**Discussion: **An ideal (i.e. frictionless and massless) pulley serves merely to redirect the one-dimensional motion of the string and attached masses, without affecting the tension in the string or otherwise interfering with the motion. With this in mind, we can view the Atwood's machine system as being driven by an external force F_{ext} equal in magnitude to the difference between the two weights. If m_{1} is the larger mass and m_{2} is the smaller mass, then F_{ext} = (m_{1}- m_{2})g. The total inertia of the system is given by the total mass (m_{1 }+ m_{2}).

The real pulleys used in the lab have some friction, which we will try to include in our analysis of the experiment. They also have a mass of 10 g, which introduces a small amount of 'rotational inertia'. Rotational inertia will be discussed later in the course, but it will be neglected in this experiment. The equation of motion for the system is,

F_{ext} - f = (m_{1} + m_{2})a ,

where a is the one-dimensional acceleration of the masses, and f is assumed to be a constant friction due to the pulley. For a specific value of F, we expect the acceleration to be a constant. Rearranging the equation of motion,

F_{ext} = (m_{1} + m_{2})a + f

We will vary F_{ext} by changing m_{1} and m_{2}, and at the same time __keep the total mass __(m_{1} + m_{2})__ constant__. A graph of F_{ext} vs. acceleration should be a straight line with slope (m_{1} + m_{2}) and intercept f.

For given values of m_{1} and m_{2}, you are going to measure the time t it takes m_{1} to fall from some known height h to the floor. If m_{1} is released with v_{0} = 0 and we assume a constant acceleration (as expected), then the equations of uniformly accelerated motion predict h = a t^{2}/2. Solving for the acceleration, a = 2h/t^{2}.

Note: Use m, kg, N, and s in your calculations throughout this experiment.

**Procedure**:

1. Set up the equipment as instructed. You should make h at least 1.00 m. It is recommended that you use a total mass of approximately 0.600 kg (i. e. 600 g), and that m_{1} and m_{2} initially differ by approximately 0.010 kg (i. e. 10 g). The small masses should initially be part of m_{2}, since you are going to gradually decrease m_{2} and increase m_{1} by transferring these small masses.

2. Measure the time t it takes m_{1} to fall the known height h. It is extremely important that you obtain accurate times. Be prepared to repeat the motion many times, until you obtain consistent results. Use t to calculate the acceleration of the system in m/s^{2}.

3. Transfer mass from m_{2} to m_{1}, and repeat step 2. The initial recommendation is that you transfer mass in 0.001 kg (i. e. 1 g) increments, which means that the difference m_{1}- m_{2} will increase in 0.002 kg (i. e. 2 g) increments. Notice that the total mass (m_{1} + m_{2}) remains constant. Continue in this way until you have obtained accelerations for eight or more mass distributions.

4. Make a graph of F_{ext} vs. acceleration. If the points on your graph appear to be on a straight line, you have verified Newton's second law in part. Draw the best line through your data.

5. Determine the slope (in kg) and intercept (in N) of your line. The slope should correspond to the total mass, and the intercept should correspond to the friction f.

6. Calculate the percentage difference between your slope and the known total mass of the system (m_{1} + m_{2}). If this percentage is small, you have further verified Newton's second law. Do not be alarmed if your slope is not too close to the expected value.

7. There is no predicted value of f, to which you can compare your experimental value.