Analysis

A.  Graphical

You want to find the value of the unknown force. In order to add vectors, we add them head to tail. On another piece of paper, draw the first known vector to scale. Add the second one with the correct value and angle to the head of the first one. We know that the net force is supposed to add up to zero, so draw a vector from the head of the second back to the tail of the first. Measure the length and angle of this force and use the same conversion to find the grams-force that corresponds to this value. (So if 1 cm=10 g force, then if your vector is 10 cm then the value is 100 g-force.) Measure the x and y components of this force to compare to the analytical answer.

B. Analytical

We will now determine the magnitude of the unknown force using analytical means. You should have a diagram from the beginning of the experiment that you will use.

Using the forces based on the masses (remember our arbitrary unit gram-force) and the angles measured from your original drawing, you can calculate the x and y components of each of the known forces. Because we know that the net force should be zero in each direction, we can find the x and y components of the unknown force (and then it's magnitude and angle).

1. But then we have to ask whether this is equal to our answer from the graphical analysis within experimental error. (In this experiment we are assuming that the graphical answer is the "accepted" answer.) To do this you must determine what you think the error is on each force and each angle measurement.  This error is due to how precisely you think you can read the markings on the protractor and - this is something you determine.  For example, on a wooden meter stick with markings down to 1 mm, I doubt I can read a number to better than 0.3 mm (in my mind I can break up the divisions into 3), so 0.3 mm would be the error I have to associate with any single measurement I make with that meter stick.  What do you estimate your error in reading the protractor is?
2. For the mass, the error is related to how accurate the masses are. For this mass set, we will assume that they are accurate to 0.01 g, so a 50 g mass would really be between 49.99 g and 50.01 g. Depending on your spring scale (there are several different types), determin how accurately you can find the value of the force.
3. You now need to propagate those error measurements through your calculations in the way we learned last week so that you end up with the components of the resultant vectors with their associated errors.  See below for an example of the error calculation on one component, and then do this for all your components.
4. When you sum the three individual components, you are again going to have to propagate the errors from each of the components to get the error on the unknown force. There is an explanation and example below.
5. Find the x and y components of the unknown force. Do they equal the values from the graphical analysis? Use the null hypothesis to decide whether the graphical and analytical values are comparable. You do not need to have error on your graphical answer. 
6. Calculate the magnitude and angle of the unknown force from your analytical answer. Compare this with your graphical analysis. You DO NOT need to do error analysis or the null hypothesis on this part.
7. What in this set up could introduce error?

Error Propagation Example

I have measured a force of 149 N.  I determine that from the scale I can only read it to within +/- 2 N.  Therefore the magnitude of the force is 149 +/-2 N.   The angle I measure is 17.5 degrees from the -x axis.  With the protractor, I figure I can discern 0.3 degrees, therefore the angle is 17.5 +/-0.3 degrees.

My x-component is then Fx = (149 +/-2 N)cos(17.5+/-0.3).  Without error, this gives Fx = 142 N.  What is the error in Fx (called DFx)?    Recall that it is the square-root of the sum of the squares of the error due to only the magnitude and then only the angle:

The error due to the magnitude and angle are calculated as such:

Therefore, my answer is Fx = 142 +/-2 N.

You need to do this for each of the components of the known forces (this means 4 error calculations!). Then we need to find the x and y components of the unknown force. To find the total error on the forces in the x direction, we need to combine them by adding them Pythagoreanally (square them, add them, and take the square root of them).

Write-Up

1. If this is a written formal lab (as indicated on the lab syllabus), you will have until the next lab to submit the write up to the appropriate assignment in Moodle. For written formal labs, remember to check the "write-up hints" page to be sure everything is included and check your write-up against the grading rubric.
2. If this is an informal lab, record your results in your lab notebook. Before the next lab you will need to complete the informal lab quiz in Moodle in which you will type in your results and/or answer some questions about the lab.
3. If this is an oral report lab, you will schedule a time to meet with your instructor over GoToMeeting to present your work. You should prepare your results and the answers to any questions in a neat and organized fashion so that you can refer to when necessary during your discussion.
4. Remember to read the next lab and do the pre-lab before the next scheduled lab session. You may work on the pre-lab with others, but each person must submit her or his own work.

Department of Physics