Calculating work for different springs Calculate the work requ...

Question

Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.

b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position

Accepted Answer

Welcome back, everyone. In this problem, it takes 4 joules of work to stretch a spring 0.15 m beyond its equilibrium. How much work is needed to stretch it 0.6 m from equilibrium? Use calculus. A says is 32 joules, B 48 joules, C 64 joules, and D 72 joules. Now what do we know using calculus that can help us to figure out how much work is needed to stretch it 0.6 m from equilibrium? Well, recall that the work done is the integral of the force with respect to X. Where our forests. Is equal to the spring constant multiplied by our distance X based on Hooke's law. This would imply then that the work done is going to be the definite integral of K X with respect to X. Now from our problem statement we were already told that it takes 4 joules of work to stretch a spring 0.15 m beyond equilibrium, and our spring constant stays the same. So if we can integrate from that scenario to figure out our spring constant, then we can use it to figure out how much work is needed to stretch it 0.6 m from equilibrium. So let's go ahead and do that. So when our work W is equal to 4 joules and the distance X is equal to 0.15 m, then. Based on our formula for the work done, 4 joules is going to be equal to the integral between our lower limit 0 from equilibrium to our upper limit 0.15, the distance the spring was stretched of KX with respect to X. And if we go ahead and integrate, that is going to be K multiplied by a half of X2d between the limits of 0 and 0.15. When we substitute 0.15 into our expression, that would be a half of 0.15 squared, which is 0. 0, well, let me read this properly here. 0.01125. OK. So, 4 equals 0.01125K. And if we were to substitute zero, we know we would end up with zero. So this would imply then that k will be equal to 4 divided by 0.01125, which will be equal to 3200 divided by 9 Newtons per meter, and we'll leave it written in this way for no so that we'll be able to use it, uh, in the next part of our problem. So now that we have our spring constant, then we can apply it when our distance X is equal to 0.6 m to find the work done. Because no, this means that the work done will be equal to the integral between 0 and 0.6 again of KX with respect to X. So that's going to be K multiplied by a half of X squared between 0 and 0.6. Since we know K is 3200 divided by 9, we can substitute it, so that we'll be multiplying a half of 0.6 squared. And no, that's gonna be 3209th multiplied by 0.18, which equals 64. Thus the work required is 64 joules. If we look back at our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.
b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position

Key Concepts

Hooke's Law

Work Done by a Variable Force

Determining the Spring Constant from Work

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