Animated Solution for Physics - Laws of Motion: A block of mass $M$ is pulled along a horizontal frictionless surface by a rope of mass $m$. If a force $P$ is applied at the free end of the rope, the force exerted by the rope on the block is

Connected bodies in mechanics can often look intimidating. When you see a block, a thick rope, and a pulling force, your brain might immediately start worrying about how the force distributes itself. But there is a beautiful, elegant two-step method to solve almost any connected-body problem: The System Approach followed by the Isolated Body Approach.

The Macro View (Finding Acceleration)

Imagine you are looking at the setup from far away. You don't see a separate block and a separate rope; you just see one giant object moving together. Because the rope is attached to the block and the surface is frictionless, they are forced to share the exact same acceleration $a$.

What is the total mass of this "giant object"? It is simply the sum of the individual masses: $m_{\text{total}}=M+m$.

What is the net external force pulling this giant object? It is just $P$. The tension between the rope and the block is an internal force right now, so it cancels out and doesn't affect the overall acceleration.

Using Newton's Second Law ($F=ma$), we can easily find the common acceleration:

The Micro View (Isolating the Block)

Now that we know how fast the whole train is accelerating, let's zoom in on just the block of mass $M$. We draw a Free Body Diagram (FBD) exclusively for the block.

What is pulling the block forward? It's not the force $P$ directly! The force $P$ is applied to the end of the rope. The block is being pulled by the rope itself. We call this pulling force Tension, $T$.

Since the surface is frictionless, $T$ is the only horizontal force acting on the block. And we already know that this block is accelerating at $a$.

Applying Newton's Second Law to just the block:

We have our equation for $T$, and we have our expression for $a$. All that is left is to substitute the acceleration into our tension equation:

Rearranging this slightly gives us our final, elegant answer:

Why does this make intuitive sense? Notice that the tension $T$ is strictly less than the applied force $P$. Why? Because the force $P$ has to do two jobs: it has to accelerate the rope, and it has to accelerate the block. By the time the force "reaches" the block, some of it has been "used up" to accelerate the mass of the rope ($m$). If the rope were massless ($m=0$), the formula would give $T=P$, meaning the force is transmitted perfectly without any loss!