
Prove that the magnitude M of the couple required to overcome the frictional resistance of a conical bearing as M=23μkPsinθR32−R31R22−R21.

Explanation of Solution
Write the equation of the magnitude of the couple as follows;
M=23μkPsinθR32−R31R22−R21
Here, the coefficient of kinetic friction is μk, the magnitude of the total axial force is P, the outer radii of the collar is R1, and the inner radii of the collar is R2.
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.
ΔA=rΔsΔϕ
Here, the distance between the point to the axis of the shaft is r, the change in distance is Δr, and the change in azimuthal angle is Δϕ.
Here, Δs=Δrsinθ
Find the normal force exerted ΔN due to the element of area ΔA using the relation;
ΔN=kΔA
Substitute rΔsΔϕ for ΔA.
ΔN=krΔsΔϕ
Substitute Δrsinθ for Δs.
ΔN=krΔrsinθΔϕ (1)
Find the change in friction force in y-axis;
ΔFy=ΔNsinθ
Substitute krΔrsinθΔϕ for ΔN.
ΔFy=krΔrsinθΔϕsinθ=krΔrΔϕ
Find the value of k using the relation.
P=∑ΔFy=∫2π0∫R2R1ΔFy
Substitute krΔrΔϕ for ΔFy.
P=∫2π0∫R2R1krΔrΔϕ=k[r22]R2R1[ϕ]2π0=k[R22−R212][2π]=πk(R22−R21)
k=Pπ(R22−R21)
Substitute Pπ(R22−R21) for k in Equation (1).
ΔN=Pπ(R22−R21)rΔrsinθΔϕ
Find the change in friction force ΔF using the relation.
ΔF=μkΔN
Substitute Pπ(R22−R21)rΔrsinθΔϕ for ΔN.
ΔF=μkPπ(R22−R21)rΔrsinθΔϕ
Find the change in moment ΔM using the relation.
ΔM=rΔF
Substitute μkPπ(R22−R21)rΔrsinθΔϕ for ΔF.
ΔM=rμkPπ(R22−R21)rΔrsinθΔϕ
Integrate the equation to find the couple M.
M=∫2π0∫R2R1ΔM
Substitute rμkPπ(R22−R21)rΔrsinθΔϕ for ΔM and integrate.
M=∫2π0∫R2R1rμkPπ(R22−R21)rΔrsinθΔϕ=μkPπsinθ(R22−R21)[r33]R2R1[ϕ]2π0=μkPπsinθ(R22−R21)[R32−R313][2π]=23μkPsinθ(R32−R31)(R22−R21)
Hence, it has been proved that the magnitude M of the couple required to overcome the frictional resistance of a conical bearing is M=23μkPsinθ(R32−R31)(R22−R21).
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Chapter 8 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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