EXAMPLES. 213 



81. Two particles of equal mass, connected by a thread of length a, lie on 

 a smooth table with the thread just straight. One of the particles is set in 

 motion at right angles to the thread with velocity v ; prove that each of them 

 describes a series of cycloids, the time of describing any one of which is najv. 



82. Two particles P, Q, are connected by a fine string which passes 

 through a small hole in a smooth inclined plane (inclination a). Q hangs 

 vertically, and P moves on the inclined plane. Show that the differential 

 equation of P's path is 



I K sin a cos 6 + sin a sin 6 (1 -f K) -=- 

 smtfsma d I __ v ' udQ\ 



~ - - 



where AC is the ratio (mass of Q : mass of P). 



83. Two particles, of masses m and m', are connected by a thread which 

 passes through a hole at the vertex of a smooth right circular cone having its 

 axis vertical and vertex uppermost. The particle of mass m' hangs vertically, 

 and m describes a circle of radius c on the cone. Prove that, if slightly 

 disturbed, it will perform small oscillations in time 



c(m'+m) 1 



3g (m' m cos a) sin a/ ' 



2a being the vertical angle of the cone. 



84. A particle of mass M is attached to a cord and is on a smooth table. 

 The cord passes over the edge of the table and supports a pulley, of mass m, 

 carrying another cord to the ends of which bodies of masses m 1} m 2 are 

 attached. Prove that the acceleration of M is 



85. A bead of mass M slides on a smooth fixed straight rod, and a thread 

 attached to it passes round a pulley, which is fixed at a point on the rod, and 

 is attached to a particle of mass m. Initially M is at rest, the two parts of 

 the thread contain a right angle, and m is projected with velocity V parallel 

 to the rod. Prove that, when m is at a distance r from the pulley, the velocity 

 of M is 



where a is the initial value of r. 



86. A ring can move on a long straight rough rod, the coefficient of 

 friction being /*, under an attraction to a fixed point (not on the rod) varying 

 as the distance. Prove that the time of oscillation is the same as if the rod 

 were smooth, and that the ring will ultimately come to rest at a point within 

 a length 2/zc? of the rod, where d is the distance of the rod from the centre of 

 force. 



