Video: Spring Force

Which of the following is a correct unit of a spring constant? [A] N ⋅ m [B] m/N² [C] N/m [D] N ⋅ m² [E] N²/m

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Video Transcript

Which of the following is a correct unit of a spring constant? 1) newton times metre, 2) metres per newton squared, 3) newtons per metre, 4) newton times metre squared, 5) newton squared per metre.

Okay, so to answer this question, we need to recall something known as Hooke’s law. Hooke’s law tells us that if we have a spring — let’s say this is our spring — and that spring has a constant of 𝑘 and we apply a force 𝐹 to that spring, then the spring ends up extending such that its extension is 𝑥. And the relationship between 𝐹, 𝑘, and 𝑥 is given by Hooke’s law. In other words, the force applied to the spring is equal to the spring constant multiplied by the extension of this spring.

Now, in this question, we’re trying to find out the units of a spring constant. So let’s rearrange Hooke’s law to find out what the spring constant is. If we divide both sides of the equation by 𝑥, the extension, then 𝑥 cancels on the right-hand side, leaving us with just the spring constant. In other words, 𝑘, the spring constant, is equal to 𝐹, the force exerted, divided by 𝑥, the extension of the spring.

And so we can use this information to work out the units of the spring constant because the unit of force or at least the standard unit of force is the newton. And we have to divide this by the unit for the extension. Now, the extension is a distance or a length. It’s how much longer or shorter in some cases the spring is compared to its natural length. So the standard unit for a distance or a length is the metre.

And so the right-hand side of this equation has the unit newtons per metre. But the right-hand side of the equation is equal to the spring constant. So the spring constant must also have the unit of newtons per metre. So this checks out. And so at this point, we can cancel out number one cause that’s not the correct unit — same with number two, same with number four, and number five.

Our final answer then is that number three newtons per metre is the correct unit of a spring constant.

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