Accelerations a₁ and a₂ of two separate objects

Problem:

The same net force causes accelerations a₁ and a₂ of two separate objects.
What will be the acceleration if these bodies are fastened together and the resultant force mentioned above is applied to this system?


Solution:

To find the acceleration of two objects when connected and subjected to the same net force, let's apply the principle F=ma correctly:

Given:

• $�$
• F is the net force applied to both objects when they are connected.
• ${�}_1$ and ${�}_2$ are the accelerations of the two objects when the force is applied separately to each.
• ${�}_1$ and ${�}_2$ are the masses of the two objects, respectively.

From Newton's second law, when the force $�$ is applied to each object separately: $�={�}_1{�}_1$ $�={�}_2{�}_2$

These equations give us the individual masses of the objects as: ${�}_1=\frac{�}{{�}_1}$ ${�}_2=\frac{�}{{�}_2}$

When the objects are connected and the same force $�$ is applied, the total mass is ${�}_1+{�}_2$. The acceleration $�$ of this combined system under the force $�$ is given by: $�=({�}_1+{�}_2)�$

Substituting ${�}_1$ and ${�}_2$ from above into the equation: $�=(\frac{�}{{�}_1}+\frac{�}{{�}_2})�$

Solving for $�$, we get: $�=\frac{�}{(\frac{�}{{�}_1}+\frac{�}{{�}_2})}$

This simplifies to: $�=\frac{1}{(\frac{1}{{�}_1}+\frac{1}{{�}_2})}$

So, the acceleration of the combined system is determined by the reciprocal of the sum of the reciprocals of the individual accelerations. This formula shows how the combined acceleration relates to the individual accelerations ${�}_1$ and ${�}_2$.