Page:Elementary Text-book of Physics (Anthony, 1897).djvu/55

§ 37] If the line $$AB$$ he rotated through the angle $$\theta$$ about any point in it, and if then another point in it be taken and the line rotated about that point through an angle $$- \theta$$, the result is a translation of the line $$AB$$. We may therefore substitute for a rotation about one point a translation and an equal and opposite rotation about another properly chosen point.

By the following construction it is always possible to find a point $$O$$ in the plane in which $$AB$$ moves, such that a pure rotation of $$AB$$ about it will bring the body from its initial to its final position.

Join $$AA'$$, $$BB'$$ (Fig. 12), and bisect the lines $$AA'$$ and $$BB'$$ at the points $$C$$ and $$D$$. At those points erect perpendiculars which will intersect at the point $$O$$. Join $$OA$$, $$OB$$, $$OA'$$, $$OB'$$. By the geometry of the figure the triangles $$AOB$$ and $$A'OB'$$ are similar, and adding to their equal angles at $$O$$ the common angle $$A'OB$$, we have $$AOA' = BOB'$$. Hence a rotation through the angle $$AOA' = BOB'$$ will transfer $$AB$$ to $$A'B'$$. The perpendicular through may be called the axis of rotation. This construction fails when the initial and final positions of $$AB$$ are parallel.

37. Kinetic Energy of a Rotating Body.— Let $$r$$ represent the distance of any particle of the body, of mass $$m$$, from the axis about which the body rotates, and $$\omega$$ its angular velocity about that axis. Then the kinetic energy of this particle is $$\tfrac{1}{2} mr^2 \omega^2$$, and the kinetic energy of the rotating body is $$\tfrac{1}{2} \omega^2 \sum mr^2$$.In § 36 we have shown that we may replace a rotation by a translation and a rotation of the same amount about another axis. Since velocities are measured by the displacements of the moving particle which occur in the same interval of time, it is also possible to replace an angular velocity by a velocity of translation and an equal angular velocity in the opposite sense about another axis. We choose for the new axis that passing through the centre of mass, at the distance $$R$$ from the original axis. The velocity of the centre of mass is then $$R \omega$$. We represent by $$l$$ the distance of the mass $$m$$ from the axis passing