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two different paths (path 1 and path 2) lead to top of a mountain. Path 1 is 4km long, while path 2 is less winding and is only 2km long. Which pass will lead to gaining more PE for the moutain climber?
Neither. Gravitational potential energy depends on the height of the object, not how it got there.
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Gravity is a conservative force. That means that the work done by gravity as an object moves between two points is independent of the path taken.
how about work done by the gravitational force of earth to keep the moon in orbit ?
Gravity is gravity. It will always be conservative. If you approximate the orbit of the moon as a circle, there would be no work done anyway, because the force would act perpendicularly to the displacement.
Right ... the important thing is work is done if a force acts on a body in the direction it moves. With uniform circular motion, the acceleration is always perpendicular to the direction in which the moon is moving. In other words, the velocity vector is perpendicular to the acceleration vector. The acceleration vector always points towards the center of the circle; the velocity vector is tangential to the circle.
Now, for elliptical orbit, it's not that the work everywhere in the orbit is zero. But in the course of one complete rotation, the net work is zero.