Video: Spring Force

Which of the following formulas correctly shows the relation between the force applied to a spring, the change in the spring’s length Δ𝑥, and the spring constant (also known as stiffness) of spring 𝑘? [A] 𝐹 = (1/2)𝑘Δ𝑥² [B] 𝐹 = 1/𝑘Δ𝑥 [C] 𝐹 = 𝑘/Δ𝑥 [D] 𝐹 = 𝑘Δ𝑥 [E] 𝐹 = 𝑥/Δ𝑘.

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Video Transcript

Which of the following formulas correctly shows the relation between the force applied to a spring, the change in the spring’s length Δ𝑥, and the spring constant, also known as stiffness, of spring 𝑘?

Okay, so in this question, we’re trying to find the relationship between the force applied, which we’ll call 𝐹, the change in the spring’s length, which is called Δ𝑥, and the spring constant, which is 𝑘. To do this, we need to recall a law known as Hooke’s law. Hooke’s law tells us that the force applied to a spring 𝐹 is directly proportional to the change in the spring’s length, which we’ve called Δ𝑥. And the constant of proportionality is 𝑘 — the spring constant. Now, that’s actually why it’s known as the spring constant because it’s the constant of proportionality.

But anyway, so we take Hooke’s law and we say that since 𝐹 is directly proportional to Δ𝑥, we say that 𝐹 is equal to something times Δ𝑥. Well, that something is the constant of proportionality 𝑘. And therefore, the relationship that we’re looking for is that 𝐹 is equal to 𝑘Δ𝑥. And that happens to be option number four. So we found our final answer.

The formula that shows the correct relation between 𝐹, 𝑘, and Δ𝑥 is 𝐹 is equal to 𝑘 multiplied by Δ𝑥.

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