State the meaning of the centre of gravity of an object.
As shown in Fig. 2.1, a non-uniform rod XY is pivoted at point P. The rod is $4.00\,\text{m}$ long and has weight $44.0\,\text{N}$. Its centre of gravity is $1.70\,\text{m}$ from end X of the rod. Point P is $1.10\,\text{m}$ from end X. A sphere is suspended from end Y by a wire. The sphere weighs $3.0\,\text{N}$. The wire’s weight is negligible. A vertical downward force $F$ acts at end X so that the horizontal rod is in equilibrium.
Calculate $F$ by taking moments about P.
Calculate the force that the pivot exerts on the rod.
In (b), the sphere is now placed in a liquid in a container, as shown in Fig. 2.2. The density of the liquid is $1100\,\text{kg m}^{-3}$. The upthrust acting on the sphere due to the liquid is $2.5\,\text{N}$. The size of $F$ is unchanged, so the horizontal rod is not in equilibrium.
Use Archimedes’ principle to work out the radius $r$ of the sphere.
Calculate the magnitude and direction of the resultant moment of the forces on the rod about P.