To solve this problem, we need to analyze the forces acting on both blocks and how they interact with each other. We have block A with mass m resting on plank B, which has a mass of 3m. The coefficient of friction between the two is 0.2, and a horizontal force of magnitude 2mg is applied to plank B. Let's break this down step by step.
Understanding the Forces Involved
First, let's identify the forces acting on both blocks:
- Force on Plank B: A horizontal force of 2mg is applied to plank B.
- Frictional Force: The frictional force between block A and plank B will act to the right on block A, trying to accelerate it along with plank B.
Calculating the Maximum Frictional Force
The maximum static frictional force that can act on block A is given by:
F_friction = μ * N
Where:
- μ = coefficient of friction = 0.2
- N = normal force = weight of block A = mg
Thus, the maximum frictional force is:
F_friction = 0.2 * mg = 0.2mg
Net Force on Plank B
When the force of 2mg is applied to plank B, we need to consider the total mass that is being accelerated. The total mass is the mass of plank B plus the mass of block A:
Total mass = 3m + m = 4m
The net force acting on plank B, considering the frictional force acting on block A, is:
Net Force = Applied Force - Frictional Force = 2mg - 0.2mg = 1.8mg
Finding the Acceleration of Plank B
Using Newton's second law (F = ma), we can find the acceleration of plank B:
a_B = Net Force / Total Mass = 1.8mg / 4m = 0.45g
Acceleration of Block A
Now, let's determine the acceleration of block A relative to plank B. Since the maximum frictional force (0.2mg) is less than the force needed to keep block A moving with plank B, block A will not accelerate with plank B. Instead, it will experience a relative acceleration due to the frictional force.
Acceleration of Block A Relative to the Ground
Block A will accelerate due to the frictional force:
F_friction = m * a_A
Substituting the frictional force:
0.2mg = m * a_A
Solving for a_A gives:
a_A = 0.2g
Relative Acceleration of Block A to Plank B
The relative acceleration of block A with respect to plank B is the difference between their accelerations:
a_A relative to B = a_A - a_B
Substituting the values we found:
a_A relative to B = 0.2g - 0.45g = -0.25g
This negative sign indicates that block A is accelerating to the left relative to plank B.
Summary of Results
In summary, the accelerations are:
- Acceleration of block A relative to plank B: -0.25g (to the left)
- Acceleration of block A relative to the ground: 0.2g (to the right)
This analysis shows how the forces interact and how the frictional force plays a crucial role in determining the motion of the blocks on the frictionless surface. If you have any further questions or need clarification on any part, feel free to ask!
