225] FORMATION OF EQUATIONS. 243 



centre of the reel is #a 2 /(a 2 + 2 ), where k is the radius of gyration of the reel 

 about its axis, and that the tension of the thread is {l + 2 /( 2 +a 2 )} of the 

 weight of the reel. 



5. A thread passes over a smooth peg and unwinds itself from two 

 cylindrical reels freely suspended from it and having their axes horizontal. 

 Prove that each reel descends with uniform acceleration. 



6. A ball is at rest in a cylindrical garden roller, when the roller is seized 

 and made to roll uniformly on a level walk; to find the motion of the ball, 

 supposing it does not slip on 



the roller. 



Let a be the radius of the 

 ball, b of the roller, 6 the 

 angle the line of centres 

 makes with the vertical, V 

 the velocity of the roller. 



Prove (i) that the an- 

 gular velocity of the roller is 



V ">' Fig. 62. 



(ii) that the angular 



velocity o> of the ball is Vja -(b-a) 0/a. 



Let k be the radius of gyration of the ball, supposed uniform, about an 

 axis through its centre of inertia, m the mass of the ball. Initially all the 

 impulsive forces acting on the ball pass through the point of contact, and 

 therefore the moment of momentum of the ball about any axis through this 

 point is zero initially. Hence obtain the equation 



mFfflo - ma {(b - a) - F} = 



for the initial values o> of o> and of ; prove that o> vanishes, and find the 

 value of Q . 



Obtain the equations of motion 



mk 2 ) - ma (b-a) 6= mga sin 6, 



m(b-a)0 2 =R- mg cos 0, 



where R is the pressure of the roller on the ball. Prove that the motion in B 

 is the same as that of a simple pendulum of length (b a), and show that 

 the value of R in any position is 



mg (if cos 6 - J^) + m F 2 /(6 - a). 



Deduce the condition that the ball may roll quite round the interior of 

 the roller. 



7. A cube containing a spherical cavity slides without friction down a 

 plane of inclination a, and a homogeneous sphere rolls in the cavity. Prove 

 that the angle 6 between the normal to the plane and the radius through the 

 point of contact of the sphere with the cavity is connected with the angular 

 velocity o> of the sphere by the equation (a b)0=b<o, where a is the radius 

 of the cavity, and b is the radius of the sphere. 



162 



