Force Analysis in an Interacting Mass System: A Detailed Guide

In physics, analyzing the forces between interacting masses is a fundamental concept that is often explored through scenarios involving multiple objects in contact. This article delves into the detailed steps of calculating the force that one mass exerts on another when both are interacting on a smooth horizontal surface. Through an example, we will explore the application of Newton's laws to derive the required force, ensuring clarity in understanding for students and educators alike.

Understanding the Problem

Consider two blocks placed in contact on a smooth horizontal surface. The masses of the blocks are 2 kg and 3 kg. A force of 100 N is applied to the 2 kg block. The objective is to determine the magnitude of the force that the 2 kg block exerts on the 3 kg block.

Step-by-Step Solution

The solution to this problem involves the following steps, based on the fundamental principles of physics:

1. Calculate the Total Mass of the System

The total mass of the system is the sum of the individual masses of the blocks.

m_{text{total}} m_1 m_2 2 text{kg} 3 text{kg} 5 text{kg}

2. Calculate the Acceleration of the System Due to the Applied Force

According to Newton's second law, the acceleration of the system can be calculated as:

a frac{F}{m_{text{total}}} frac{100 text{N}}{5 text{kg}} 20 text{m/s}^2

3. Calculate the Force Exerted by the 2 kg Block on the 3 kg Block

The 2 kg block and the 3 kg block will have the same acceleration (20 m/s2) since they are in contact and moving together. The force exerted by the 2 kg block on the 3 kg block can be calculated as:

F_{2 text{on} 3} m_2 cdot a 3 text{kg} cdot 20 text{m/s}^2 60 text{N}

Explanation and Verification

Let us now verify this calculation:

Verification Steps

The net force acting on the 2 kg block can be calculated as:

F_2 100 - F_{2 text{on} 3} 100 - 60 40 text{N}

The acceleration of the 2 kg block can be found using Newton's second law:

a frac{F_2}{m_2} frac{40 text{N}}{2 text{kg}} 20 text{m/s}^2

This confirms that the acceleration of both blocks is 20 m/s2, as initially calculated. This consistency verifies the correctness of our force calculation.

Application of Newton's Third Law

Newton's third law of motion states that for every action, there is an equal and opposite reaction. In this scenario, the 2 kg block exerts a force of 60 N on the 3 kg block, and by Newton's third law, the 3 kg block exerts an equal and opposite force of 60 N on the 2 kg block.

Friction and Its Impact

It is important to note that if there were friction between the blocks or the surface, the force analysis would change. However, in this problem, the surfaces are assumed to be smooth, so there is no frictional force involved. If friction were present, the force applied to the 2 kg block would be partially balanced by the static friction force acting on the 2 kg block, and the net force on the 3 kg block would be adjusted accordingly.

Conclusion

This detailed analysis demonstrates the application of Newton's laws in solving problems involving interacting masses. The force of 60 N that the 2 kg block exerts on the 3 kg block is a quintessential example of the interplay between mass, force, and acceleration. Understanding these principles is essential for grasping more complex physics problems and is highly relevant for students and educators in the fields of physics and engineering.