Concept explainers
The gears A and D are rigidly attached to horizontal shafts that are held by frictionless bearings. Determine (a) the couple M0 that must be applied to shaft DEF to maintain equilibrium, (b) the reactions at G and H.
Fig. P6.159
(a)
The couple
Answer to Problem 6.159P
The couple
Explanation of Solution
Take all vectors along the
Radius of gear
Consider the projection of the gears on
The free body diagram of the Gear
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 and the point.
The moment at
Thus, write the complete expression of anticlockwise moment
Here,
At equilibrium, the sum of the moment acting at
Write the expression for the total anticlockwise moment acting at
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
Thus, write the complete expression of anticlockwise moment
Here,
At equilibrium, the sum of the moment acting at
Write the expression for the total anticlockwise moment acting at
Calculation:
Substitute
Substitute
Since the rotation is in the yz plane , the direction of couple is in
Therefore, the couple
(b)
The reaction at
Answer to Problem 6.159P
The point
Explanation of Solution
Free body diagram of Projection on
Here, is the tangential force acting on the gear,
From figure 2, write the equation of net moment about
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
Thus, write the complete expression of anticlockwise moment
Here,
At equilibrium, the sum of the moment acting at
Write the expression for the total anticlockwise moment acting at
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.
At equilibrium, the sum of the moment acting at
Write the expression for the total anticlockwise moment acting at
Here,
The free body diagram of the projection on
Here,
From figure 3, write the equation of net moment about
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
Thus, write the complete expression of anticlockwise moment
Here,
At equilibrium, the sum of the moment acting at
Write the expression for the total anticlockwise moment acting at
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.
At equilibrium, the sum of the moment acting at
Write the expression for the total anticlockwise moment acting at
Here,
Consider the projection at
The free body diagram of the Bracket
Here,
Write the expression for the total force along
Since in this direction net force is equal to zero. Equate above equation to zero.
Since total moment of force about
Write the equilibrium moment of force about
The free body diagram of the Bracket
Here,
Write the expression for the total force along
Since in this direction net force is equal to zero. Equate above equation to zero.
Since total moment of force about
Write the equilibrium moment of force about
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
The negative sign indicate that it is directed along
Substitute
Substitute
The positive value indicate that it is directed along
Substitute
Therefore, the net force at
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Chapter 6 Solutions
Vector Mechanics for Engineers: Statics
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