For the frame and loading shown, determine the reactions at A E when (a) α = 30º, (b) α = 45 º.
Fig. P4.27
(a)
The reaction at
Answer to Problem 4.27P
The reaction at
Explanation of Solution
Forces acting upward and rightward are considered as positive and the torque acting counter clockwise is considered as positive.
Let
The free body diagram is sketched below as figure 1.
Here,
Write the expression for the moment at
Here,
Above equation implies that net moment at any point is the sum of product of each force acting on the system and perpendicular distance of the force about that point.
The moment at
The angle
The complete expression of moment
Here,
From figure 1, write the expression for the
From figure 1, write the expression for the
At equilibrium, the sum of the moment acting at
Write the expression for the total moment acting at
The forces along the
Therefore, write the expression for the net force along the
Here,
At equilibrium, the net force along the
The forces along the
Therefore, write the expression for the net force along the
Here,
At equilibrium, the net force along the
Write the expression for the magnitude of net reaction at
Here,
Let
Therefore, write the expression for the
Calculation:
Substitute
Substitute
Substitute
The negative sign indicates that direction of
Substitute
Substitute
The negative sign indicates that
Therefore, the reaction at
(b)
The reaction at
Answer to Problem 4.27P
The reaction at
Explanation of Solution
Take all vectors along the
Let
The free body diagram is sketched in figure 1.
Write the expression for the moment at
Here,
Above equation implies that net moment at any point is the sum of product of each force acting on the system and perpendicular distance of the force and the point.
The moment at
The angle
Thus, the complete expression of net anticlockwise moment
Here,
From figure 1, write the expression for the
From figure 1, write the expression for the
At equilibrium, the sum of the moment acting at
Write the expression for the total moment acting at
The forces along the
Therefore, write the expression for the net force along the
Here,
At equilibrium, the net force along the
The forces along the
Therefore, write the expression for the net force along the
Here,
At equilibrium, the net force along the
Write the expression for the magnitude of net reaction at
Here,
Let
Therefore, write the expression for the
Calculation:
Substitute
Substitute
Substitute
The negative sign indicates that direction of
Substitute
Therefore, the reaction at
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Chapter 4 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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