Problem 153 - A Complete Resource Book in Physics for JEE Main

Question

A light spring balance hangs from the hook of the other light spring balance and a block of mass \(M \mathrm{~kg}\) hangs from the former one. Then the true statement about the scale reading is (A) Both the scales read \(M \mathrm{~kg}\) each (B) The scale of the lower one reads \(M \mathrm{~kg}\) and of the upper one zero (C) The reading of the two scales can be anything but the sum of the readings will be \(M \mathrm{~kg}\) (D) Both the scales read \(M / 2 \mathrm{~kg}\)

Step-by-Step Solution

Answer

Both the scales read \(M \mathrm{~kg}\) each.

Step 1: Analyze forces acting on the mass M

Consider the block of mass M hanging from the lower spring balance. There is a gravitational force acting on the mass, which can be represented as \(Mg\), where \(g\) is the acceleration due to gravity. The lower spring balance exerts an upward force on the mass, which we can call \(T_1\). In equilibrium, these forces must balance each other, i.e., \(T_1 = Mg\).

Step 2: Analyze forces acting on the lower spring balance

Now, we will analyze the forces acting on the lower spring balance. There is an upward force \(T_2\) exerted by the upper spring balance on the hook of the lower spring balance, and a downward force \(T_1\) exerted on it by the mass M. In equilibrium, these forces must balance each other, i.e., \(T_2 = T_1\), or \(T_2 = Mg\) from Step 1. The scale of the lower spring balance measures the force it experiences from mass M, hence it reads \(M \mathrm{~kg}\).

Step 3: Analyze forces acting on the upper spring balance

Finally, let's analyze the forces acting on the upper spring balance. There is a downward force \(T_2\) exerted on it by the lower spring balance, and an equal and opposite upward force exerted by the hook it hangs from. In equilibrium, these forces balance each other. The scale of the upper spring balance measures the force it experiences from the lower spring balance, so it will read a value corresponding to the force \(T_2\), which we already determined as \(Mg\) in Step 2. Therefore, the upper spring balance scale also reads \(M \mathrm{~kg}\). Based on the analysis above, the correct statement about the scale reading is: (A) Both the scales read \(M \mathrm{~kg}\) each.

Key Concepts

Forces in Equilibrium

When we're looking at scenarios involving forces in equilibrium, we are describing situations where all the forces acting on an object are balanced, meaning the net force is zero. In the case of the spring balances and the block, we apply this concept to ensure the system is at rest or moves with a constant velocity.

For the forces to be in equilibrium, the force due to the block's weight, which acts downward, must be exactly balanced by the tension force from the spring balance, acting upward. Hence, the lower balance reads the actual mass of the block because it is accounting for the force required to hold the block in place against gravity. Similarly, the upper balance experiences equal tension because it sustains the combined weight of the lower balance and the block. It's like a game of tug-of-war with neither side gaining ground—each force perfectly counters the other.

Gravitational Force

The concept of gravitational force is fundamental to understanding why the spring balances register the mass they do. This force, which we often symbolize as 'mg' where 'm' is mass and 'g' is the acceleration due to gravity, describes the force with which the Earth attracts any object with mass towards its center.

Now, remember that a spring balance measures force, not just mass. However, since Earth's gravity is a well-known constant (approximately 9.81 meters per second squared at the surface), we can easily translate this force to an equivalent mass value. That’s why the scales can display kilograms rather than newtons, which are the standard unit of force. Both the upper and lower balances indicate the same mass because both are experiencing the same gravitational force — one directly from the block, and one indirectly as it supports the entire system.

Newton's Laws of Motion

To grasp the full picture of our spring balance problem, it's crucial to understand Newton's laws of motion. Particularly, Newton's first law (also known as the law of inertia) states that an object at rest stays at rest and an object in motion remains in motion at a constant velocity unless acted upon by an unbalanced force.

Applied here, the block and the spring balances remain in a state of no net motion (at rest) because the gravitational force pulling down is matched by the tension in the springs pulling up. If we were to apply Newton's second law, it would tell us that the net force on an object is equal to its mass times its acceleration (F=ma). Since there is no acceleration in the system (because the forces are balanced, and thus, equilibrium is established), the net force is zero. Finally, Newton's third law highlights action and reaction pairs—it's why the upper balance reads the same as the lower. It is responding to the action of the lower balance's force, exerting an equal force in return.