A horizontal force P is applied on the 60 kg block resting over a 100 kg block which is resting on atable. The static coefficient of frictions is 0.4 between the blocks and 0.12 between the 100 kg block and the table. Determine the magnitude of P required to initiate impending motion.

To determine the magnitude of the horizontal force \( P \) required to initiate impending motion of the 60 kg block resting on the 100 kg block, we need to analyze the forces acting on both blocks and consider the frictional forces involved. Let's break this down step by step.

Understanding the Forces at Play

We have two blocks: the top block (60 kg) and the bottom block (100 kg). The static friction between the blocks and between the bottom block and the table will play a crucial role in determining the force required to move the top block.

Step 1: Calculate the Normal Forces

The normal force acting on each block is equal to its weight. The weight can be calculated using the formula:

• Weight = mass × gravitational acceleration (g = 9.81 m/s²)

For the 60 kg block:

• Normal Force (N₁) = 60 kg × 9.81 m/s² = 588.6 N

For the 100 kg block:

• Normal Force (N₂) = 100 kg × 9.81 m/s² = 981 N

Step 2: Calculate the Maximum Static Friction

The maximum static friction force can be calculated using the formula:

• Friction = coefficient of friction × Normal Force

For the friction between the two blocks (coefficient of static friction = 0.4):

• Friction between blocks (F₁) = 0.4 × N₁ = 0.4 × 588.6 N = 235.44 N

For the friction between the 100 kg block and the table (coefficient of static friction = 0.12):

• Friction with table (F₂) = 0.12 × N₂ = 0.12 × 981 N = 117.72 N

Step 3: Determine the Total Frictional Force

To initiate motion, the applied force \( P \) must overcome the total frictional force acting on the system. The total frictional force is the sum of the frictional forces acting on both blocks:

• Total Friction = F₁ + F₂ = 235.44 N + 117.72 N = 353.16 N

Step 4: Calculate the Required Force \( P \)

Since the force \( P \) must overcome the total frictional force to initiate motion, we find that:

• Magnitude of \( P \) = Total Friction = 353.16 N

Final Thoughts

Therefore, the magnitude of the horizontal force \( P \) required to initiate impending motion of the 60 kg block over the 100 kg block is approximately 353.16 N. This analysis illustrates how frictional forces between surfaces can significantly impact the forces needed to initiate motion in a system. Understanding these principles is crucial in physics, especially when dealing with real-world applications involving motion and forces.