Fear of RIGID BOD DYNAMICS (R.B.D.)

 Hello readers,

Do you know that the effect of several parallel /antiparallel forces acting on a planar rigid body can be reproduced by a single equivalent force only?

Consider a planar rigid body on which several parallel forces act as shown.



Sign convention:

(anticlockwise)

So, in the given situation,

Equivalent force

Equivalent torque

τ =

So effective height (h) below center of mass (CM) for F to act:



Therefore, applying the equivalent force (F) at effective height (h) produces the same effect.

Example 1:

Consider a disc of mass m and radius R as shown below. The forces acting on the disc is also shown. Find the effective height below center of mass for the equivalent force to act.



Solution:

The equivalent force acting on the disc,



Equivalent torque acting on the disc,



So, the effective height (h) below center of mass (CM) for F to act,



How is this result useful?

Remember this!
Angular momentum of the body is conserved about any point at this effective height

Let’s consider the disc mentioned in example 1, assume it is also given an initial angular velocity (ωo), in clockwise direction. What will be its translational velocity just when it attains pure rolling?



Solution:

Above we had calculated the effective height .



Angular momentum about P is conserved.

Initial angular momentum,



Final angular momentum:



Equating,



So, knowing the value effective height helped us to find the value of translational velocity quickly.