Why do we not apply force on all the particles of a door to open or close it?

The direction of the angular velocity vector is along which of the following?

$A$ hemisphere of radius $R$ and mass $4m$ is free to slide with its base on a smooth horizontal table. $A$ particle of mass $m$ is placed on the top of the hemisphere. The angular velocity of the particle relative to the hemisphere at an angular displacement $\theta$ when the velocity of the hemisphere is $v$ is:

$A$ wheel of radius $1.5 \ m$ rotates with a constant angular acceleration of $10 \ rad/s^2$. Its initial angular speed is $(\frac{60}{\pi}) \ rpm$. What will be its angular speed and angular displacement at $t = 2.0 \ s$?

$A$ body is in pure rotation. The linear speed $v$ of a particle,the distance $r$ of the particle from the axis,and the angular velocity $\omega$ of the body are related as $\omega = \frac{v}{r}$. Thus,

$A$ mass $m$ is supported by a massless string wound around a uniform cylinder of mass $M$ and radius $R$. On releasing the mass from rest,it will fall with an acceleration of: