My dad is known for being an entertaining dancer who stops and starts as he pleases and bops up and down or side to side. This past weekend it became even more apparent while we were at one of his college friends daughter’s wedding. While he was doing this I thought it would be interesting to see how his potential energy due to gravity changes as he bops up and down versus side to side.
Let's set the ground as y=0. When my dad is standing up straight, since he is 5’11’, we set 71 inches as h. When he crouches his knees and bends down to “bop” his height drops to 65 inches lets say. Since the equation for potential energy is PEg = mgh, when he is standing still his potential energy is m x 9.8 x 1.8034 m. When he is crouched his potential energy will be less since his height is 6 inches lower. It would be m x 9.8 x 1.651m. The potential energy my dad has when he is crouched is less than when he is standing up straight and not moving. Theoretically this makes sense because potential energy is related to the energy of an object's position, not the objects motion. When he begins moving and starts lowering his h value, the potential energy decreases because he is moving his position. Since his position is lowering from the baseline, his "h" value decreases, lowering his potential energy value.
Now let's consider the change in potential energy due to gravity as he shifts from side to side. When he shuffles from side to side, his height is still staying the same, as well as his mass and the affects of gravity which are still the same as before, so his potential energy does not change. If you think about this theoretically it makes sense. The potential energy due to gravity is not being affected by his movement in the x direction, just the y direction since it is the potential energy due to gravity. When using the equation PEg = mgh, the value of PEg will be the same no matter what the x component of my dad's location is. Thus, the change in potential energy is apparent when dancing in the y direction, but not from side to side in the x direction.