Q.While rotating an object or while opening a door or a water tap we apply a force or forces. Under which conditions is this process easy for us? Why? Define the vector quantity concerned. How does it differ for a single force and for two opposite forces with different lines of action?

[!TLDR] Easiest when the force is applied far from the axis, perpendicular to the position vector. The vector quantity is TORQUE, $\vec{\tau}=\vec{r}\times\vec{F}$; for a single force it depends on the (fixed) axis's location, while a COUPLE (two opposite forces on different lines) gives a torque independent of the axis, producing pure rotation wit …

Rotating an object, or opening a door or a tap, is EASIEST when the applied force is (a) as far as possible from the axis of rotation (larger r), and (b) applied as close to PERPENDICULAR to the position vector r as possible (angle theta near 90 degrees, maximising $\sin\theta$) -- this is why door handles are placed far from the hinge and pushed roughly perpendicular to the door. The relevant VECTOR quantity is TORQUE (moment of a force), $\vec{\tau}=\vec{r}\times\vec{F}$, magnitude $\tau=rF\sin\theta$ (4.10).

For a SINGLE force acting on a body pivoted at a fixed axis, the torque explicitly depends on both the point of application (via r) AND the choice/location of that fixed axis -- change the axis, and the same force generally produces a different torque; additionally, if the axis is not truly fixed (or friction is insufficient), a single force can also produce translational acceleration alongside rotation. …

Recall the conditions that maximise torque (large r, theta near 90 degrees), define torque as $\vec{r}\times\vec{F}$, and contrast the axis-dependence of a single …

Treating a single force's torque and a couple's torque as governed by identical rules, missing the key …