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If gravitational force is a conservative force, which means it doesn't change the total amount of mechanical energy of a body, why is it that when we lift an object its potential energy increases?

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You have misunderstood the definition of a conservative force. No work is done against a conservative force if an object is taken in a closed loop. However, if the object moves from one place to another, work can be done by or against a conservative force, and so the mechanical energy of the object can change.

An equivalent definition of a conservative force is that the work done by or against the force depends only on the initial and final positions of the object, and not on the path taken between those positions.

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While we are lifting the object, the force we are doing on it is not conservative. On the other hand, if we throw up an object, after it is released, each $\Delta h$ is paid with a loss of kinetic energy. Because in this case the only acting force is gravity, that is conservative.

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You're right-- the potential energy of the object-Earth system changes when you lift the object. Why? Because work is done on the object by the Earth's gravitational field. The potential energy associated with a given conservative force is always defined so that $$W_C = -\Delta U. \tag{1}$$ Think of a book (for concreteness, I've assumed that your "object" is a book) being lifted from an initial height $y_0$ to a final height $y_f$ (relative to an arbitrary $y = 0$ line). The gravitational force on it is nearly constant, $\mathbf{F_g} = -mg\hat{y}.$ The work done by $\mathbf{F_g}$ is therefore $$W_g = \int_{y_0}^{y_f}{-mg\,dy} = -mg(y_f-y_0)=-mg\Delta y.$$ By $(1)$, $$\Delta U_g = mg\Delta y.$$ So we see that any change in the height of the book produces a change in potential energy.

Now, you might be wondering where the "extra" energy comes from in this example. After all, the kinetic energy of the book is constant (assuming that it is lifted at uniform speed) and the Earth-book system's potential energy is increasing. It seems like mechanical energy is being pulled from nowhere. But it's not. The external agent (you) is doing work against the gravitational force, pumping energy into the system.

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When you lift a mass you do work on the mass to increase its gravitational potential energy. You do the work, not gravity. When you release the mass it falls, releasing the potential energy you put into it. So gravity itself has done no work on the mass. You did work on it lifting it, which was spent as it fell.

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Conventionally, we consider the potential energy of any object within the Earth's vicinity to be negative. Rather that going for a mathematical solution, we can understand it with a general sense. When we lift an object upwards, we are moving that object away from the vicinity of the Earth. It is like, we are reducing the debt of energy that the object was in. As for the Mathematical part, it has already been explained by the others.

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