# Generalized Forces

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## Generalized Forces

quantities that play the role of ordinary forces when the configuration of a mechanical system is defined by generalized coordinates in an investigation of the equilibrium or motion of the system. The number of generalized forces is equal to the number *s* of degrees of freedom of the system; here, each generalized coordinate *q _{i}* has a corresponding generalized force

*Q*. The value of

_{i}*Q*

_{1}corresponding to the coordinate

*q*

_{1}can be found by computing the virtual work δ

*A*

_{1}performed by all the forces for a virtual displacement of the system in which only the coordinate

*q*

_{1}is changed, receiving an increment δ

*q*

_{1}. Then δ

*A*

_{1}=

*Q*

_{1}δ

*q*

_{1}that is, the coefficient for δ

*q*

_{1}in the expression for δ

*A*

_{1}will be the generalized force

*Q*

_{1}. The generalized forces

*Q*

_{2},

*Q*

_{3}, …,

*Q*are calculated in a similar manner.

_{s}For example, suppose we have a winch (see Figure 1) that is lifting a load *P* on a cable, which is a system having one degree of freedom. If the generalized coordinate *q*_{1} is taken to be the angle of rotation ø of the shaft of the winch and if a torque *M _{f}* and a frictional moment

*M*are applied to the shaft, then neglecting the weight of the cable we obtain δ

_{f}*A*

_{1}= (

*M*–

_{t}*M*–

_{f}*Pr*)δø, where

*r*is the radius of the shaft. Consequently, for this system, the generalized force corresponding to the coordinate ø will be

*Q*

_{1}=

*M*–

_{t}*M*–

_{f}*Pr*.

The dimensions of a generalized force depend on the dimensions of the generalized coordinate. For example, if the *q _{i}* are lengths, then the

*Q*will have dimensions of an ordinary force; if the

_{i}*q*are angles, then the

_{i}*Q*will have dimensions of a moment of a force. In an investigation of the motion of a mechanical system, generalized forces appear instead of ordinary forces in the Lagrange equations of mechanics; when the system is in equilibrium, all the generalized forces are equal to zero. As an example, for the winch discussed above, when the load is being lifted uniformly, we must have

_{i}*Q*

_{1}= O; that is,

*M*=

_{t}*M*+

_{f}*Pr*.

S. M. TARG