312 BOOK OF MATHEMATICAL PROBLEMS. 



1493. A lamina, having its centre of gravity fixed, is at rest 

 and is struck by a blow at the point (a, b) perpendicular to its 

 plane ; prove that the equation of the instantaneous axis is 

 Aax + by = Q, the axes of co-ordinates being principal axes at 

 the centre of gravity, and A, B the principal moments of inertia. 

 If (a, b) lie on a certain straight line, there will be no impulse on 

 the fixed point 



1494. A uniform beam revolves uniformly about one ex- 

 tremity in such a manner as to describe uniformly a cone of 

 revolution about a vertical axis ; determine the pressure on the 

 fixed extremity and the relation between the angle of the cone 

 and the time of revolution. If 0, < be the angles which the 

 vertical makes with the rod and with the direction of pressure 

 respectively, then will 



4 tan < = 3 tan 6. 



1495. A fine string, of length 26, is attached to two points in 

 the same horizontal line, at a distance 2a, and carries a particle p 

 at its middle point ; a uniform rod, of length 2c and mass m, has 

 a ring at each end through which the string passes, and is let fall 

 from a symmetrical position in the same straight line as the two 

 points; prove that in order that the rod may reach the particle 



(a + b - 2c) (m 2 -f 2mp) > 2 (2c - a) p*. 



1496. A circular disc rolls on a rough cycloidal arc whose 

 axis is vertical and vertex downwards, the length of the arc being 

 such that the curvature at either extremity is equal to that of 

 the circle; prove that, if the contact be initially at one extremity, 

 the point on the auxiliary circle of the cycloid corresponding to 

 the point of contact will move with uniform velocity ; and that 

 this velocity will be independent of the radius of the disc. 



1497. A sphere rolls from rest down a given length I of a 

 rough inclined plane, and then traverses a smooth portion of the 

 plane of length ml; find the impulse which the sphere sustains 

 when perfect rolling again commences, and prove that the subse- 

 quent velocity is less than if the whole plane had been rough. 



