326 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



about a point distant x from its middle point, where x is the positive root of 

 the equation 



a? - ( J - 2P/,* TT) o*x - %Po?ln W= 0, 



where /* is the coefficient of friction. 



100. A thin uniform rectangular plank of mass M is suspended from 

 four points in the same horizontal plane by four parallel chains of equal 

 length and negligible mass attached to the corners, and a uniform cylinder of 

 mass m is on the plank with its axis parallel to an edge and its centre of 

 inertia vertically over that of the plank. The whole system is drawn aside 

 in the vertical plane at right angles to the axis of the cylinder till the chains 

 make an angle a with the vertical, and is then let go. Prove that the initial 

 tension of each chain is 



J (M+m) (3J/+m) g cos a/{3 (&amp;gt;/+m) - 2w cos 2 a}, 

 or J M (M+ m} g cos a/{M+ m sin 2 a - mp sin a cos a}, 



according as /z the coefficient of friction is greater or less than 

 (M+m) tan a/(3M+m). 



101. A uniform circular disc (mass M] rotates in a horizontal plane with 

 angular velocity o&amp;gt;. Close round it moves a ring of mass m and radius c 

 rotating about its centre with angular velocity i&amp;gt;(&amp;lt;o&amp;gt;). The ring carries a 

 massless smooth spoke along a radius, and a bead of mass p can move on the 

 spoke under the action of a force to the centre of the ring equal to 

 ^/(distance) 2 , and the bead is in relative equilibrium at a distance a from the 

 centre. Prove that, if a slight continuous action now begins between the 

 disc and the ring, of the nature of friction and proportional to the relative 

 angular velocity, the distance of the bead from the centre, and the angular 

 velocity of the ring, will at first increase, and their values after a short time t 

 will be 



a + t 3 av\ (&amp;lt;o v)l 

 and 



* + pa?} - \ X* 2 [X (o&amp;gt; - *)/( we 2 + pa*)] [2/1/c 2 + 1 /(w 

 where X0 is the frictional couple when the relative angular velocity is 6. 



102. A series of 2n equal uniform rods each of mass m are hinged 

 together and held so that they are alternately horizontal and vertical, each 

 vertical rod being lower than the preceding; the highest rod is horizontal 

 and can turn freely round its end which is fixed. Prove that, when the rods 

 are let go, the horizontal component JT 2r and the vertical component Y 2r of 

 the initial action between the 2rth and the (2r-fl)th rods are given by 



&amp;gt; + C -5- 



the constants B, C, , C&quot; being determined by the equations 

 A r 2n = 0, F 2n =0, .Y 2 + 2^ =0, 2 



