This is a classic introductory physics problem. Basically, you have a cart on a frictionless track (call this *m*_{1}) with a string that runs over a pulley to another mass hanging below (call this *m*_{2}). Here's a diagram.

Now suppose I want to find the acceleration of the cart after it is let go. The string that attaches the two carts does two things. First, the string makes the magnitude of the acceleration for both carts is the same. Second, the magnitude of the tension on cart 1 and cart 2 has the same value (since it's the same string). This means I can draw the following two force diagrams for the two masses.

So, how do you find the acceleration of cart 1? It seems clear, right? You just need to find the tension in the string since that's the only force in the horizontal direction. You could write:

If I know the tension, I can calculate the acceleration. Simple, right? Even simpler, the tension would just be equal to the gravitational force on the hanging mass (*m*_{2}). WRONG! This is not the correct way to solve this problem — I actually remember making this exact mistake when I was an undergraduate student. But why is it wrong?

If I just solved the problem the correct way, it wouldn't be that much fun. Instead, I am going to set up an experiment to show that the tension in the string isn't just the weight of mass 2. You can see the setup in the image above (it's called a half Atwood machine in case you want to google the solution). Basically, I have a mass on a low friction track with a string connected over a pulley to a hanging mass. Both masses have force sensors on them so that I can measure the tension in the string. Also, I can measure the position, velocity and acceleration of the cart using the Vernier motion encoding system. This is basically just series of lines on the track that the cart uses to determine its position (and thus velocity and acceleration).

With a cart mass of 1207 grams (mass 1) and a hanger mass of 145 grams (mass 2), I get the following data.

Here is what you should look at:

- First, the forces. Yes, there are two force probes — one on each mass. Both of the force sensors give the same force. This means the tension on mass 1 has the same magnitude as the tension on mass 2.
- The first part of the graph shows the two masses at rest (I'm holding it with my hand). In this case, the acceleration is zero m/s
^{2}(as seen in the graph showing the acceleration) and the tension in the string is just the weight of mass 2 (0.145 kg)*(9.8 N/kg) = 1.421 N. - The masses accelerate around a time of
*t*= 1.4 seconds. Ok, this is wrong. They actually accelerate before that, but the first acceleration was caused by me pushing the cart away from the pulley. The cart is accelerating the way you would expect from time of 1.4 seconds to 2.6 seconds. - During this acceleration, the tension in the string drops from 1.421 Newtons to just 1.285 Newtons.