Object slides down a circular hill and loses some of its initial mechanical energy due to friction. As an object slides down, friction increases and force in direction of motion decreases until the object reaches bottom of the hill. Intial height is equal to the radius of the circular path. How can you solve this problem and find the final velocity at the bottom using calculus.
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Generally you want to integrate the friction force over the whole path using a line integral (with a circular hill you probably want to use spherical coordinates). The integral gives you the energy lost to friction, while you can get the freed potential energy easily from the height difference.
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3$\begingroup$ One needs to compute the friction force dynamically, as it is increased by the centripetal force. The solution can be tricky, as the current speed depends on the path integral of the friction force, while the friction force depends on the current speed. $\endgroup$– PoutnikCommented Mar 23, 2019 at 11:49
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2$\begingroup$ With several inter-related variables, I would set this problem up as a numeric integration on a spreadsheet, which would let you choose values for the constants. $\endgroup$ Commented Apr 5, 2021 at 15:39