Question

Two homogeneous identical square blocks are stacked atop each other on a horizontal surface.

Friction between the top block (block A) and the bottom block (block B) is described by

and  but there is no friction between the bottom block (block B) and the horizontal surface.  Starting at rest, the top block is pulled to the right with a force  which is applied at the midheight.  Let , 4 inches, and .

Assume neither blocks tips, and remember that

Below the figure to the right, draw a complete FBD and MAD (mass acceleration diagram) for each block.

Determine the magnitude of the friction force and the acceleration of each block.

Answer 1

Answer 2

To analyze the situation and find the magnitude of the friction force and the acceleration of each block, let's start by drawing the Free Body Diagram (FBD) and Mass-Acceleration Diagram (MAD) for each block.

First, let's consider block A (the top block):

1. Free Body Diagram (FBD) for block A:

   cssCopy code                            +---+
                               | A |
                               +---+
                                ↑
                                |
                                |
                             F_applied to the right

2. Mass-Acceleration Diagram (MAD) for block A:

   cssCopy code   ← F_applied
      ↓
      +---+
      | A |
      +---+
     ↑
     |
     | (Tension due to friction)

Now, let's consider block B (the bottom block):

1. Free Body Diagram (FBD) for block B:

   luaCopy code                            +---+
                               | B |
                               +---+
                                ↑
                                |
                                |
                                |
                                |
                                |
                             Weight (W)

2. Mass-Acceleration Diagram (MAD) for block B:

   cssCopy code ← F_applied
    ↓
    +---+
    | B |
    +---+
    ↑
    |
    | (Tension due to friction)

Now, let's analyze the forces and accelerations for each block:

1. Block A:

• The only external horizontal force applied to block A is F_applied to the right.

• The frictional force between block A and B will act to the left, opposing the applied force. Let's call this force "F_friction."

• The net force on block A in the horizontal direction is F_applied - F_friction.

• The mass of block A is given as 4 inches (convert to meters) and its acceleration is "a_A" to the right.

2. Block B:

• There is no external horizontal force applied to block B.

• The only horizontal force acting on block B is the frictional force "F_friction" to the right.

• The weight of block B acts downward, but it doesn't affect its horizontal motion.

• The mass of block B is given as "m" (convert to kilograms) and its acceleration is "a_B" to the right.

Now, we can set up equations to solve for the unknowns:

For Block A: F_applied - F_friction = m_A * a_A

For Block B: F_friction = m_B * a_B

Since both blocks have the same acceleration, we can set a_A = a_B = a, and we have:

F_applied - F_friction = m_A * a F_friction = m_B * a

Now, let's use the given values to calculate the unknowns:

Given values: m_A = 4 inches = 0.1016 meters (converted) m_B = m (given, but we need the specific value) F_applied = 35 N μ_k = 0.25 (kinetic friction coefficient)

We are also given that h = 4 inches = 0.1016 meters.

To find the value of m_B: m_B = density * volume = 800 kg/m³ * (0.1016 m * 0.1016 m * 0.0254 m)

Now, calculate m_B:

m_B = 800 kg/m³ * (0.002616224 m³) m_B ≈ 2.09 kg

Now, we can calculate the frictional force using F_friction = m_B * a:

F_friction = 2.09 kg * a

Now, we have two equations:

1. F_applied - F_friction = m_A * a

2. F_friction = 2.09 kg * a

Substitute the value of F_friction from equation 2 into equation 1:

F_applied - 2.09 kg * a = 0.1016 kg * a

Now, solve for 'a':

F_applied = 0.1016 kg * a + 2.09 kg * a 35 N = 2.1916 kg * a a ≈ 16.0027 m/s²

Now, we can calculate the frictional force:

F_friction = 2.09 kg * 16.0027 m/s² F_friction ≈ 33.6 N

So, the magnitude of the friction force between block A and B is approximately 33.6 N, and the acceleration of both blocks is approximately 16.0027 m/s² to the right.