Why does a ballet dancer extend her hands in order to reduce her spinning motion?

A car of mass $m$ in heading towards a pole with a velocity $v$. What is the angular momentum of the car with reference to the pole?

State with reasons whether the following statements is true or false:

Two particles of masses $1 \mathrm{kg}$ and $3 \mathrm{kg}$ move towards each other under the mutual forces of attraction. No other force acts on them. When the relative velocity of approach of the two particles is $2 \raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right.$, their centre of mass has a velocity of $05 \raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right.$. When the relative of approach becomes $3 \raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right.$, the velocity of the centre of mass is $0.75 \raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right.$.

A child sits stationary at one end of a long trolley moving uniformly with speed $v$ on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, then, what is the effect on the speed of the centre of mass of the trolley-child system?

What is the significance and measure of the moment of a force about a point? What are positive and negative moments?