Even though I grew up in northern New Jersey, I never went to the New Year’s ball drop in Times Square. Every year on top of the One Times Square building, a ball starts at the top of a pole at 11:59pm, and travels downward until it hits the bottom of the pole at 12pm. How much applied force is needed to make sure that it travels the length of the pole in the required time? Ignore friction and assume that the ball moves at a constant speed for the entire trip.
The ball, like everything else on Earth, has a force acting downwards due to gravity. We will choose to label this force as positive since it’s in the direction of motion. This force of gravity would cause the ball to fall too fast on its own. Thus, the force that we need is a force that acts upwards to slow the ball down just enough so that it travels the length of the pole in exactly one minute. We’ll choose to label this force as negative since it opposes motion.
We can find this force by using the concept of work. Work is defined as the energy transferred when applying a force over a distance. Both forces mentioned above transfer energy via work as the ball moves up and down the pole. The total work on the system is the
Total Work = Work due to gravity force – Work due to applied force
Total Work = Gravity force * distance of pole – Applied Force * distance of pole
The key to solving this problem is the work energy theorem, which states that the total work done on a system is equal the total change in the energy of a system as the ball travels from the top to the bottom of the pole. The energy can be broken down into kinetic energy (proportional to speed) and partially potential energy (proportional to height above ground).
Top of the pole – The ball has is moving so it has kinetic energy, and it also has potential energy
Bottom of the pole – The ball has the same kinetic energy as the top since it’s moving at a constant speed. The potential energy is lower because the ball is at a lower height.
Using this, we can also write our total work in terms of potential energy:
Total Work = Potential Energy at bottom of pole – Potential Energy at top of pole
We can then set our equations for Total work in 2 and 5 together and solve for our applied force.
If you’re interested in the actual math, the solution is below.