Video: Finding the Elastic Potential Energy Stored in an Elastic String

A ball of mass 0.5 kg hangs in equilibrium from the ceiling by a light elastic spring of natural length 1.2 m and a modulus of elasticity 9.8 N. How much energy is stored in the spring? Take 𝑔 = 9.8 m/s².

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Video Transcript

A ball of mass 0.5 kilograms hangs in equilibrium from the ceiling by a light elastic spring of natural length 1.2 meters and a modulus of elasticity 9.8 newtons. How much energy is stored in the spring? Take 𝑔 equals 9.8 meters per second squared.

So to find 𝑘, what we’re gonna do is divide 9.8, which is our modulus elasticity, by 1.2, the natural length of the spring. And this is going to give us 49 over six. And the units are going to be newtons per meter. And that’s because we saw that we had 9.8 newtons divided by 1.2 meters. Okay, so now what’s the second step?

The next step is to find the change in length of the spring. And we can do that using the equation 𝑚𝑔 is equal to 𝑘 multiplied by 𝑦, where 𝑦 is the change in length. So if we rearrange this, we can say that 𝑦 is gonna be equal to 𝑚𝑔 over 𝑘. So this is gonna be equal to 0.5 multiplied by 9.8 divided by 49 over six. And that’s because that’s what we found for 𝑘 in the first part. And this is gonna give us result of three over five meters.

So now the final stage is to calculate the potential energy, so the elastic potential energy, cause that’s gonna tell us how much energy is stored in the spring. So the elastic potential energy is gonna be equal to a half multiplied by 𝑘, which is our spring constant, multiplied by 𝑦 squared, which is our change in length squared. So this is gonna be equal to 0.5 multiplied by 49 over six, which is our 𝑘, multiplied by three over five, which is our change in length. And this is gonna be all squared.

And then, finally, when we calculate this, we’re gonna get the potential energy, so our elastic potential energy. It’s gonna be 1.47 joules. So therefore, we can say the amount of energy stored in the spring is going to be 1.47 joules.

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