# 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.

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.