Animated Solution for Physics - Laws of Motion: A block of mass $m$ is connected to another block of mass $M$ by a spring (massless) of spring constant $k$. The blocks are kept on a smooth horizontal plane. Initially the blocks are at rest and the spring is unstretched. Then, a constant force $F$ starts acting on the block of mass $M$ to pull it. Find the force on the block of mass $m$.

The world of classical mechanics is filled with fascinating puzzles, but few are as conceptually illuminating as connected body problems. Imagine you are observing a train engine pulling a series of carriages. The engine exerts a massive force, but how much of that force is actually felt by the very last carriage? This is the exact physical scenario we are exploring in this problem, albeit scaled down to two blocks and a spring.

I know this setup might look a bit intimidating at first glance. We have a block of mass $m$, connected to another block of mass $M$ by a massless spring with a spring constant $k$. They are resting peacefully on a smooth horizontal plane. Suddenly, a constant force $F$ starts pulling on the larger block $M$. Our mission? To find the exact force acting on the smaller block $m$. Let's take a breath and break this down step by step.

Before we dive into the equations, let's visualize the physical reality. We have a frictionless surface, which is a huge relief because it means we don't have to worry about energy being lost to heat or sound. The only external force acting in the horizontal direction is the force $F$ pulling on block $M$.

Now, what happens the moment this force is applied? Block $M$ wants to move forward. But it's tethered to block $m$ by the spring. As block $M$ inches forward, the spring begins to stretch. This stretching creates a restoring force—a tension—that pulls back on $M$ and pulls forward on $m$. It is this very spring force that causes block $m$ to accelerate.

The System Approach

Here is a powerful secret in physics: when multiple bodies are moving together, you can often simplify your life by treating them as a single, unified system.

Let's draw an imaginary box around both blocks and the spring. What is the total mass inside this box? It is simply $(m+M)$.

Now, what are the forces acting on this entire box? The spring force is pulling $M$ to the left and $m$ to the right. Because these forces are equal and opposite, they cancel each other out perfectly when we look at the system as a whole. They are internal forces.

The only external force acting on our system in the horizontal direction is $F$.

According to Newton's Second Law, the net external force on a system equals the total mass of the system multiplied by its acceleration.

Substituting our specific values, we get:

Rearranging this to solve for the acceleration $a$, we find:

This is the acceleration of the center of mass of our system. Even though the blocks might be oscillating back and forth relative to each other due to the spring, their collective center of mass marches forward with this exact, constant acceleration.

Now that we know how fast the entire system is accelerating, let's zoom in on our target: the smaller block $m$.

Imagine you are sitting on block $m$. What forces do you feel? The surface is smooth, so there is no friction. The only thing touching block $m$ horizontally is the spring. Therefore, the force pulling block $m$ forward is entirely provided by the spring. Let's call this force $F^{\prime }$.

Since block $m$ is part of the system, its average acceleration must be the same as the system's acceleration, $a$.

Let's apply Newton's Second Law specifically to block $m$:

Since the only force acting on it is $F^{\prime }$, we can write:

We are now at the finish line. We have an expression for the force $F^{\prime }$ in terms of $a$, and we already found the value of $a$ in our system analysis. All that is left is a simple substitution.

Let's plug our expression for $a$ into the equation for $F^{\prime }$:

Which simplifies beautifully to:

And there we have it! This is the force acting on block $m$. Notice how elegant this result is. The force $F$ is distributed between the two masses in proportion to their mass. Block $m$ gets a fraction of the total force, specifically the fraction $\frac{m}{m+M}$.

The Reality of the Spring

While we have solved the problem, it is worth taking a moment to appreciate the deeper physics at play. We assumed that block $m$ accelerates at a constant rate $a$. In a perfectly rigid system (like if they were connected by a solid rod), this would be exactly true at every instant.

However, because they are connected by a spring, the reality is a bit more dynamic. When force $F$ is first applied, block $M$ accelerates faster than block $m$, causing the spring to stretch. As the spring stretches, the force it exerts on $m$ increases, causing $m$ to accelerate more. Eventually, $m$ might even accelerate faster than $M$, causing the spring to compress again!

The blocks will actually oscillate back and forth relative to their center of mass. The force $F^{\prime }$ we calculated is the average force or the force required to maintain the steady-state acceleration of the system. It is a brilliant example of how macroscopic principles (like the motion of the center of mass) can cut through complex microscopic details (like the oscillations of the spring) to give us a clean, accurate answer.