Hints for the Lab "Roller Coaster of Statistically Likely Doom"

This lab is another of our "use-the-physics-you-know-to-make-a-prediction-and-the-best-one-wins-a-prize" labs.

In it, you will be shown the loop-the-loop apparatus that will be used for the competition in the second week. It looks something like shown on the right. Your goal is to roll a golf ball down the incline so that it completes the loop without leaving the track. Clearly, the higher up you start on the incline you start the ball, the faster it will go round the loop and so the more certain it will be to stay on the track.

The team that keeps their test ball on the track and starts at the LOWEST height will win two bonus points on their lab.

You will be allowed to make measurements on it before the competition, but only static ones -- radius of the loop, angle of the incline, etc. You will NOT be allowed to run a test on it before the competition.

You can get a pretty good idea of where to start by using energy conservation and the centripetal force rules. These are sketched at the bottom of these hints.

The law of mechanical energy conservation we wrote down in class assumes there is no energy lost to friction and no other forms of mechanical energy available other than kinetic energy and gravitational potential energy (or spring PE, but there is none here). That is likely not to be very accurate for this apparatus. It might be pretty good, but understanding the deviation from the ideal is where this competition will be won!

During the first week you will be given pieces of track of the same material as the competition apparatus. Your job is to determine how close the apparatus comes to the ideal and to figure out a way how to extrapolate your measurements to the large apparatus on which you will be competing. Be sure to document your hypotheses and how you test them for your lab report!

One of the components of your test will certainly involve a measurement of the speed of your ball after it has rolled down a piece of track from a certain height. A good hint at how to do that is to let the ball roll off the end of the table and use the relation between height, distance traveled off the edge of the table, and speed that you needed for the previous lab ("No such thing as free launch.") This is also outlined briefly at the bottom of this hint.

Mathematical Hints

1. The centripetal rule of circular motion

When an object is traveling along a circular arc, its velocity is changing direction. Newton's second law tells us that this changing the direction of velocity requires a net force. We will work out in lecture that if an object is changing its direction on a circular arc of radius R and is moving with a velocity v, then changing its direction requires a net force that points towards the center of the circle and has a magnitude

Fnet = mv2/R

When the ball is at the top of the loop, the net force is a combination of gravity and the normal force of the track pushing it down. If the speed is too slow, the ball will need a net force that is smaller than that provided by gravity alone. This would mean that the track would have to pull up on it. A track can do this on a cart with rails, but not on our golf ball. If the velocity is so slow that it need N from the track to point up, it will fall off. From this, you should be able to figure out the criitical speed.

2. Velocity of a ball rolling off a table

If a ball rolls off a table with a velocity, v0,its motion will be determined by Newton's second law (N2) with the net force given by gravity alone (free fall). Since N2 is a vector law, it means that the acceleration in the x direction and the y directions are determined by the forces in the x and y directions.respectively. Since there is no force in the x direction, the x velocity, initially equal to v0, will remain the same. The horizontal distance traveled by the ball before it hits the ground will depend on how long it stays in the air. This is determined by the y-motion. It starts with 0 y-velocty and accelerates downward due to the force of gravity.

If the height of the table is h, we want to develop an equation that tells us the distance, d, the ball will travel before it hits the ground. We therefore want to find an equation for v0 that expresses it in terms of h, d, and g (and nothing else). This is fairly straightforward to do if you