$A$ rod of length $3 \; m$ has a linear mass density that is directly proportional to the distance $x$ from one of its ends. Find the position of its centre of gravity from that end. (in $; m$)

Two particles of mass $5\, kg$ and $10\, kg$ respectively are attached to the two ends of a rigid rod of length $1\, m$ with negligible mass. The centre of mass of the system from the $5\, kg$ particle is nearly at a distance of $..........\, cm$.

Write a short note on the centre of gravity.

Three identical spheres,each of mass $1 \ kg$,are placed touching each other with their centres on a straight line. Their centres are marked $K, L$,and $M$ respectively. The distance of the centre of mass of the system from $K$ is

Distinguish between the center of mass and the center of gravity.

The figure below shows a shampoo bottle in a perfect cylindrical shape. In a simple experiment,the stability of the bottle filled with different amounts of shampoo volume is observed. The bottle is tilted from one side and then released. Let the angle $\theta$ depict the critical angular displacement resulting in the bottle losing its stability and tipping over. Choose the graph that correctly depicts the fraction $f$ of shampoo filled ($f=1$ corresponds to completely filled) versus the tipping angle $\theta$.