Animated Solution for Physics - Laws of Motion: A light spring balance hangs from the hook of the other light spring balance and a block of mass $M$ kg hangs from the former one. Then, the true statement about the scale reading is

A Tale of Two Balances

Imagine you are in a physics lab, and you decide to hang two spring balances one below the other, finally attaching a heavy block of mass $M$ at the very bottom. A common intuition might trick you into thinking that the upper spring balance will read a higher value, or perhaps the weight gets distributed. But physics, as always, demands a rigorous look at the forces involved.

Let's break down this classic mechanics problem step by step.

The key to unlocking this problem lies in a single word used in the question: "light". In the realm of physics, a "light" object implies that its mass is negligible, effectively zero.

Why is this important? Because according to Newton's Second Law, the net force on an object is $F_{net}=ma$. If the mass $m$ is zero, the net force on that object must also be exactly zero, regardless of its acceleration. This simple fact will guide our entire analysis.

Let's start from the bottom and work our way up.

First, consider the block of mass $M$. It is hanging peacefully at rest. The forces acting on it are gravity pulling it downwards with a force of $Mg$, and the tension from the lower spring balance ($S_2$) pulling it upwards, which we will call $T_1$. Since the block is in equilibrium, these forces must perfectly balance each other:

Now, let's move up to the lower spring balance, $S_2$. What forces are acting on it? The tension $T_1$ is pulling it downwards from the bottom hook, and the tension from the upper spring balance ($S_1$), let's call it $T_2$, is pulling it upwards from the top hook.

Because $S_2$ is a light spring balance, its mass is zero. Therefore, the net force acting on it must be zero. This means the upward force must equal the downward force:

Since we already established that $T_1=Mg$, it naturally follows that $T_2=Mg$ as well.

Finally, let's look at the upper spring balance, $S_1$. It is also light and massless. It experiences a downward pull of $T_2$ and an upward pull from the ceiling support, let's call it $T_3$. By the exact same logic of zero net force:

What does a spring balance actually measure? It measures the tension force acting across it.

For the lower spring balance $S_2$, the tension is $T_1$, which is equal to $Mg$. Therefore, it will display a reading corresponding to a mass of $M$ kg.

For the upper spring balance $S_1$, the tension is $T_2$, which is also equal to $Mg$. Therefore, it too will display a reading corresponding to a mass of $M$ kg.

Both scales read $M$ kg each. The tension is transmitted uniformly through the massless spring balances, just like it would through an ideal, massless string.

Always remember: whenever you see the word "light" or "massless" in a mechanics problem, it's a massive hint that the tension will remain constant across that component!