# What's Gravitational Potential Energy of a Balanced Mass on a single Point?

Anyone know how to calculate the gravitational potential energy of a balanced Mass such as an upside down 2d triangle balanced on a single point. Is there any way to calculate the gravitational potential energy of the Tilt motion of the upside down triangle, as it falls to one side.

Thanks guys

• Is it balanced or is there a tilt-motion? I feel those two messaged are counteracting. – Steeven Oct 2 '18 at 16:06

$$x_{com}=\frac{\sum mx}{\sum m}\quad , \quad y_{com}=\frac{\sum my}{\sum m}\,.$$
Similar for the z-coordinate. If you have individual particles, it is fairly easy to sum them all up. If you have an extended body, you must integrate $$\int$$ instead of summing $$\sum$$; it may be tricky depending on the exact geometry.
Regardless of how you find the CoM, you can consider this point as where gravity pulls. As if the entire object's mass was actually only concentrated at this point. This is therefore easy to input into a gravitational potential energy formula, such as $$U=mgh\, ,$$
depending on your purpose. Just remember that the height $$h$$ in this particular formula is not the distance of the CoM from the ground, but the distance between where the CoM starts out and where it ends after tilting over. Only this amount of potential energy is released.