266 



MOTION UNDER A VARIABLE FORCE 



To find the curve in which the particle must be constrained to 

 move, let us go back to equation (94), namely 



dh 



which is the equation of motion of a particle constrained to move 

 in any curve, provided is the angle which the tangent to the 

 curve at a distance s along it makes with the horizontal. For this 

 equation to represent simple harmonic motion, the acceleration 

 d 2 s 



dt 2 



must be equal to k 2 s. Thus we must have 



(99) 



g sin = tfs, 



so that sin 6 must be proportional to s. 



211. This relation expresses a property of the cycloid, i.e. of 

 the curve described in space by a point on the rim of a circle which 

 rolls along a straight line. For, in fig. 132, let P be a point on 

 a cycloid which is formed by a circle rolling along the line EF. 

 When the point on the rim of the moving circle is at P, let A be 



the point of the circle which is in contact with the line EF, and 

 let AB be the diameter of the circle which passes through A. 



At the instant considered, we know that the motion of the point 

 P on the rim of the circle is perpendicular to the line AP (see 

 example 1 on p. 9). Thus since APB is a right angle, the motion 

 must be along BP. Thus BP is the tangent to the cycloid. 



If EF is supposed horizontal, the angle 6 between the tan- 

 gent at P and the horizontal is equal to the angle PAB, so that 



