Forces, acting on the mass m are shown in the figure. As N⃗=m g⃗ the net torque of these two for

step 1 Hi friends, as shown in the figure, force is acting on mass M and is equal to MG. Net torque is

Forces, acting on the mass m are shown in the figure. As...

Forces, acting on the mass $m$ are shown in the figure. As $\vec{N}=m \vec{g}$ the net torque of these two forces about any fixed point must be equal to zero. Tension $T$, acting on the mass $m$ is a central force, which is always directed towards the centre $O$. Hence the moment of force $T$ is also zero about the point $O$ and therefore the angular momentum of the particle $m$ is conserved about $O .$ Let, the angular velocity of the particle be $\omega$, when the separation between hole and particle $m$ is $r$, then from the conservation of momentum about the point $O$, : $m\left(\omega_{0} r_{0}\right) r_{0}=m(\omega r) r$ or Now, from the second law of motion for $T=F=m \omega^{2} r$ Hence the sought tension;

Forces, acting on the mass $m$ are shown in the figure. As $\vec{N}=m \vec{g}$ the net torque of these two forces about any fixed point must be equal to zero. Tension $T$, acting on the mass $m$ is a central force, which is always directed towards the centre $O$. Hence the moment of force $T$ is also zero about the point $O$ and therefore the angular momentum of the particle
$m$ is conserved about $O .$ Let, the angular velocity of the particle be $\omega$, when the separation between hole and particle $m$ is $r$, then from the conservation of momentum about the point $O$, :
$m\left(\omega_{0} r_{0}\right) r_{0}=m(\omega r) r$
or
Now, from the second law of motion for $T=F=m \omega^{2} r$
Hence the sought tension;

Key Concepts

Conservation of Angular Momentum

This concept states that if no external torque acts on a system, its angular momentum remains constant. In the problem context, because the net torque about a chosen point is zero due to the nature of the forces involved, the product of the mass's angular velocity and the radial distance (which together relate to the angular momentum) is conserved.

Central Forces

Central forces always act along the line joining the center of force and the object, meaning that the torque associated with such forces about the center is zero. This property is significant in the problem as it ensures that forces like tension, which are central, do not change the system's angular momentum.

Centripetal Force

Centripetal force is the required inward force that causes an object to move in a circular path. It is provided by the tension in the context of the problem, and it is mathematically expressed as the product of mass and the square of the angular velocity times the radius (m?²r), ensuring that the object maintains its circular motion.

Torque and Its Relation to Angular Momentum

Torque is the measure of the force causing an object to rotate about an axis. The problem highlights that when the net external torque on a system is zero, there is no change in angular momentum. This principle is used to derive relations between forces, distances, and angular speeds in rotational dynamics.

Please give Ace some feedback