
Prove that the magnitude M of the couple required to overcome the frictional resistance of a conical bearing as

Explanation of Solution
Write the equation of the magnitude of the couple as follows;
Here, the coefficient of kinetic friction is
Show the free-body diagram of the conical bearing as in Figure 1.
It has been assumed that the normal force per unit area is inversely proportional to r.
Here, the distance between the point to the axis of the shaft is r, the change in distance is
Here,
Find the normal force exerted
Substitute
Substitute
Find the change in friction force in y-axis;
Substitute
Find the value of k using the relation.
Substitute
Substitute
Find the change in friction force
Substitute
Find the change in moment
Substitute
Integrate the equation to find the couple M.
Substitute
Hence, it has been proved that the magnitude M of the couple required to overcome the frictional resistance of a conical bearing is
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Chapter 8 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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