The maximum value of F which can be applied on the system shown in figure so that both blocks move together with same acceleration is

To determine the maximum value of the force $F$ that can be applied to a system of blocks so that both blocks move together with the same acceleration, we need to consider the forces acting on each block and their masses. This scenario typically involves analyzing the forces, friction, and the relationship between mass and acceleration as per Newton’s laws of motion.

Understanding the System

Assuming we have two blocks, Block A and Block B, with masses $m_A$ and $m_B$, respectively, placed on a surface. Let’s say Block A is on top of Block B, which is on a horizontal surface. We also consider the friction between Block A and Block B, and between Block B and the surface. The goal is to ensure that both blocks move together, meaning they must have the same acceleration $a$.

Forces Acting on the Blocks

The force $F$ is applied to Block A. The friction between Block A and Block B must be enough to prevent Block A from slipping off Block B. The maximum static friction force $f_{\text{max}}$ that can act between them is given by:

• Friction Force: $f_{\text{max}}={\mu }_s\cdot m_A\cdot g$, where ${\mu }_s$ is the coefficient of static friction, $g$ is the acceleration due to gravity.

Applying Newton’s Second Law

According to Newton's second law, the net force acting on each block is equal to the mass of the block multiplied by its acceleration. Thus, we have:

• For Block A: $F-f_A=m_A\cdot a$, where $f_A$ is the friction force acting on Block A.
• For Block B: $f_A-f_B=m_B\cdot a$, where $f_B$ is the friction force between Block B and the surface.

Combining the Equations

To ensure both blocks move together, we can express the friction forces in terms of the applied force $F$. First, substituting $f_A$ from the equation of Block A into the equation for Block B, we get:

Substituting $f_A$ gives us:

• From Block A: $f_A=F-m_A\cdot a$
• Substituting into Block B's equation yields: $F-m_A\cdot a-f_B=m_B\cdot a$

Finding the Maximum Force $F$

Now we can express $F$ in terms of the maximum static friction force:

Setting $f_B={\mu }_k\cdot m_B\cdot g$ (where ${\mu }_k$ is the coefficient of kinetic friction if Block B is on the verge of slipping), we arrive at:

• Combining everything leads to $F=(m_A+m_B)\cdot a+f_B$.

To find the maximum $F$ while ensuring both blocks accelerate together, we need to solve for $a$ based on the maximum static friction between Block A and Block B.

Conclusion

In essence, to find the maximum value of $F$, you need to ensure that the static friction is enough to hold Block A on Block B without slipping, while also considering the friction between Block B and the surface it rests on. By applying Newton's laws and analyzing the forces at play, you can derive the conditions necessary for both blocks to move together with the same acceleration.