Physics of a Falling Christmas Tree

My family celebrates Christmas, and our tree goes up the day after Thanksgiving to get maximum holiday cheer out of the season. Every year, we tie the tree to the wall in order to prevent it from tipping.

But what if it were to fall? Why are trees easy to tip over and what are some physics concepts involved in the tipping of the tree?

First, a tree is basically an uneven pole with its center of mass more towards the bottom due to the general cone shape of the branches and trunk. One might think this would make the tree stable. However, despite the wide sweeping base of the branches, the trunk is very narrow so one only needs to tip the tree over the edge of its trunk in order to get the tree to fall.

Second, there are many physics concepts involved with the tipping of the tree. For our purposes, the tree can be called the lever arm with the base of the tree acting as the pivot point. Now if a cat were to begin climbing up the trunk of the tree, it would not start tipping until the cat climbed farther from that point. The farther away from the pivot point, the more likely the tree will fall due to the force of the cat acting farther away. This relates to torque, or the rotational equivalent of linear force. 

Fig. 1. Diagram of Christmas tree with an ornament demonstrating the force of gravity and arrows demonstrating the place where a cat climbs and the relative amount of force required to generate enough torque to tip the tree. The cat would have changed the force of gravity applied at the point due to the mass change.  The torque would be greater farther away from the pivot point (bigger r), and the tree will be easier to tip.

Once the tree starts tipping (which is quite easy as discussed above), the tree can be said to have rotational motion. If considering the tree as an uneven pole, we can say that the tip of the tree will have a much greater linear velocity at the tip than at the base when each hit the ground. Their angular velocities will be the same.

But a Christmas tree also has branches which can provide cushioning so much so that the tip of the tree may not even hit the ground.

But lest you think your ornaments can be saved at the top, they will likely fall off due to the force of gravity acting on a different side of the ornament, and they will hit the ground with the linear velocity that the tip of the tree would have felt.