CALC A clever engineer designs a 'sprong' that obeys the force law Fx=−q(x−xeq)³ , where xeq is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, we'll let xeq=0 m .Then Fx=−qx³. b. Find an expression for the potential energy of a stretched or compressed sprong.

Video transcript

Welcome back, everybody. We are told that scientists have made a new device called a Twis Matic. And we are told that its force law is given by this equation right here. Now we are going to assume that its equilibrium position is X not equal zero, which just makes its equation for force, simply negative C which is some constant times X over two cubed. And we are tasked with driving an expression for potential energy of the twist O Matic. Let's look at our answer choices here. Answer choice A is that our potential energy is equal to C X to the 6/8 answer. Choice B is C X to the fourth over eight answer. Choice C is C X to the fourth over 32 D is C times X to the second over 16. So what do we do here? Well, we know that any equation for force is just equal to the negative of the derivative of the potential energy with respect to X. Now, I'm going to multiply both sides by negative D X. And what this gives us is that our derivative for potential energy is equal to the negative of the force times D X. And what we will go ahead and do is we will integrate both sides. And what this yields is that our potential energy that we are looking for is just going to be equal to the negative of the integral of our force equation with respect to X. Let's go ahead and plug in our force equation and find that we have that our formula for potential energy is equal to the negative integral of negative C Iams X over two cubed with respect to X. Now, what we can do here is we can take out the negative C and the 1/2 cubed giving us C over eight times the integral of X cubed D X. And we're able to take it out because it's simply a constant. When we evaluate the integral here, uh using the reverse of the power rule, we get that C over eight, we multiply times X to the fourth over four. And then finally multiplying straight across, we get C X to the fourth over 32 which corresponds to our final answer. Choice of C. Thank you all so much for watching. Hope this for your help. We will see you all in the next one.