The radius R of the steering wheel in Fig. 4.50 is 200 mm. The distance from O to C is 1 m. The center C of the steering wheel lies in the x-y plane. The force F = sin α(10i + lOj — 5k) N. Deteimine the value of α at which the magnitude of the moment of F about the shaft OC of the steering

Question

The radius R of the steering wheel in Fig. 4.50 is 200 mm. The distance from O to C is 1 m. The center C of the steering wheel lies in the x-y plane. The force F = sin α(10i + l0j – 5k) N. Deteimine the value of α at which the magnitude of the moment of F about the shaft OC of the steering the steering wheel is a maximum. What is the maximum magnitude?

Strategy
We will determine the moment of F about OC in terms of the anglea α and obtain a graph of the moment as a function of α.

Accepted Answer

In terms of the vector [latex]r_{CA}[/latex] from point C on the shaft to the point of apphcation of the force, and the unit vector [latex]e_{OC}[/latex] that points along the shaft from point O toward point C (Fig. a), the moment of F about the shaft is

[latex]M_{OC} = \left\lgroup e_{OC} . \left(r_{CA} imes F ight) ight group e_{OC} [/latex]

From Fig. a, the unit vector [latex]e_{OC} [/latex] is

[latex]e_{OC} = \cos 20º i + \sin 20º j . [/latex]

and the z component of [latex]r_{CA}[/latex] is -R sin α. By viewing the steering wheel with the z axis perpendicular to the page (Fig. b), we can see that the x component of [latex]r_{CA}[/latex] is R cos α sin 20° and the y component is -R cos α cos 20°, so

[latex]r_{CA} = R(\cos α \sin 20° i - \cos α \cos 20° j - \sin α K) [/latex]

The magnitude of [latex]M_{OC}[/latex] is the absolute value of the scalar

[latex]e_{OC} . (r_{CA} imes F ) = \left|\begin{matrix} \cos 20° & \sin 20° & 0 \ R \cos α \sin 20° &- R\cos α \cos 20° & - R \sin α \ 10 \sin α & 10 \sin α & - 5 \sin α \end{matrix} ight| = R \left[5 \sin α \cos α + 10(\cos 20° - \sin 20°) \sin ^{2} α ight] [/latex] .

Computing the absolute value of this expression as a function of α, we obtain the graph shown in Fig. 4.51. The magnitude of the moment is an extremum at values of α of approximately 70° and 250°. By examining the computed results near 70°,

α	[latex] \left\|M_{OC} ight\|(N-M) [/latex]
67°	1.3725
68°	1.3749
69°	1.3764
70°	1.3769
71°	1.3765
72°	1.3751
73°	1.3728

we can see that the maximum value is approximately 1.38 N-m. The value of the moment at α = 250° is also 1.38 N-m.

Question 4.22: The radius R of the steering wheel in Fig. 4.50 is 200 mm. T...

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