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## Main Question or Discussion Point

Hello Everyone,

I am having a very difficult time intuitively understanding the formula -dU/dx = Force(x). I don't want help deriving it, I'm simply looking for an intuitive understanding about why this might be true. An example with gravity would really help.

My thoughts so far: I find working with -Δ(U)/Δ(x) = avg Force easier for understanding. So lets say a ball is falling from initial height xi and ends up at final height xf. Keeping this same displacement, if a ball loses 1,000,000,000,000 J of potential energy in that displacement, then the avg Force over that displacement must be very big........why? what force? net force? gravitational force? Instead, if the ball loses .000000001 J of potential energy in that exact displacement, then the force acting on the ball must have been.....small? why?

Instead, let's say that over a displacement, the ball loses 100 J of potential energy. If we make the displacement particularly big, say 1,000,000,000 m, then, man, it took that much displacement for the ball to lose 100 J? so the force must be ... very small? Makes a little more sense... if the ball loses 100 J in .001m, then the force was very large? What does the force have to do with how much energy is lost?

Lastly, why the negative sign? Bonus points for connecting this with ∫f(x)dx = -U

Thanks

I am having a very difficult time intuitively understanding the formula -dU/dx = Force(x). I don't want help deriving it, I'm simply looking for an intuitive understanding about why this might be true. An example with gravity would really help.

My thoughts so far: I find working with -Δ(U)/Δ(x) = avg Force easier for understanding. So lets say a ball is falling from initial height xi and ends up at final height xf. Keeping this same displacement, if a ball loses 1,000,000,000,000 J of potential energy in that displacement, then the avg Force over that displacement must be very big........why? what force? net force? gravitational force? Instead, if the ball loses .000000001 J of potential energy in that exact displacement, then the force acting on the ball must have been.....small? why?

Instead, let's say that over a displacement, the ball loses 100 J of potential energy. If we make the displacement particularly big, say 1,000,000,000 m, then, man, it took that much displacement for the ball to lose 100 J? so the force must be ... very small? Makes a little more sense... if the ball loses 100 J in .001m, then the force was very large? What does the force have to do with how much energy is lost?

Lastly, why the negative sign? Bonus points for connecting this with ∫f(x)dx = -U

Thanks