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## Q. 19.12

Figure (a) shows a uniform sphere of radius R and mass m that is welded to the rod AB of length L (the mass of AB may be neglected). The clevis at B connects the rod to the vertical shaft BC. The assembly is initially rotating about the vertical at the angular velocity ω, with the sphere resting against the shaft. Assuming that ω is gradually increased, determine the critical angular velocity $ω_{cr}$ at which contact between the sphere and rod is lost. Neglect friction.

## Verified Solution

The free-body diagram of the sphere and the rod, drawn at the instant when the assembly is rotating at ω = $ω_{cr}$, is shown in Fig. (b). The xyz-axes are assumed to be attached to rod AB with the origin at B (the x-axis is out of the paper). In addition to the weight mg of the sphere, the FBD also contains the reactions provided by the clevis at B: the pin force B and the two moment components $M_y$ and $M_z$ ($M_x$ = 0 because the pin of the clevis is frictionless). If the angular velocity were less than the critical angular velocity, the FBD would also contain the normal force N that is exerted on the sphere by the vertical shaft. However, when ω = $ω_{cr}$, then N = 0.
It is convenient to choose the Z-axis to coincide with the vertical shaft, as shown in Fig. (b). The Euler angle θ, which was defined as the angle between the Z- and z-axes, is also shown in the figure. When the sphere is about to lose contact with the vertical shaft, its motion consists of a rotation about the Z-axis at the rate $ω_{cr}$. The spin rate is zero, because the sphere cannot rotate relative to the rod AB. Therefore, the motion of the sphere can be described as a steady precession with no spin; in other words, $\dot \phi = ω_{cr}, \dot ψ = \dot θ = 0$. As a result, the steady precession equation, Eq. (19.46), becomes

$\begin{matrix}∑M_x = (I_z − I)\dot \phi^2 \sin θ \cos θ + I_z\dot \phi\dot ψ \sin θ\\∑M_y = 0\qquad ∑M_z = 0\end{matrix}$               (19.46)

$∑M_x = (I_z − I)ω_{cr}^2 \sin θ \cos θ$               (a)

The inertial properties of the sphere about point B are

$I_z = \frac{2}{5}m R^2$      and      $I = I_y = \frac{2}{5}m R^2 + m (L + R)^2$

from which we obtain

$I_z − I = −m (L + R)^2$               (b)

Referring to Fig. (b), we find that the moment of the external forces (the weight) about the x-axis is

$∑M_x = mg R$               (c)

From the same figure we also deduce that

$\sin θ = \sin(π − θ ) = \frac{R}{L + R}$               (d)

and

$\cos θ = − \cos(π − θ ) = − \frac{\sqrt{(L + R)^2 − R^2}}{L + R}$               (e)

Substituting Eqs. (b)–(e) into Eq. (a) and solving for the critical angular velocity, we obtain

$ω_{cr} =\frac{\sqrt{g}}{\sqrt[4]{(L + R)^2 − R^2}}$