Question 11.2: Total moment of gravity forces A system S moves under unifor...

Classical Mechanics

Total moment of gravity forces

A system S moves under uniform gravity. Show that the total moment of the gravity forces about any point is the same as if all the mass of S were concentrated at its centre of mass.

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Without losing generality, let the point about which moments are taken be the origin O. Under uniform gravity, $F _{i}=-m_{i} g k$ , where the unit vector k points vertically upwards, so that

$\begin{aligned}K _{O} &=\sum_{i=1}^{N} r _{i} \times\left(-m_{i} g k \right)=\left(\sum_{i=1}^{N} m_{i} r _{i}\right) \times(-g k )=(M R ) \times(-g k ) \\&= R \times(-M g k )\end{aligned}$

where M is the total mass of S and R is the position vector of its centre of mass. This is the required result. Note that it is only true for uniform gravity.

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