EXAMPLES. 249 



13. A uniform sphere is placed on the highest generator of a rough 

 cylinder which is fixed with its axis horizontal. Prove that, if slightly 

 displaced, the sphere will roll on the cylinder until the plane through the 

 centre of the sphere and the axis of the cylinder makes with the vertical 

 an angle a satisfying the equation 



17 fj. cos a 2 sin a = 10/*, 

 where /* is the coefficient of friction. 



14. A system consisting of a rough uniform circular wire of mass 3/, 

 and a straight uniform rod of mass m^ whose ends can slide on the wire, 

 moves in one plane under no forces, the rod subtending an angle 2a at the 

 centre of the wire. Prove that if neither of the expressions 



(J/"-f-m) sin 2 a + 3Jt/"cos 2 a p sin a cos a (m - 3Jf) 



is negative (p being the coefficient of friction), and, if initially the rod has an 

 angular velocity Q about the centre while the wire is at rest, the rod will 

 come to rest relatively to the wire after a time 



sin 2 a ~ 2/i 2 M sin 2 a] 



jimG [( Jf +m)l*m 2 a - 



15. A flat circular disc of radius a is projected on a rough horizontal 

 table which is such that the friction on an element a is c F 3 ma, where V is the 

 velocity of the element and m the mass of a unit of area. Prove that, if U Q 

 and o> are the initial velocity of the centre of inertia and angular velocity of 

 the disc, the corresponding velocities u, o> at any subsequent time satisfy the 

 equation 



(3w 2 - a 2 o> 2 ) 2 /(3 uf - a V) 2 = (^ 2 a>)/(w 2 o)- 



16. A uniform circular ring moves on a rough curve under no forces, the 

 curvature of the curve being everywhere less than that of the ring. The 

 ring is projected from a point A of the curve, and begins to roll at a point B. 

 Prove that the angle between the normals at A and B is /i" 1 log 2, where p. is 

 the coefficient of friction. 



17. A locomotive engine of mass J/has two pairs of wheels of radius a 

 such that the moment of inertia of either pair with its axle about its axis of 

 rotation is A. The engine exerts a couple G on the forward axle. Prove 

 that, if both pairs of wheels bite at once when the engine starts, the friction 

 between one of the forward wheels and the line capable of being called into 

 play must not be less than ^6r(A + Ma 2 )/a(2A+Ma 2 ). Prove also that, if 

 the only action between an axle and its bearings is a couple varying as the 

 angular velocity of the axle, the final friction called into play between either 

 forward wheel and the line is G/la. 



18. A homogeneous solid hemisphere of mass M and radius a with a 

 smooth base, is placed with its vertex lowest on a rough horizontal plane, 

 and a particle of mass m is placed on the base at a distance c from the 

 centre. Prove that the hemisphere begins to roll or slide on the plane 

 according as 



/> or <25ac/{26(^/+w)a 2 +40mc 2 }, 

 where /n is the coefficient of friction between the hemisphere and the plane. 



