S cornwell asked 1 year ago
A hexagonal plate i.s acted upon by the force P and the couple shown. Determine the magnitude and the direction of the smallest force P for which this system can be replaced with a single force at E.
Sazid Ahmad answered 1 year ago
Step: 1
Draw the schematic of the hexagonal plate.
Step: 2
Identify that the triangle 
is an equilateral triangle. Hence, the distance 
is equals to 
.
Step: 3
Determine the net resultant force.
…… (1)
Calculate the distance 
Substitute 
for 
, and 
for 
.
Determine the magnitude of the net moment about point 
.
…… (2)
Step: 4
Represent the net resultant force, and the net moment at point 
on the figure.
Fig. 2
Now let us consider that the moment,
and force,
may be replaced by force 
making an angle of 
with the horizontal axis at point 
as shown below.
Write the vector notation of force 
…… (3)
Calculate the distance 
Determine the magnitude of the net moment about point 
(see fig. 3)
…… (4)
Step: 5
Equate the 
components of equations (1) and (3).
Therefore, the magnitudes of the angles 
and 
are equal.
Equate the equations (2) and (4).
Substitute 
for 
.
…… (5)
Step: 6
Observe the equation (5).
The magnitude of the force 
is minimum when the value of 
is maximum.
The maximum value of sine of an angle is 1.
Equate the 
to 1.
Substitute 
for 
in the equation (5).
Therefore, the smallest force, 
is 
making an angle of 
with the horizontal.