Change in Potential energy corresponding to a conservative force is defined as $$\Delta U = U_f – U_i=-W_f$$ and gravitational potential energy is $$\Delta U = U_f-U_i = -W_g $$ Suppose a mass $m_1$ is kept at a fixed point $A$ and a second mass $m_2$ is displaced from point $B$ to point $C$ such that $AB = r_1$ and $AC = r_2$.
$\therefore$ , $$\Delta U = -W_g = \int{\frac{Gm_1m_2}{r^2}}dr$$ $$U(r_2)-U(r_1) = Gm_1m_2\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$$ Now I am free to choose any reference point thus if I take potential energy at $U(r_1) = 0$ and $r_2 = \infty$ Then I will get potential energy at infinity as $$U(\infty) = \frac{Gm_1m_2}{r_1}$$ which I think is wrong as a reference point at $r_1$ the potential energy at infinity should be infinite.
So where I am wrong, is my concept of gravitational potential energy wrong itself.
Newtonian Gravity – Concept of Gravitational Potential Energy
conventionsdefinitionnewtonian-gravitypotential energy
Best Answer
The potential energy at infinity is only infinite if it takes an infinite amount of work to get to infinity. However, because the gravitational force decreases rapidly with distance, a projectile rapidly reaches a space where the force of gravity is not strong enough to reverse its velocity. Potential energy keeps increasing, but there is a maximum value that is approached asymptotically.
We have actually built space probes that have escaped Earth's gravity and the Sun's gravity. Voyager 1 and 2 have exited the Solar System and are never coming back. If they don't run into anything, they will reach arbitrarily far distances. The only thing stopping them from actually reaching infinity is the infinite time it would take to get there. To reach such a state, it only took the energy in the rocket fuel and some kinetic energy stolen from planets during planetary slingshots.
There are systems with a potential energy that reaches infinity at infinite distance. The spring potential energy $U = \frac{1}{2}kx^2$ is an example. When $x=\infty$, $U=\infty$ because the spring force keeps getting stronger as the spring is stretched.