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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

Abhishek Dogra , 6 Years ago
Grade 11
anser 1 Answers
Eshan

Last Activity: 6 Years ago

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 mA and mB, 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 fmax that can act between them is given by:

  • Friction Force: fmax=μsmAg, where μs 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: FfA=mAa, where fA is the friction force acting on Block A.
  • For Block B: fAfB=mBa, where fB 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 fA from the equation of Block A into the equation for Block B, we get:

Substituting fA gives us:

  • From Block A: fA=FmAa
  • Substituting into Block B's equation yields: FmAafB=mBa

Finding the Maximum Force F

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

Setting fB=μkmBg (where μk is the coefficient of kinetic friction if Block B is on the verge of slipping), we arrive at:

  • Combining everything leads to F=(mA+mB)a+fB.

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.

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