Potential Energy EX 6

# Particles	System	# Pairs	Total Potential Energy
3		3	$U=U_{12}+U_{13}+U_{23}$
4		6	$U=U_{12}+U_{13}+U_{14}+U_{23}+U_{24}+U_{34}$
5		10	$U=U_{12}+U_{13}+U_{14}+U_{15}+U_{23}+U_{24}+U_{25}+U_{34}+U_{35}+U_{45}$
6		15	$U=U_{12}+U_{13}+U_{14}+U_{15}+U_{16}+U_{23}+U_{24}+U_{25}+U_{26}+U_{34}+U_{35}+U_{36}+U_{45}+U_{46}+U_{56}$

(c) For $n$ interacting particles, there are $\frac{1}{2}n(n-1)$ pairs of interactions and therefore, there are that many terms in the expression for the total gravitational potential energy.