A treatise on gyrostatics and rotational motion . ame values of k2 and m2 for a=±7r. Hence thetwo results agree, as they ought, when a = 0, a= ±7r. In this case of the motioncomplete revolutions are just achieved. Theangular speed at the lowest point is then given ],y n2tanh2A/i. 10. Example: Z/iquid filament in arevolving vertical circular tube. We now consider an example which forms an interesting variant of the problem of the Watt governor. A filament of mercury is enclosed in a uniform circular glass tube, the plane of which is vertical and revolves with uniform angular speed about tie- ve
A treatise on gyrostatics and rotational motion . ame values of k2 and m2 for a=±7r. Hence thetwo results agree, as they ought, when a = 0, a= ±7r. In this case of the motioncomplete revolutions are just achieved. Theangular speed at the lowest point is then given ],y n2tanh2A/i. 10. Example: Z/iquid filament in arevolving vertical circular tube. We now consider an example which forms an interesting variant of the problem of the Watt governor. A filament of mercury is enclosed in a uniform circular glass tube, the plane of which is vertical and revolves with uniform angular speed about tie- vertical, «>/.. through the centre of the circle. It is required to find the motion of the mercury in the tube. Let (Fig. 93) a line drawn from the centre of the circle to the centre C of the filament make an angle with the vertical, and the line drawn to an element of the filament make an angle + 6 with the vertical. Let m be the mass of the filament per unit of length, and / its radius. Take three axes of. 348 GYROSTATICS chap. coordinates OC, OD, drawn from the paper upwards, and OE in the plane of the diagramperpendicular to OC. If \l be the angular speed about the vertical, we have ficoB, //sin. Componentsor 2. About OC=l <f)S2ddO = mP[;£(a-isin 2a). 3. AboutOE=l ?nr^sin<£cos2#CE0 = »i(3/xsin(a + isin2a). •(1) Hence the rate of growth of about OD is -2ml3aif>, due to the time-rate ofincrease of the first component, - mP^aiD (/>cos coscf>(a + i sin 2«t), due to the turning of OEabout OC. The total rate of growth of about OD is therefore - mP(2a(f> — /z2sin <£ cos sin 2a). The moment of forces about OD is easily found by integration to be lamPg sin a sin ?
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