Why aren't the $mg$ and tension equal in an Atwood machine? Why in an Atwood machine is $mg$ not equal to the tension force when the mass is accelerating? According to third law of motion they should be equal, as force applied by the block on the rope should be equal to the force applied by rope on the block, which are $mg$ and the tension respectively.
 A: The force applied to the block by the rope is equal to the force applied to the rope by the block, by Newton's Third Law.  Both are equal to the tension in the rope.
But neither one is necessarily equal to the weight of the block, which is the force applied to the block by the Earth.  Newton's Third Law says that the force applied to the block by the Earth is equal to the force applied to the Earth by the block;  there is not necessarily any relationship between these forces and forces applied to (or by) the rope.
A: There are two forces on the mass, tension $T$ pulling upward and the weight (gravitational force) $-mg$ downward.  Newton's Second Law says that $ma$ is the net force on the mass.  In this case, the net force is $T-mg$.  If the mass is stationary, in equilibrium, then $a=0$, so $T-mg=0$, and the weight and tension are equal (and opposite).  However, if the mass is accelerating and falling, then $T\neq mg$.
A: This is the most common confusion with Newton's third law.
Tension in the rope and $mg$ on block doesn't form the action - reaction pair of Newton's law.
The main thing to note that $mg$ acts on the block due to earth and tension acts on the block due to the rope. Both have different sources and act on the same body and hence we can't just say that they form action- reaction pairs.
But from Newton's third law, two things can be confirmed that :
The body pulls the earth with same force $mg$ upward.
The body applies a force on the rope equal in magnitude to the tension force acting on the body itself downward.
Hope it helps .
A: If the suspended mass is still relative to Earth, then tension = mass times gravity. To accelerate the mass upwards the tension must increase. If the mass and rope were in freefall where gravity was 0 and tension was 0, you would still have to apply tension to the rope to accelerate the mass.
