To solve this problem we have to cut the rod at a distance of "x" from point P. The main thing here is that even if we cut the rod into two parts (whether equal or not) the acceleration of each part will be same as that of the original rod before cutting. The tension developed here is nothing but the restoring force which is caused due to the elastic nature of the rod. 

**The Problem Setup**

We have a rod PQ of mass m, length l, and cross-sectional area A, lying on a smooth table. A force F is applied at point P on the rod. We need to find the tension at a distance x from point P.

**Understanding the Problem**

To find the tension at distance x from point P, we need to analyze the forces acting on the rod. The applied force F will create a tension force at point P, which will be transmitted along the rod. At any point on the rod, there will be two forces acting: the tension force from the left side and the tension force from the right side. We need to find the magnitude of the tension force at point x.

**Analysis of Forces**

Let's consider a small section of the rod between points P and x. This section has a length dx and a mass dm. The force F applied at point P will create a tension force T at point P, which will be transmitted along the rod.

**Tension Force Calculation**

To calculate the tension force at point x, we can consider the equilibrium of forces for the small section of the rod between points P and x.

The tension force at point P can be written as:
T = F

The tension force at point x can be written as:
T + dT = Tension force from the right side

Since the rod is in equilibrium, the sum of forces acting on the small section should be zero.

**Sum of Forces**

The sum of forces acting on the small section of the rod can be written as:
dT - T = 0

Simplifying the equation, we get:
dT = T

**Integration to Find Tension**

To find the tension force at point x, we need to integrate the equation dT = T over the length of the rod.

∫ dT = ∫ T

Integrating both sides, we get:
T - T0 = ∫ T dx

T - T0 = ∫ F dx

T - T0 = Fx

Therefore, the tension force at a distance x from point P can be calculated as:
T = T0 + Fx

**Conclusion**

In conclusion, the tension force at a distance x from point P on the rod is given by the equation T = T0 + Fx, where T0 is the tension at point P and F is the applied force at point P. This equation is derived by analyzing the forces acting on a small section of the rod and considering the equilibrium of forces.