Using a tilted Atwood's machine

The goal of today's experiment is for you to determine

Clean your track gently with a damp paper towel, then dry it off. Tilt it at an angle of between 8 and 12 degrees; any value in that range is okay, just make sure you measure it properly and stabilize the track so it won't move. Place a cart on the track. Connect this cart to a hanging mass (use the blue mass sets) via a pulley.

Add mass to the cart in the following manner:

Measure the mass of the cart and all its cargo on a scale. We'll call this cart mass mc.

Add weights to the hanger so that its total mass is somewhere between 100 and 150 grams; we'll call the hanging mass mw. Adjust this hanging mass so that the cart will slowly roll up the ramp, taking about 5 seconds to cover a distance L = 60 cm.


Measurements

Your job is to measure the acceleration of the cart up the track. In each case, you'll measure the time it takes for the cart to travel L = 60 cm, and you should make three trials for each case. The trick is that you will make measurements while varying the mass of both the cart and the hanger, by transferring pennies, two at a time, from cart to hanger. So, you should make measurements with:

Note that the total mass, mc + mw, remains the same. All you are doing is moving some mass from the cart to the hanger.

In every case, the time it takes for the cart to roll L = 60 cm should be more than 2 seconds. Check with the instructor if you can't make this happen.

Make a table of your all your measurements, showing the cart mass mc and hanging mass mw as well as the times and accelerations.

Preliminary analysis, which you can do on the sheet handed out in the class.

  1. Draw a free-body diagram for the cart.
  2. Draw a free-body diagram for the hanging mass.
  3. Make tables showing all forces acting on each object.
  4. Write down two equations, one for the forces acting on the cart parallel to the ramp, one for the forces acting on the hanging mass vertically.
  5. Eliminate the tension force T from the two equations, leaving one equation. This equation should include acceleration and coefficient of rolling friction, as well as other terms.


Use a graph to find an approximate value for "g" (and its uncertainty)

Use algebra to put your equation into a form that looks like this:

Now, the first term on the right-hand side changes when you add pennies to the hanger, because mw gets bigger; but the second term doesn't change (much) at all as you move pennies. That means that if you make a graph which has hanging mass mw on the horizontal axis, and acceleration a on the vertical axis, the slope of a line through your data should be equal to the "(stuff)" term.

Your value may not be very accurate, due to a not-quite-valid assumption made in the paragraph above. Can you explain?


Use calculations to find a value for coefficient of friction

Use algebra to put your equation into a form that looks like this:

In this case, you won't make any approximations. Adopt the proper value for g = 980 cm per second per second , and make a neat table. For each trial of your measurements, compute a value of mu. I had three trials each for seven different mass combinations, so a total of 21 values.

After you have finished, compute a mean value and an uncertainty for this coefficient of rolling friction. The uncertainty in my value was about 15 percent; can you do as well?


Each group must submit a single lab report at the start of our next class (it's okay to hand it in earlier, of course!). The lab report must contain

I will judge each report on a number of criteria, which include grammar, spelling, and neatness.