Question 11.6: Rigid body moving under uniform gravity A rigid body is movi...
Rigid body moving under uniform gravity
A rigid body is moving in any manner under uniform gravity. Show that its motion relative to its centre of mass is the same as if gravity were absent.
Under uniform gravity, the total moment of the gravity forces about any point is the same as if they all acted at G, the centre of mass of the body (see Example 11.2). It follows that K _{G}= 0.
The rigid body equations (11.15) therefore take the form
\boxed{\begin{matrix} \text{Rigid body equations} \\M \frac{d V}{d t}= F \quad \frac{d L _{G}}{d t}= K _{G}\end{matrix} } (11.15)
M \frac{d V }{d t}=-M g k , \quad \frac{d L _{G}}{d t}= 0.
Hence, when a rigid body moves under uniform gravity, G undergoes projectile motion (which we already knew), and the equation for the motion of the body relative to G is the same as if the body were moving in free space.