Moment of Force and Rotational Mechanics

While opening a tap with two fingers, a couple should be applied which is defined as a pair of parallel equal forces acting in opposite direction about two different axis.

A constant torque applied to a rigid body results in constant angular acceleration, which causes the angular velocity to change over time.

Two equal and opposite forces are applied about different axis, which is termed as couple, to unwind a screw.

Torque acting about any point O         $\tau _o = rF$                 for $sin\theta = 1$

We know, 

See-saw, Swing and Toy horse are the playground equipment make use of the turning effect of forces.

spanner with longer handle needs less force to generate same torque than with spanner with short handle .

if the base area will be heavy than the centre of gravity will be near to ground and if the base area is curved than the object will come in its balanced position everytime due gravitational torque.

cycling , swimming , badminton and hockey all these sports makes use of turning effect  of forces while play. 

In the Earth-Sun system, the gravitational force acts on the Earth, and because the Earth is not a point mass but an extended body, the gravitational gradient creates a torque that contributes to the Earth's axial precession and spin dynamics.

Just as linear momentum is conserved when there is no net external forces$,$

Here we apply two forces of same magnitude but opposite in direction.Hence we have F and - F which makes resultant force as zero.

The formula for the angle of twist in a cylindrical shaft is theta = (2 * torque * length) / (pi * shear_modulus * radius^4).

Due to unequal forces, the net force will act on the body, so the forces are unbalanced and the unbalanced forces cause a change in motion. This means body having translational motion.
There also be torque acting on the body due to unbalanced force. So there will be rotational motion too.

As we have one meter rule pivoted at midpoint there will be a perpendicular distance of 50 cm from the contact of force of 50gf.
 Here we have moment of force of $r \times F$.
After substituting we get $moment\ of\ force = 50\times50 gf cm$.
Here this must be balanced by 100gf's moment of force.
 Let us think that there would be x perpendicular distance.
On equating we get 
$ 50\times50  gf cm = 100\times x gf cm$
$ x = 50\times 50/100 = 25 cm$. 

Here, we are applying two force in same magnitude but in opposite that is one is F and -F. Hence resultant force is $F + (-F) = 0 $. 

Moment of inertia of a body depends  on the body's mass distribution and the axis of rotation.
The moment of inertia of a body is directly proportional to its mass and increases as the mass is moved further from the axis of rotation.
Answer (A) Mass & (C) Distance from rotating axis.

As 100 g is balanced at fulcrum it will have a perpendicular distance of 40 cm from itself and mass m will have $100 - 20 = 80 cm $.

The forces acting on the see-saw and swing produces torque and make them to rotate. See-saw and Swing make use of the effect of forces.

Due to striking the space junk, the asteroid will exert a force on the junk piece.

A pair of equal and opposite forces acting on a body is called a couple.

If principle of moments hold good, then the net torque about a given point is zero (usually CM or the pivoted point is zero). Hence the object does not rotate and is said to be in equilibrium

The whole system does not continue to rotate about P because the moment is balanced. Thus option B is wrong. And the center of mass of the system lies at pivot point P. Thus option C and D are wrong. As the force applied at the two points of suspension is different $\tau$ is different.

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This relates to the critical angular velocity required for a ring to maintain contact or prevent slipping in a rotating system. The derivation leads to the expression in option C.

$K.E=\cfrac{1}{2}mv^2\rightarrow(1)\v=r\omega$

For a disc rolling down an incline, the final linear velocity v = sqrt(2gh / (1 + k^2/R^2)). For a uniform disc, k^2/R^2 = 0.5. Since the velocity depends only on height h and the shape (moment of inertia factor), the mass and radius do not affect the final velocity. Thus, the ratio is 1:1.