In Case A the mass of each block is 4.0 kg. In Case B the mass of block 1 (the block behind) is 8.0 kg, and the mass of block 2 is 4.0 kg. No frictional force acts on block 1 in either Case A or Case B. However, a kinetic frictional force of 7.8 N does act on block 2 in both cases and opposes the motion. For both Case A and Case B determine (a) the magnitude of the forces with which the blocks push against each other and (b) the magnitude of the acceleration of the blocks
To determine the magnitude of the forces with which the blocks push against each other in Case A and Case B, and the magnitude of the acceleration of the blocks, let's break down the problem step by step.
First, let's consider Case A:
Given:
Mass of each block (m₁ and m₂) = 4.0 kg
Frictional force acting on block 2 (F_friction) = 7.8 N
(a) The magnitude of the forces with which the blocks push against each other:
Since there is no friction acting on block 1, the force exerted by block 1 on block 2 is the only force pushing them together.
Using Newton's third law, we know that the force exerted by block 1 on block 2 is equal in magnitude but opposite in direction to the force exerted by block 2 on block 1. Therefore, the magnitude of the force with which the blocks push against each other in Case A is equal to the force exerted by block 1 on block 2.
(b) The magnitude of the acceleration of the blocks:
To determine the magnitude of the acceleration, we need to consider the net force acting on the blocks.
The net force acting on block 2 is given by:
Net force = Force pushing blocks together - Frictional force
Since the force pushing the blocks together is equal to the force exerted by block 1 on block 2, we can calculate the net force as follows:
Net force = Force exerted by block 1 on block 2 - Frictional force
Using Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration, we can rewrite the equation as:
m₂ * acceleration = Force exerted by block 1 on block 2 - Frictional force
Now, let's calculate the force exerted by block 1 on block 2 using Newton's second law:
Force exerted by block 1 on block 2 = m₁ * acceleration
Substituting this back into the equation for the net force, we have:
m₂ * acceleration = m₁ * acceleration - Frictional force
We can rearrange the equation to solve for the acceleration:
acceleration = Frictional force / (m₁ + m₂)
Now that we know the acceleration, we can calculate the requested values.
Now, let's consider Case B:
Given:
Mass of block 1 (m₁) = 8.0 kg
Mass of block 2 (m₂) = 4.0 kg
Frictional force acting on block 2 (F_friction) = 7.8 N
(a) The magnitude of the forces with which the blocks push against each other:
Similar to Case A, since there is no friction acting on block 1, the force exerted by block 1 on block 2 is the force pushing them together.
(b) The magnitude of the acceleration of the blocks:
Using the same approach as in Case A, we can calculate the acceleration using the equation:
acceleration = Frictional force / (m₁ + m₂)
Substitute the given values into the equation to find the acceleration for Case B.
So, by following these steps, you can determine the magnitude of the forces and the acceleration for both Case A and Case B.