What is the significance of torque actually we call it a rotational analogue of force or a twist to the object but how as it is always with respect to a reference point? Why was it defined this way what is the physical intuition behind it??
Torque has two forms really.
One is the idea of a force couple (or pure torque $\vec{\tau}$) having no net force. This will cause rotation about the center of mass. If the center of mass moved it would have linear momentum, and this requires a net force to acquire. There is no point of application of pure torque. It is applied to the entire rigid body.
The other is an equipollent torque, which represents a force at a distance. This is calculated by $\vec{M} = \vec{r} \times \vec{F}$. The line of action of the force is described by this torque vector. The location of the force is back-calculated by $\vec{r} = \frac{\vec{F} \times \vec{M}}{\| \vec{F} \|^2}$ (where $\times$ is the vector cross product).
In reality both can be present making the torque applied on a point described as $\vec{M}_A = \vec{M}_B + \vec{r}_{B/A} \times \vec{F}$.
The intuition here is that equipollent torque is a force at a distance, and a pure torque is a zero force at infinity.
This is enitirely analogous to motion where linear velocity $\vec{v}$ describes a rotation at some distance and a pure translation is equivalent to a body rotating about infinity with zero angular velocity. Remember what is the rule for finding the velocity of a point if we know the motion of another point $\vec{v}_A = \vec{v}_B + \vec{r}_{B/A} \times \vec{\omega}$.
This is the same law is with torques
If we have a rigid body translating with constant velocity $v$ not being acted upon by an external torque about a point O but having a net external torque about point P. Then will the body rotate?
- An applied torque does not have a point of application. In the case above the net torque will make the body accumulate angular momentum. The net torque about the center of mass equals the rate of change of angular momentum at the center of mass.
What will happen if these forces are applied at the center of mass ?
What forces? Your question talks about torques and not forces. In the absense of net forces linear momentun remains unchanged and the center of mass will continue to move with $\vec{v}$.
If the net torque about the center of mass is zero (line of action of forces through center of mass) then the body will translate. This means that every part of the body will have the same linear velocity and the angular velocity is zero.
