. . . g o i n g o n e s t e p f u r t h e r
UE102030
Mechanic s / Force
Parallelogram of forces
The vector addition of forces can be demonstrated in a clear and
simple manner on the force table. The point of action of three
individual forces in equilibrium is exactly in the middle of the
table. Determine the magnitude of the individual forces from the
suspended weights and, using a protractor, note the angle of each
force vector (the direction of each force).
In a state of equilibrium, the sum of the three individual forces is
given by:
(1)
-
F
3
is therefore the sum of individual forces
F
1
and
F
2
(also see Fig. 2):
(2)
The parallel vector components for sum
F
are given by
(3)
and the vertical components are given by
(4)
Equations (3) and (4) provide a mathematical analysis of the vector
addition. For the experiment, it is advisable to align force
F
3
at an
angle of 0°.
For analytical observations, the equilibrium of forces can alterna-
tively be investigated on a graph. To do so, draw lines representing
all three forces diverging from the central point of action. Note the
magnitude and angle of each force. Subsequently, displace forces
F
2
and
F
3
along a parallel path till the point of origin is at the end
of the preceding vector. The resultant vector is 0 (also see Fig. 3). In
the experiment, carry out this procedure for three arbitrary forces,
making sure to maintain the state of equilibrium every time.
In the experiment, the analytical observation is restricted to the
special situation that the two forces
F
1
and
F
2
are symmetric to
F
3
.
0
3 2 1
= + +
F F F
2 1
3
F F F F
+ == −
2
2 1
1
3
cos
cos
α ⋅
+ α ⋅ == −
F
F F F
2
2 1
1
sin
sin
0
α
⋅
+ α ⋅ =
F
F
EVALUATION
Equation (4) is satisfied in a symmetric case (
F
1
=
F
2
and
α
1
= -
α
2
.
From equation (3) we get the characteristic equation applied in
Fig. 4 (for describing the measurement data).
1
1
cos
2
α ⋅ ⋅ =
F F
Fig. 1: Vector sum of forces (parallelogram of forces).
Fig. 2: Determining the sum of vectors of two forces
F
1
and
F
2
from equilib-
rium force
F
3
.
Fig. 3: Graphic investigation of the equilibrium of three arbitrary forces
acting in different directions.
Fig. 4: Measured and calculated sums of two symmetric forces in relation to
the angle
α
1
.
