Springs & Elastic Potential Energy: Study with Video Lessons, Practice Problems & Examples

Elastic potential energy, denoted as $U^{\mathrm{elastic}}=\frac{1}{2}k{x}^2$, is stored in springs when they are compressed or stretched. This energy is similar to gravitational potential energy, which is calculated as $U{g}_0=mgy$. In energy conservation equations, both types of potential energy can be combined, allowing for the analysis of systems involving springs and blocks using Hooke's law and energy principles.

Energy in Horizontal Springs

Hey, guys. So we've already seen one type of potential energy, which was called gravitational. And in this video, I want to introduce you to the other type of which is called elastic or spring potential energy here. Alright? So let's take a look here. The idea is the same. Potential energy, remember, is just stored energy. So just like you store energy when you lift something, springs store energy when you compress or stretch them. So this energy is called elastic potential energy. So let's take a look here. Right? So if you're at the ground, for gravitational potential, your gravitational potential energy is 0. But if you raise it to some height of y, then your U g = m g y here. Well, it's the same idea elastics for springs. Basically, the ground is like the equilibrium position. When springs are relaxed, they have no stored energy so their elastic potential energy is 0. But what happens is that when you push against them and you deform them by compressing or stretching them, then you have some applied forces and some spring forces. And the spring force here depends on your deformation, which is this x here. Alright. So remember that we said the relationship between work and gravitational potential energy was that W g = - δ U g . It's the same idea for springs. The work that's done by springs is going to be the change in spring or elastic potential energy, this - δ U e . So what we saw here is that if this - δ U g = - m g δ y , then the equation for gravitational potential was just U g = m g y . It's the same exact thing we can do for springs. We can basically cancel out these negative signs here, and we can say that the elastic potential energy is really equal to 1 2 k x 2 . So this is the equation that we're going to use in our energy conservation equations. Now how does this change our energy conservation equation? It actually really doesn't. We're still going to write K+U+work done=K+U. The only thing is that up until now, we've only been focused on gravitational potential energies. But now we're actually just going to include elastic potential energies because these things are the same type. They're both potential energies, so we can just combine them. So our potential energy is going to be U g + U e . So all you have to do now is just keep track or keep on the lookout for any springs in our problems. Let's go ahead and take a look here. We have a block that's attached to a horizontal surface. We have the spring constant k and we're going to push the block with a force of 100 newtons. So I've got my applied force. The magnitude is 100, which means that the magnitude of the spring force that pushes back is also 100. So what we want to calculate is the compression distance, how far we've actually compressed the spring. So that's actually this distance right here. This is x . So how do we solve this? Do we use energy? Do we use something else? Basically, the idea here is that whenever you have spring problems in which objects are stationary like we have in this first part here, we're still just holding the block up against the compressed spring. Then we're going to solve this by using forces. And the idea here is if we want to solve the compression distance, remember, which is just x , we can solve this by using Hooke's law, which says that the compression, sorry, the absolute value of the spring force equals k x . So we actually have the spring force and the applied force. They're both 100. And we have the spring constant. So we can figure out what our x is by just rearranging for this. Let's go ahead and do that. So x is really just going to be equal to the magnitude of your applied force divided by k which is just 100 divided by 500

Springs in Rough Surfaces

Hey, guys. Hopefully, you got a chance to work this one out on your own. Let's go ahead and check it out together. So we have a 4 kilogram box that is moving to the right with an initial speed of 20, and it's going to collide with the spring. The force constant of the spring is 600. What happens is the spring is going to compress. So this box is pushing up against the spring as it collides with it, and then it's going to compress. So the box comes to a stop due to maximum compression when the spring is coiled up a little more.

Now, we want to write our energy conservation equation. We can't solve this using forces because the force that occurs as you're compressing the spring isn't constant; it varies. We have to use energy conservation. Our equation is: K i + U i + W nonconservative = K f + U f . The initial kinetic energy is significant as the box is moving with some speed. There's no initial gravitational potential energy, and since the spring isn't compressed initially, there's no elastic potential energy. There's no work done by non-conservative forces, and the final kinetic energy is zero, so all its kinetic energy has transformed into elastic potential energy.

We write our expressions: 1 2 m v initial 2 = 1 2 k x final 2 . We move k to the other side to get m v initial 2 k = x final 2 . Take the square root to find: x final = m v initial 2 k . Plug in the numbers to get a compression distance of 1.63 meters.

So, after the block has stopped, it has transferred all of its energy and compressed the spring by 1.63 meters. That's the answer. Let me know if you guys have any questions, and I'll see you in the next one.

A 4-kg block moving on a flat surface strikes a massless, horizontal spring of force constant 600 N/m with a 20 m/s. The block-surface coefficient of friction is 0.5. Calculate the maximum compression that the spring will experience.

• A. Δx = 0.63 m
• B. Δx = 1.60 m
• C. Δx = 1.63 m
• D. Δx = 1.67 m

Here’s what students ask on this topic:

What is elastic potential energy and how is it calculated?

Elastic potential energy is the energy stored in a spring when it is compressed or stretched. It is similar to gravitational potential energy, which is stored when an object is lifted to a height. The formula for elastic potential energy is given by:

$U_{\mathrm{elastic}}=\frac{1}{2}k{x}^2$

where $k$ is the spring constant, and $x$ is the displacement from the equilibrium position. This energy is stored as long as the spring is deformed and can be released when the spring returns to its equilibrium position.

How does elastic potential energy relate to energy conservation?

In energy conservation equations, elastic potential energy can be combined with other forms of potential energy, such as gravitational potential energy. The total mechanical energy of a system is conserved, meaning the sum of kinetic energy and potential energy (both gravitational and elastic) remains constant if no non-conservative forces (like friction) are doing work. The energy conservation equation can be written as:

$K+U_g+U_{\mathrm{elastic}}=K+U_g+U_{\mathrm{elastic}}$

This allows for the analysis of systems involving springs and blocks using Hooke's law and energy principles.

What is Hooke's Law and how is it used in spring problems?

Hooke's Law describes the relationship between the force exerted by a spring and its displacement from the equilibrium position. It is given by:

$F=kx$

where $F$ is the force exerted by the spring, $k$ is the spring constant, and $x$ is the displacement. In spring problems, Hooke's Law is used to determine the force exerted by the spring when it is compressed or stretched. This force can then be used in energy conservation equations to analyze the motion and energy of objects attached to the spring.

How do you calculate the launch speed of a block released from a compressed spring?

To calculate the launch speed of a block released from a compressed spring, you use the principle of energy conservation. Initially, the block has elastic potential energy stored in the compressed spring and no kinetic energy. When the spring is released, this potential energy is converted into kinetic energy. The energy conservation equation is:

$\frac{1}{2}k{x}^2=\frac{1}{2}m{v}^2$

where $k$ is the spring constant, $x$ is the initial compression distance, $m$ is the mass of the block, and $v$ is the final velocity. Solving for $v$ gives:

$v=\sqrt{\frac{k}{m}{x}^2}$

Plug in the values for $k$, $x$, and $m$ to find the launch speed.

When should you use energy principles instead of force principles in spring problems?

Energy principles should be used in spring problems when the force exerted by the spring is not constant, such as when an object is moving between two points on a spring. In these cases, the force changes as the spring compresses or stretches, making it difficult to use force principles and motion equations. Instead, energy conservation equations can be used to analyze the system. For example, when calculating the launch speed of a block released from a compressed spring, energy principles are used because the force exerted by the spring changes as the block moves. By using energy conservation, you can account for the conversion of elastic potential energy into kinetic energy.