# Atwood Machine and Newton’s 2nd Law

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#### Updated on Dec 10, 2013

An Atwood Machine is a very simple device invented by George Atwood in 1794 as a way to demonstrate Newton’s Laws of Motion. Newton’s Second Law of Motion says that the force required to move something equals the object’s mass times it’s rate of acceleration: F = ma. When Earth’s gravity is the force, you use 9.8 m/s2 for a. This is gravitational acceleration, the rate at which gravity pulls everything towards the center of the Earth.

You can rewrite Newton’s Second Law to solve for acceleration by dividing both sides by m: a = F / m.Acceleration is just what happens when you push on a mass, m, with a force, F.

In this project, you’re going to build an Atwood Machine and see this law in action. You will use the Atwood Machine to verify for yourself the acceleration due to gravity.

### Problem

Build a simple Atwood Machine to understand Newton’s Second Law and estimate the pull of gravity.

### Materials

• Ring stand
• Pulley (can be found at a hardware store)
• Length of string
• Several masses of different weights (two which should be the same) to which the string can be tied
• Ruler
• Stopwatch

### Procedure

1. Attach the pulley to the top of the ring stand.
2. Tie two masses of equal weight to opposite ends of the string and run the string over the top of the pulley. The string should be long enough so that one mass can rest on the table/ground with the other dangling near the top of the stand.
3. Position the masses at different heights from the ground and let go. Do the masses move? What do you think is going on?
4. Replace one mass with another that is slightly heavier. Drag the lighter mass down as far as it will go and release. Do the masses move this time? What’s different from before?
5. Replace the same mass with one that is slightly heavier still and repeat Step 4. Do the masses move differently? Why or why not?
6. Go back to the masses you used in Step 4 – one slightly heavier than the other. Pull the lighter mass down as far as it will go.
7. Use a ruler to measure the height of the other mass above the table.
8. Grab a stopwatch. Release the mass and start the watch at the same time. As soon as you hear the other mass hit the ground, stop the timer. Do this at least three times, and record your times in a table.
9. For each drop, calculate the acceleration of the masses using the equation: a = 2h / t2, where h is the height of the mass before being dropped (in meters), t is the time it took to fall (in seconds), and a is the acceleration in m/s2. Record this in the table as well.
10. Calculate the acceleration due to gravity with the equation:g = a (m1+m2/m1 -m2),where m1 and m2 are the heavier and lighter masses, respectively, and a is the acceleration from step 7.
11. Calculate the average g from all your trials.
12. Replace the mass with at least two other heavier masses and repeat Steps 6-9.
 m1 = m2 = Height (m) = Time (s) Acceleration (m/s2) g (m/s2) Trial 1 Trial 2 Trial 3 Average

### Results

With equal masses, the weights will not move. When the weights are unequal, the masses will move so that the heavier one falls. As the difference in the masses increases, the speed with which it hits the ground increases as well.

When calculating g, you should get something close to the accepted value of 9.8 m/s2, though it will most likely be a bit less because of friction in the pulley.

### Why?

There are three forces at work here: the forces of gravity on both masses and the tension in the string connecting them. With equal-weight masses set up, all the forces are in balance. Gravity pulls on each mass equally. The tension on mass 1 (m1) is caused by the gravitational force on mass 2 (m2). Since the gravitational forces are the same, that means m1, for example, is feeling the same force pulling up as it does pulling down. This is true no matter how high or low the masses are. Everything is in equilibrium—no force means no motion.

Once the masses are unequal, the forces become out of balance. Mass 1 feels its own weight pulling down, and the weight of the other mass pulling up. The net force pulling down is the amount that would be felt if gravity were pulling on something that weighed the same as the difference between the two masses: F = (m1-m2)g.

But the force is pulling on both masses simultaneously, so the acceleration equals what you would get when applying this force to something that weighed the same as the sum of the masses: a = F / (m1+m2).

Putting those two effects together, the final acceleration equals: a = ((m1-m2) /(m1+m2)) g.

Solving for g gives you the equation you used to figure out the gravitational acceleration above.

The accepted value for g is 9.8 m/s2. How close did you get? You probably notice that you got something a little different every time and that your final average value is lower than the “accepted value.”

This is because your Atwood Machine is not a perfect system. There are many places where energy gets lost, mostly in the pulley. There is friction in the bearings and it takes energy to get something rotating. You can get an idea of how much friction is involved by figuring out how small a mass difference you can have without the masses falling. If there’s not a large enough difference, there’s not enough net force to overcome the friction. You can expand your procedure by using different pulleys to see how they affect your value for g.