Ruff Ruffman and Inertia Makes it Happen

Dear Ruff Ruffman,

My kids really like your show. However, there is a problem. You promote it like it is science, but the content keeps having mistakes in it. Previously, I pointed out your mistake about in infrared thermometer (if can't remember, you said the thermometer measures the temperature with a laser. In fact, the laser is just used to aim.)

So, you see, I don't just like to randomly attack people. The problem is that you are saying "hey look at science" but your science is wrong. I suggest you either a) stop pretending to be a science show or b) get a science advisor (I am available).

The most recent problem I noticed (I don't usually watch the shows, so I have no idea how many errors there are) was in the episode where some kids were trying to design a roller coaster. The kids wanted to make the car do a loop. At first, they did not start off the car at a high enough position and it didn't make it through. The next time it started higher and DID make it around the loop. Your comment was something along the lines of:

"Gravity gave it energy and inertia kept it going through the loop"

Or something like that. So, what is my issue with this? Well, you can think of the car loosing gravitational potential energy and gaining kinetic energy, but inertia?? What do you mean by inertia? The common definition of inertia is:

INERTIA: the resistance of an object to a change it its state of motion (wikipedia - the source of all truthiness). In most definitions, more mass means more inertia.

Ruff, if this is indeed the definition you used - I am not sure if it fits in this case. Maybe you meant momentum, although I think that is not quite right either. How about I solve this problem?

Suppose I have the following track:

Here a car (say of mass m) starts a distance h above the top of the track of radius r. How high does it have to start to make it around the loop? There are two things here. First, how fast does it have to be when at the top of the track? Second, how high does it have to start to get that speed? Let me start with the first part. Here is the car at the top of the track with some possible forces on it.

Here, there could be two forces on the car. Gravity pulling down and also maybe there is a force from the track (labeled as N here). At the lowest speed, there is no normal force. Remember that if the object is moving in a circle, it needs a net force pointing towards the center of the circle. I can still use Newton's second law and put the acceleration as centripetal acceleration:

Here, I lost the vector notation because I am just talking about the net force in the r-direction (towards the center of the circle). So, what is the lowest speed such that only gravity is pulling down?

Ok - that is the minimum speed to be upside down moving in a circle. If it were moving slower, it wouldn't stay on the track. Now, how high above the track does it have to be to get to that speed (assume little or no friction)? Here I can use the work-energy theorem taking the cart plus Earth as the system.

There is no external work. The initial kinetic energy is zero, and I will take the top of the loop as y=0 such that the final gravitational potential energy is zero. This gives:

Notice that the mass cancels (see Ruff - mass doesn't really matter). Also, I can put in the expression for the minimum speed to make it through the loop:

So, for the car to make it around loop, it must start 1/2 the radius of the loop higher than the top of the loop.

Ruff, I know what you are going to say. "Well, then how would you explain this idea to these kids?" Couldn't you just say "oh look, it started high enough to make it to the top of the loop, but it still fell down." And then when they started it higher and it worked, you could say "Wow! It worked. I guess it is not enough to just get back to the top of the loop, it had to be still going fast to not fall".

Or how about: "wow, gravity gave it energy and the Flying Spaghetti Monster made it stay on the loop".

Ruff, I still like you. I mean, how many dogs do you see that can talk?