82 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



116. Prove that the velocity and acceleration of a particle describing an 

 equiangular spiral as a central orbit about the pole are at any instant 

 the same as those of a particle describing a certain ellipse with centre at the 

 pole as a central orbit about the centre, the axes of the ellipse being propor- 

 tional to the distance from the pole. 



117. If an equiangular spiral whose pole is is described as a central 

 orbit about any point $, prove that the acceleration at P is inversely propor- 

 tional to OP . SP 2 . sin 3 <, where (p is the angle the radius vector 8P makes 

 with the tangent at P. 



118. Prove that the acceleration towards the centre of the fixed circle 

 with which a particle can describe an epicycloid is proportional to r/p*, where 

 r is the radius vector and p the perpendicular from the centre to the 

 tangent. 



119. The curve r=a+bd is described as a central orbit about the origin 

 with initial distance a and initial velocity V in a direction making an angle 

 ^TT with the initial radius vector. Find the formula for the acceleration. 



120. Prove that the acceleration with which the curve r=a sin n6 can be 

 described as a central orbit about the origin is proportional to 



121. Prove that the curve r=a (l + ^Q cos 6} is a central orbit about the 

 origin for acceleration inversely proportional to the fourth power of the 

 distance. 



122. If the curve r 2n +& 2n + 2a w r n cosn#=0 is described as a central orbit 

 about the origin with areal velocity |A, prove that the central acceleration is 



123. If any curve is described as a central orbit about a point the 

 velocity of the foot of the perpendicular from on the tangent varies inversely 

 as the chord of curvature through 0. 



124. A particle is describing a central orbit about a point S t and h 

 is twice the rate at which the radius vector describes areas. Another particle 

 moves so that at any instant its distance (r) from S is equal to that of 

 the first particle, and the angular velocity of its radius vector is less than that 

 of the first particle in the ratio sin a : 1. Show that the second particle has 

 an acceleration to S less than that of the first particle by h 2 cos 2 a/r 3 . 



125. A series of particles are describing the same curve as a central orbit 

 about a point with an acceleration whose tangential component is h 2 /p 2 <p'(p}. 

 Prove that if the line density at any time is constant and =p , the line 

 density p at any subsequent time t is given by 



P, being the rate of description of areas about 0, and p the perpendicular 

 from on the tangent. 



