#2 The Study of Concurrent Forces with the Force Table
Apparatus: Force table with 4 pulleys, centering ring and string, 50 g weight hangers,
slotted weights, protractors, and rulers.
Discussion:
The force table is designed to help you study the properties of forces at known angles.
Only when forces are along the same line do they add by ordinary algebra. If two or
more forces on the same body form angles with each other, it is necessary to use
geometry to find the amount and direction of their combined effect.
Prior to Lab:
Complete the calculations in the following.
The component method of adding vectors is given here for three sample forces as
follows:
A = 2.45 N @ 40o
B = 3.92 N @ 165o
C = 3.43 N @ 330o
Overview of the component method of vector addition.
First make a neat drawing, not necessarily to exact scale, but reasonably accurate as
to sizes and angles, placing the three forces on a diagram with a pair of x and y axes.
Find the angle of each force with the x-axis. This angle is called the reference angle,
and is the one used to calculate sines and cosines. Next compute the x- and
y-components of the three forces, placing like components in columns. Place plus or
minus signs on the various quantities according to whether an x-component is to the
right or left of the origin, or whether a y-component is up or down relative to the
origin. Add the columns with regard to sign (subtracting the minus quantities), and
place the correct sign on each sum. The resulting quantities are the x- and
y-components of the resultant. Since they are forces at right angles to each other, they
are combined by the Pythagorean Theorem. The angle of the resultant is found by the
fact that the ratio of the y-component to the x-component is the tangent of the angle of
the resultant with the x-axis. Make a new drawing showing the two components of the
resultant along the x- and y-axis, and showing their resultant. This drawing will make
clear whether the reference angle obtained by use of the Arctan (tan-1) is the actual
angle with the x-axis. (If the resultant is in the second, or third quadrant the reference
angle must be added to 180o to obtain the actual angle of the resultant.) The
procedure is illustrated below:
Draw a sketch of the vectors and a xy coordinate system here:
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Fill in the following table:
Ax = 2.45 N cos 40o = ______________
Ay = 2.45 N sin 40o = ______________
Bx = 3.92 N cos ___ = ______________
By = 3.92 N sin ___ = ______________
Cx = ______ cos ___ = ______________
Cy = ______ sin ___ = ______________
Rx = _______________
Ry = _______________
Draw a sketch of the components of the resultant in an xy coordinate system here
(include the magnitude and direction of the resultant also):
The magnitude of the resultant is R 
Rx2  Ry2  _________________________
The angle the resultant makes with the x – axis (reference angle) is
  tan 1
Ry
Rx
 ______________
According to your sketch above does the angle place the resultant in the correct
quadrant? If not, what is the correct angle for the resultant vector?
Write the resultant in both polar and unit vector notation.
R = __________________________________ = ___________________________________
Overview of the experiment:
This experiment has three parts. For the first part of the experiment you will take two
forces at right angles to each other and find their resultant. Next you will find the
components of a force. Finally, in the third part of the experiment you will combine
several forces using their components.
Procedure:
Part 1.
Consider two forces at right angles to each other, one equal to the weight of 300 g and
the other to the weight of 500 g (2.94 and 4.90 Newtons, respectively). Let the 4.90 N
weight be at 0o, and the 2.94 N weight be at 90o. Set these up on the force table and
find by experiment the magnitude and direction of the force on a third string which
will be equal and opposite to the effect of the two forces working together. This force is
called the equilibrant of the first two forces. It is equal and opposite to their resultant.
Write down the magnitude and directions of both the equilibrant and the resultant,
expressing the magnitude in Newtons.
Next compute the resultant of these same two forces by use of the Pythagorean
theorem, which applies in the case where the forces are at right angles to each other.
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Draw a force parallelogram (a rectangle in this case) to scale letting 1 Newton equal a
length of 1 cm and, using one triangle, find the angle of the resultant, using:
sin  
2.94 N
R
How do the results compare with those of the force table? Calculate the per cent
difference in magnitude and angle.
Part 2.
Consider a force of 9.80 newtons (the weight of 1 kg.) acting at an angle of 35°. Is it
possible for two forces acting at 0o and at 90o to produce the same effect when acting
together as this force when acting along? If so, then this 9.80 n. force can be
represented by a diagonal of a rectangle with the two component forces represented by
the sides from the same vertex. Compute the component along the zero degree line;
i.e., the x-component, as follows:
Using the lower triangle in the figure,
Fx
 cos 35o
9.80 N
Fx  9.80 N cos 35o  _____________
Compute the y-component, using:
Fy
 sin 35o
9.80 N
Fy  9.80 N sin 35o  _____________
Fy
F = 9.80 N
35o
Fx
If these two forces acting together are actually equivalent to the original force, than
their opposites acting together should be opposite and equal to the original force. Set
up the 9.80 N force at 35o on the force table and also the forces equal in magnitude
and opposite in direction to the two computed components. Do they counteract each
other to produce equilibrium?
Part 3.
Finally, the resultant of three given forces is to be found by the method of
components. The three forces you will use will be given by your instructor.
1. Set up the forces given you on the force table.
2. Using the fourth string, determine by trial and error the direction that the force
should act to balance the vector sum of the first three forces.
3. Clamp a fourth pulley at the location of the equilibrant force found in step 2.
4. Using a weight hanger and slotted weights determine the magnitude of the
equilibrant required to balance the three forces.
5. Show your lab instructor your result.
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6. Using the method of components (see example at beginning of instructions)
calculate the vector sum of the three forces. What should the angle of the
equilibrant be?
7. Calculate the per cent difference between the direction (angle) of the calculated
equilibrant and the equilibrant measured using the force table.
8. Repeat 7 for the magnitudes of the calculated and measured equilibrants.
Questions:
A. What do you conclude about the nature of vector sums? Can you use simple
algebra to determine a vector sum?
B. What may have contributed to your experimental error?
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