Experiment 3
Experiment 3: Equilibrium of Forces
Purpose
(1) To become familiar with vectors, their components, their addition and subtraction.
(2) To study the equilibrium of coplanar forces intersecting at a point.
Apparatus
Force table, slotted weights
Theory
There are basically two types of quantities in physics: scalars quantities and vector quantities.
A scalar quantity is simply a number that has no direction. Examples of scalars are 32 ft, 10 kg, and 8.6 s.
A vector quantity has both a magnitude (which is just a scalar) and a direction (which tells you which way the vector is pointing). Forces, velocity, and acceleration are examples of vectors. They are often designated by putting an arrow over the symbol like this representation of a force vector: F.
The symbol used for drawing a vector is an arrow. The length of the arrow represents the magnitude (after choosing a scale such as 1 inch = 1 newton). The angle the arrow makes with the x-axis gives us the direction of the vector. The angle is measured from the positive x axis, counter-clockwise from 0° to 360°. A force F of magnitude 90 N acting at angle q = 200° would be drawn using an arrow as shown in Figure 1.
Another way of describing a vector V is by decomposing it into its components Vx and Vy in the x and y directions as shown in Figure 2.
BOTH way of expressing the vector V tell us the magnitude and direction of the vector.
1) magnitude V and angle q
2) the components Vx and Vy
We can use trigonometry to go back and forth between these two expressions:
1) From the magnitude and angle to the components: Vx = V cos q and Vy = V sin q (1)
2) From the components to the magnitude and angle: V = Ö Vx 2 + Vy 2 and (2)
q = tan-1 (Vy / Vx).
WARNING: There are TWO solutions of the equation tan q = Vy / Vx.
To find the correct value of q use these facts:
(1) both x and y are positive in quadrant 1
(2) x is negative and y is positive in quadrant 2
(3) both x and y are negative in quadrant 3
(4) x is positive and y is negative in quadrant 4
Vector Addition
You can add vectors in both representations - either as components or graphically.
Adding Vectors As Components
To add two vectors as components, add the components which are in the same direction. for example, to add vector A + vector B to obtain the resultant vector C, you would add Ax + Bx to obtain the value of Cx and the value of Ay + By to obtain the value of Cy. We now know the two components Cx and Cy of the resultant vector C.
You can add more than one vector the same way: The sum of adding three forces 1, 2, 3, is found by summing all three components and using Equation (2) to find the magnitude and direction of the sum:
Sx = åFx = sum of all Fx (3)
Sy = åFy = sum of all Fy
S = magnitude of the sum = Sx2 + Sy2
q = angle of = tan-1 (Sy /Sx) (0° q 360°)
Adding Vectors Graphically
Figure 3 shows how to add vector A and vector B graphically. Draw vector A at the correct angle on the graph, its length calculated according to the scale. Then vector B is placed at the arrow end of A. Draw a line parallel to the x axis to measure the angle of B. The sum is the vector which goes from the beginning of A to the end arrow of B. Measure the line and use your scale to find the magnitude of the vector. Use a protractor to find its angle from a line parallel to the x axis.
Preliminary Procedure
(a) Your instructor will explain how to operate the force table.
(b) Do the holders have mass? This must be taken into account when calculating the total applied force of the masses at the ends of the strings.
(c) When all of the applied forces sum to zero (ie the system is in equilibrium), the plastic ring will remain centered. If the ring is pulled in some direction that means there is an unbalanced force in that direction.
Using only two holders (let other two strings hang free), set one pulley at 127° and the other at 307°. Load each holder with 100 grams. After all adjustments, verify that the plastic ring is centered on the central pin. This shows that the two forces are equal and opposite and that S F = 0 since the ring is not pulled in any direction (Newton’s first law).
(d) Remove 50 grams from one mass holder. Note that the ring is immediately pulled by the heavier force (105 gms > 55 gms). This is due to the imbalance of the two forces since when the sum of forces is not zero a net force is present, and an acceleration (i.e. a motion) begins to occur. This shows Newton's second law, namely:
S F = m a
UNITS NOTE: Grams are units of mass, not force. You would have to multiply masses by the acceleration of gravity g to find their weight (W = mg, g = 9.8 m/s2) which is the force due to gravity. We are going to simplify our calculations and use grams as a proxy for the force.
Procedure Part I: Finding the Opposing Force by Adding Components
In Part I, we will add vectors F1 and F2 using the component method and then calculate their opposing force. Copy this table onto your data sheet:
PART I DATAMagnitude (grams) / ϴ (º) / Fx (grams) / Fy (grams)
F1
F2
Sum
(1) Set a pulley at 0° and load the string with any mass you choose unless directed otherwise by your instructor. This mass + massholder is the magnitude of force vector F1. Set the second pulley at 90° and load it with a different mass. This is force vector F2. Calculate the vector components of F1 and F2 using Equation (1). Sum the vector’s components in both x and y directions.
(2). Copy the table below onto your data sheet and enter the components of the sum. Then find the magnitude and angle q of the sum using Equation (2). Show your calculations (including units) to three significant digits. Write the magnitude and direction of the opposing force. It has the same magnitude as the sum but the opposite angle qOPPOSING = qSUM + 180°
PART I CALCULATIONSSum of Fx (gm) / Sum of Fy (gm) / Magnitude (gm) / ϴ (º) / Does it Check? (Y/N)
Sum / ×
Opposing Force / × / ×
(3) Check if your calculations are correct by setting up the third pulley at the angle ϴOPPOSING and load it with the magnitude of the opposing force. Verify that the ring is perfectly centered on the pin as shown in Fig. 4 and note this observation on your data sheet (Y/N).
DO NOT CONTINUE UNTIL YOU HAVE THE RING CENTERED. If you can’t center it, and you can’t find your mistake, CALL YOUR INSTRUCTOR.
Procedure Part II: Finding The Opposing Force by Trial And Error
We find the opposing force for two vectors F1 + F2. Copy this data table onto your data sheet.:
PART II DATAMagnitude (g) / ϴ (º)
F1
F2
OFII (Opposing Force for Part II)
(4) Set two pulleys at angles q1 and q2 that are not at 90° to each other, load them with F1 and F2 grams (choose new values for q1 and q2, F1 and F2 unless directed otherwise by your instructor).
Pulling the third string, by trial and error find the direction which will oppose the force due to the sum of F1 and F2 and leave the ring centered. By experimentation, find the mass that will balance the two forces. This is the opposing force OFII of the vector sum S of F1 and F2. Record its magnitude and angle. No calculations required.
(5) Now that the three forces are in equilibrium, increase the load on the opposing force string OFII in steps of 1 gram. Record the smallest change that will cause the ring to move. Call this value DOFII. This is the uncertainty in the opposing force caused by friction in the pulleys.
(6) Return to the original magnitude of OFII. Shift the opposing force pulley and find the smallest change in the angle qII of the opposing force that causes an observable change in the angle. Measure and record this small change as DqII.
Procedure Part III: Finding the Opposing Force of Three Forces
by Trial And Error
7) Make a data table PART III DATA for 3 vectorsF1, F2, F3 like the one you made for Part II. Load
three strings with masses of your choice (see Fig. 5).
Using trial and error as you did in Part II, find the
fourth force that will balance the three forces.
Record the magnitude and angle of the opposing
force OFIII.
No calculations required.
(8) Repeat (5) and (6) above to find DOFIII and DqIII.
. /
ADVICE: BEFORE YOU LEAVE THE LAB
You will have to add forces graphically, as well as mathematically, in your report. See page 9. Ask your instructor for help if you are not sure how to do this.
Lab Report
Part I
(1) Answer the following: If F1 was acting at 45° (instead of 0°) and F2 at 135° (instead of 90°), would either the angle or the magnitude, or both, of their sum change? Can you answer this question without any calculations?
Part II
(2) Copy the Part II Table to your Lab Report and insert your Part II experimental values. From your values of F1, F2, q1, and q2, calculate the components of F1 and F2, and their sum and write them in the table. Round to 3 significant figures.
PART II TABLEMagnitude (grams) / ϴ (º) / Fx (grams) / Fy (grams)
F1
F2
SumII
(3) Calculate the magnitude and angle of the vector sum SumII of Fx and Fy using Equation (2). The calculated magnitude of the opposing force OFII = magnitude of SumII and its angle
qOP = qII + 180˚. Write in Part II Analysis Table your Part II measured values of the opposing force OFII from your data and your calculated values from (2).
PART II ANALYSIS TABLEmagnitude (grams) / q (º)
OFII (measured)
OFII (calculated)
(4) Calculate for Part II:
experimental % uncertainty
u = D OFII x 100% (Find the experimental % uncertainty in the magnitude of ΔFopposing using
OFII your experimental value from (5) above and your measured magnitude OFII)
% discrepancy
d = (OFII measured- OFII calculated) x 100% (% discrepancy between the measured and calculated magnitudes
OFII measured OFII)
On the basis of the quantity u (your experimental uncertainty in the magnitude OFII), explain what could be the major reasons of your discrepancy d between the measured and calculated values of OFII. If your error is small, still consider what can cause discrepancies.
Part III
Magnitude (g) / θ (º) / Fx (g) / Fy (g)
F1
F2
F3
SumIII
(5) Complete the Part III Table using the experimental data and calculations.
(6) Calculate the magnitude and angle of the sum SumIII of Fx and Fy using Equation (2).
(7) Calculate quantities u and d using your experimental value for ΔOIII and and calculated value for OIII.
Comment on your results.
(8) Forces F1, F2 and F3 can be added graphically on graph paper as shown in Fig. 6. First write the scale you will use for your graph in terms of grams per centimeter. Draw your diagram using a pencil, a ruler, a triangle, and a protractor. Calculate the lengths of your vectors using your scale. Remember all angles are calculated from the positive x axis. (See page 9 for help.)
Scale_____x direction
F2
F3 x direction
Sgraph
F1 x direction
Fig. 6
(9) Measure the length of the sum Sgraph on your graph and calculate its magnitude using your scale. Enter it in Table 4 along with the other two values listed. State which one of the three values you think is most reliable and why.
TABLE 4: PART IIIQUANTITY / MAGNITUDE (g)
OFIII (from your experimental data)
SUMIII (from calculation(6))
Sgraph (from your graphical sum)
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