Page:Elementary Text-book of Physics (Anthony, 1897).djvu/60

celeration of the centre of mass is given by $F=Ma$, and the angular acceleration about the centre of mass by $F\,.\,R=I\alpha$, where $R$ is the distance from the centre of mass to the line of the force. The actual acceleration of any point of the body about the centre of mass, due to this angular acceleration, is $\alpha x$, where $x$ is the distance of the point from the centre of mass. The total acceleration of any point on the line of $R$ is $a\pm \alpha x={\frac {F}{M}}\pm {\frac {F\,.\,R}{I}}x,$ the positive or negative sign being used according as the point lies in $R$ itself, or in its prolongation through the centre of mass. If $a-\alpha x=0$, the point considered is at rest. The condition that the point is at rest is therefore ${\frac {1}{M}}-{\frac {Rx}{I}}=0$, or

$x={\frac {I}{MR}}\cdot$

(32)

The movement of the body will, therefore, not be altered if a fixed axis be passed through this point. If the body be considered as free to rotate about this axis, the point of application of the force, which is distant $R+x$ from the axis, and which is such that the force there applied will occasion no stress on the axis, is called the centre of percussion. We have

$x+R={\frac {I}{MR}}+R={\frac {I+MR^{2}}{MR}}\cdot$

(33)

By § 38, $I+MR^{2}$ is the moment of inertia of the body about the axis of rotation, so that the distance from the axis of rotation to the centre of percussion is equal to the moment of inertia of the body divided by $MR.$ The product $MR$ is sometimes called the static moment.

45. Mechanical Powers.— There are certain simple cases of the combination of forces in accordance with the foregoing principles which are of especial importance because of their application in the construction of machines. They are generally called the mechanical powers.

They are all designed to enable us, by the application of a certain force at one point, to obtain at another point a force, in general