# 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.

Question-AnswerCategory: Engineering MechanicsA 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.
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.

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.