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I am currently reading Fundamentals of Physics (10th edition) and am stumped on a question (Chapter 7, Question 7) regarding the work done by gravity.

The question is as follows: there is a pig that goes down three slides, all three slides are at the same height from the ground, but of different lengths, they are therefore positioned at different angles. I'm asked to rank them by the amount of work the gravitational force does on the pig during each descent.

At this point of the chapter the book has given two apparently relevant formulas for calculating this work: one is that the work done by the gravitational force is equal to $mgd\cos\phi$ (where $m$ is the mass of the object, $g$ is the gravitational acceleration, $d$ is the displacement and $\phi$ is the angle between the force and the displacement), and the other is that $\Delta K = K_{f} - K_{i} = W$ (where the difference in kinetic energies is equal to the work).

Now the book says that along all three slides the amount of work done by the gravitational force is the same, but I fail to see how that is possible applying the first formula. I thought that somehow the cosine would make it so all three works are the same but trying with some plugged in values I get different values for different slides. Applying the second formula, since the kinetic energy of the pig is the same at the beginning and the end I can see how all three works could be zero, but then the works should still be zero if one of the slides happened to be lower that the others, right?

I am truly stumped, I actually know that the work done by gravity does not depend on the path of the object, through the conservation of energy formulas, but I can't understand how you can justify it with the formulas given.

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  • $\begingroup$ What do you consider or define displacement as, in this particular question? physicsclassroom.com/class/vectors/Lesson-3/Inclined-Planes $\endgroup$ – user140606 Jan 10 '17 at 1:53
  • $\begingroup$ What direction is the gravitational force? How far do the pigs move parallel to the gravitational force? The cosine factor simply picks out the parallel distance. $\endgroup$ – Bill N Jan 10 '17 at 2:15
  • $\begingroup$ Also, the kinetic energies at beginning and end are NOT the same unless you let the pig actually hit the ground, but then you have the ground doing work on the pig, too. $\endgroup$ – Bill N Jan 10 '17 at 2:17
  • $\begingroup$ Build some right angled triangles. See what happens if you change one of the acute angles, keeping the altitude a constant. $\endgroup$ – UKH Jan 10 '17 at 2:27
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    $\begingroup$ @AlexVlasov Why don't you answer your own question? That way your thought process will be useful to future readers. $\endgroup$ – rob Jan 10 '17 at 3:10
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The mistake I made was calculating the distance of each slide incorrectly, because of my own inexperience with trigonometry. The formula that made me understand was $dcos\phi = h$, whereby $d = h/cos\phi$. Thank you very much to everyone who answered.

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