CONCURRENT FORCES

The correct mathematical manipulation of vector quantities (such as forces, velocities , etc.) is a necessary feat for you to learn as we try to resolve the complex behavior of objects into more simple actions. Thus, a vector may be resolved into component vectors; or if we wish, vectors may be added together to give a sum  vector known as the resultant. Vector forces can be easily used and measured on a horizontal circular "force-table". Knowing that forces cancel one another if they have equal magnitudes and act in opposite directions gives a physical requirement for studying how vectors interact. In this document vectors are printed in bold-face type rather than having an arrow over them.

The conditions for a body to be in a state of rest or in a state of uniform motion are known as the conditions for equilibrium. For translational equilibrium, which requires no change in translational speed or no change in the direction of motion, the vector sum of all the forces acting on the body must be zero.

Today you will study the case for several forces acting on an object at the same point (concurrent forces). Three strings are attached to the circular ring positioned at the center of the force table. Each string runs over a pulley and attaches to a mass hanger. The force of gravity on the mass is called its weight (force) and is not the same as its mass - the two have different units and differ numberically by a multiplicative factor called 'g'.  The magnitude of the force applied to the center ring by each of these is the mass times the acceleration of gravity (as we will learn later). The string and pulley redirect the direction of the weight force such that it pulls the ring to the side in a direction along the string. To save time you may record only the mass without calculating the force.  But keep in mind that 0.200 kg is not a force (the force is 0.200 kg x 9.82 m/sec2 = 1.96N).

Set up the force table for forces A and B as 0.15 kg at 0 degrees and 0.20 kg at 120 degrees. By trial and error, determine force C (what mass at what angle) is required to center the ring.  If necessary, slide the strings along the ring, so that an imaginary line drawn along the string would pass near the center of the ring.  When the ring is centered we say tht the ring is in equilibrium. Once you are close to equilibrium the peg in the center of the force table can be removed so that you can better judge when the ring is centered. It is helpful to give the ring a little "nudge" so that it moves a little.  Once in motion it is more likely for the ring to come to the true equilibrium position.  The ring is in equilibrium when the ring returns to a centered position of the peg (or its hole).  Carefully draw the equilibrium arrangement into your notebook including the names A, B, and Sum, along with the numberical values and units (kg or o). How is C related to the sum of A and B?

All measurements of physical quantities involve uncertainty. Uncertainties are a result of limitation on accuracy, not mistakes. No measurement is perfect. So when a scientist states a measured value he or she also states the possible range of uncertainty (sometimes called error). For instance, if I measured a book with a meter stick the result might be 35.2 cm ± 0.05 cm, which means that my best estimate of the length of the book is 35.2 cm, but it might be as small is 35.15 cm or as large as 35.25 cm. While trying to find the angle which would produce equilibrium you probably found that there was a small range of angles over which the forces were balanced rather than just one position. Determine this range of angles by starting at the angle you recorded in the last part, then move the pulley very slowly to the right until the ring is no longer centered after giving the ring a nudge. Record this angle and repeat the process while moving to the left. Record your measurement of the angle with uncertainty. For instance, if you found that equilibrium could be achieved anywhere between 43o and 47o, your answer should be stated as 45o ± 2o. In order to determine the uncertainty in the mass reposition the pulley at the center of the range of angles, then add and subtract mass a little at a time until you have found a range of mass values for which the system is in equilibrium. State the mass with uncertainty.

Repeat the procedure for the other two cases given in the table.  (You have already done Case 1.)

 Mass A (kg) Angle A (degrees) Mass B (kg) Angle B (degrees) Case 1 0.15 0 0.20 120 Case 2 0.25 340 0.22 90 Case 3 0.19 250 0.17 20

Scientists often check their experimental results by calculating what theory predicts the result should be.  In order to do that we need to analytically add vectors together which involves several steps:

• Calculate the x and y components of A and B. Let 0o correspond to the x-axis and 90o to the y-axis.
• Find the magnitude and direction of the sum of these two forces.   Call this vector S for sum. With the axes defined as above, vector addition can be reduced to a recipe as shown in the table below - which is started for Case #1.
 Name Magnitude Direction x-comp [mag * cos(dir)] y-comp  [mag * sin(dir)] A 0.15 0 0.15*cos(0) = 0.15*sin(0) = B 0.20 120 0.2*cos(120) = 0.2*sin(120)= Sum
Once you perform the calculations for the components, add the two x-component values together and place in empty space below them.  Do the same for the y-components.  All values in a column have the same units!
Now you know the components of the sum vector, to find the sum vector's magnitude just use the Pythagorean Theorem on the x- and y- components of the sum vector.   Place this value under the magnitude.
To find the direction of the sum, you need to use the inverse tangent, arc tangent or tan-1 function on your calculator. The calculator is designed ot give you an angle between -90o and 90o, so you might need to add or subtract 180o to get the proper angle.
note: It is simple to add multiple vectors just by inserting more rows into the table above.
• Now find the magnitude and direction of the force which will when added to S will give zero.   This Is the predicted (or theoretical) vale of the force C that you measured.
• Compare this theoretical result with your experimental results. Is the theoretical value within the range of values found in the experiment? In other words, do the theoretical and experimental results agree to within the accuracy limits of the experiment?  Explain in your notebook, using sentences, how your experimental and theoretical values compare.  Do the uncertainties in your experimental values encompass your theoretical values?  Explain.

(Hint: place all of your data and the results of theoretical calculations in a single table. Each row in the table should be for a given experimental setup. Be sure to include column headings similar to the following.)

MA(kg) AngA(deg) MB(kg) AngB(deg) Exp.Mass(kg) Exp.Angle(deg) Theory M(kg) Theory Ang(deg)

Remember to include a summary about what you have learned as your conclusion in your laboratory notebook!

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Beverley Taylor
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Miami University - Hamilton
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taylorba@miamioh.edu