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(II) If $U=3 x^{2}+2 x y+4 y^{2} z,$ what is the force. $\vec{\mathbf{F}}$ ?

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$\overrightarrow{\mathbf{F}}=\hat{\mathbf{i}}(-6 x-2 y)+\hat{\mathbf{j}}(-2 x-8 y z)+\overrightarrow{\mathbf{k}}\left(-4 y^{2}\right)$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

Rutgers, The State University of New Jersey

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Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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all right. Some calculus in this problem. Uh, you're given of potentially you now. Two break form. You is so us three x squared plus two x y plus for y squared Z, and you have to find the force f which is vector. So we want something that will have, uh I had component or ex component jihad compartment. Why? Component and Kay had a bar. Zika Uh So what we have to do, of course, is remember that defector force is the, uh, derivative of you. With respect, Tio are where are is a position cord. So in other words, we have to take a set of partial derivatives one for each of these dimensions, So f the vector is just negative. D u x partial ofyou with respect us Plus, do you d wide a partial ofyou wants respect to Why, and do you see partial of you? Let's respect easy Now let's do each of these individually So D u d X is just taking the partial derivative of packs with respect to be it squared. Plus two x y us for y squared. See themselves looking at Each of these trumps the derivative with respect facts of three X squared is six X right. You pull out the two in front and explains, goes down by one. And so you have six x two ex wives two times why exponents of so explain so X to the one becomes zero on then the why, of course, remains self us too I and and then this one has no ex chairman it so there's no so that zero That's so six experts to wise a partial Let's respect X then the partial with respect Why is, um do you do? Why of all of this stuff? Not going to write it down again, But, uh, something cure. So first term has no why? So that doesn't count. Second term miss to X Uh, and third term is four times to, uh times y to the one So two X plus 818 Why see the four z times two son's wife, what remains and finally do you Dizzy is much to say where partial derivative with respect to see of all of that stuff. And so first term has those. The second term was nosy. Third time's nearly woman had C and they're no power saw you have us for why Square, Um And so being armed with all those we can finally write down F iss S O. S is minus six x plus two. Why, uh, in, uh, I had directions. Us two x plus eight y z in, uh, j have direction plus for y squared in the K had direct. And so that is what f is there. That is what the force is in better form and that's it.

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