CHAPTER *XI. 



MOTION OF A RIGID BODY IN TWO DIMENSIONS. 



214. IN this Chapter we propose to discuss the motion of a 

 rigid body in cases where every particle of the body moves parallel 

 to a fixed plane, for example the plane (x, y) of a frame of refer 

 ence. In such a case the x and y of a particle of the body vary 

 with the time, but the z of each particle remains constant 

 throughout the motion. The motion is said to be &quot; in two dimen 

 sions,&quot; or &quot;in one plane.&quot; Now we saw in Article 114 that to 

 determine the position of a rigid body it is requisite and sufficient 

 to determine the positions of a particle of the body, of a line of 

 particles passing through that particle, and of a plane of particles 

 passing through that line. In the case now under discussion we 

 may take the line and plane in question to be parallel to the 

 plane (x, y), then the position of the plane is invariable, and the 

 position of the line is determined by the angle it makes with a 

 fixed line in the plane, for instance the axis of x } further the 

 position of the chosen particle is determined by its coordinates x 

 and y. Thus the determination of the position of the rigid body 

 (moving in two dimensions) requires the determination of three 

 numbers, representing the coordinates of the position of one of the 

 particles, and the angle which a line of the body through that 

 particle and in the plane of its motion makes with a fixed line. 



We can now see what is meant by the angular velocity of a 

 rigid body moving in one plane. Suppose, in fact, that one line 

 of particles in the body and parallel to the plane makes an angle 

 6 at time t with a line fixed in the plane. Then this angle is 

 increasing at a rate 0. Let any other line of particles be drawn 

 also parallel to the plane, and let a be the angle it makes with 



152 



