i6 4 .] 



PLANE MOTION. 



163. Exercises. 



(1) Show that the sectorial velocity (Art. 135) is constant whenever 



J9 = " 



(2) Show that the normal component of the acceleration is the 

 product of the radius of curvature into the square of the angular 

 velocity about the centre of curvature. 



(3) Show that the velocity is the mean proportional between the 

 acceleration and half the chord intercepted by the direction of the 

 acceleration on the osculating circle. 



(4) If the acceleration of a point P be always directed to a fixed 

 point A y show that the radius vector ^/* describes equal areas in equal 

 times. 



(5) Show that in uniform circular motion the acceleration is directed 

 to the centre and proportional to the radius. 



(6) A wheel rolls on a straight track; find the acceleration of its 

 lowest point. 



4. APPLICATIONS. 



164. Inclined Plane. Imagine a body sliding down a smooth 

 plane inclined at an angle 6 to the horizon. In addition to the 

 assumptions made in the case of falling bodies (see Art. 112) 

 we assume that the motion takes place along a " line of greatest 

 slope," i.e. in a vertical plane at right angles to the intersection 

 of the inclined plane with a horizontal plane. A " smooth " 

 plane means one that offers no fric- 



tional resistance. The body is there- 

 fore subject only to the acceleration 

 of gravity, g\ and it is sufficient to 

 consider the motion of a single point 

 of the body. 



Resolving g into two components, 

 gcosO perpendicular to the plane 

 and -'sin0 along 'the plane (Fig. 41), 



Fig. 41, 



it will be seen that the former component, being at right angles 

 to the velocity, cannot change the magnitude of this velocity. 



