Explain the torque acting on a rigid body.

Explanation

As shown in the figure a rigid body rotates about a fixed axis OZ.
`vec(F_(1)),vec(F_(2))….vec(F_(n))` are the forces acting on the particles with position vectors `vec(r_(1)),vec(r_(2)),vec(r_(3))…..vec(r_(n))`. Respectively.
The force `vec(F)_(n)` is acting on the particle with position vector `vec(r_(n))`. The torque on it is given by,
`vec(tau_(n))=vec(r_(n))xxvec(F)_(n)`
`=|(hati,hatj,hatk),(x_(n),y_(n),z_(n)),(F_(nx),F_(ny),F_(nz))|`
`therefore vectau=hati[y_(n)F_(nz)-Z_(n)F_(ny)]+hatj[Z_(n)F_(nx)-x_(n)F_(nz)]+hatk[x_(n)F_(ny)-y_(n)F_(nx)]....(1)`
The torque acting on the rigid body can be obtained by taking the vector sum of such torques.
`therefore vectau=underset(n)sumhati[y_(n)F_(nz)-Z_(n)F_(ny)]+hatj[Z_(n)F_(nx)-x_(n)F_(nz)]+hatk[x_(n)F_(ny)-y_(n)F_(nx)]`
The Z component of this torque is responsible for the torsional motion of the rigid body about Z-axis. Similarly X and Y components of torque are responsible for the motion about X and Y axis respectively.
In general if rotational motion is about a fixed axis with `hatn` as the unit vector on it, the component of the torque responsible for rotational motion is `vectau.hatn`.
In the rotational motion of a rigid body it is not necessary to apply forces on all the particles of the body.
Since the relative positions of the particles of a rigid body are invariant, a torque appliedon any particle can be considered to be the torque on the entire body. So if `vecF` is a force on a particle and `vecr` is the position vector of the particle the torque on the body can be taken as. `vectau=vecrxxvecF`