spring constant, the distance the spring is compressed or stretched, and the mass of the object attached to the spring. A stiffer spring with a higher spring constant will store more potential energy, and a larger mass will require more energy to be launched upward.

4. How is the energy transformation affected by the height of the drop?

The height of the drop can affect the amount of energy transformed by a vertical spring. A higher drop will result in a greater amount of potential energy being stored in the spring, which will then be released as kinetic energy when the spring launches the object upward. However, this also depends on the mass of the object and the spring constant.

5. What is the conservation of energy law and how does it relate to vertical spring energy transformation?

The conservation of energy law states that energy cannot be created or destroyed, only transformed from one form to another. This means that the total amount of energy in a system remains constant. In the case of vertical spring energy transformation, the potential energy stored in the spring is transformed into kinetic energy, but the total energy in the system remains the same. This law also applies to the reverse transformation, where the kinetic energy of an object is transformed into potential energy as it is launched upward by the spring.

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