To find the acceleration of the upper block, we will follow these steps:
### Step 1: Identify the given values
- Mass of the lower block (m1) = 4 kg
- Mass of the upper block (m2) = 2 kg
- External force (F) = 18 N
- Coefficient of friction between the lower block and the ground (μ1) = 0.2
- Coefficient of friction between the lower block and the upper block (μ2) = 0.4
- Acceleration due to gravity (g) = 10 m/s²
### Step 2: Calculate the normal force on the lower block
The normal force (N) acting on the lower block due to the upper block is given by:
\[ N = m_2 \cdot g = 2 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 20 \, \text{N} \]
### Step 3: Calculate the frictional force acting on the lower block
The frictional force (f1) between the lower block and the ground is given by:
\[ f_1 = \mu_1 \cdot N_{\text{ground}} = \mu_1 \cdot (m_1 + m_2) \cdot g \]
\[ f_1 = 0.2 \cdot (4 \, \text{kg} + 2 \, \text{kg}) \cdot 10 \, \text{m/s}^2 = 0.2 \cdot 6 \cdot 10 = 12 \, \text{N} \]
### Step 4: Calculate the net force acting on the lower block
The net force (F_net) acting on the lower block is given by:
\[ F_{\text{net}} = F - f_1 = 18 \, \text{N} - 12 \, \text{N} = 6 \, \text{N} \]
### Step 5: Calculate the acceleration of the lower block
Using Newton's second law (F = ma), we can find the acceleration (a) of the lower block:
\[ a = \frac{F_{\text{net}}}{m_1 + m_2} = \frac{6 \, \text{N}}{4 \, \text{kg} + 2 \, \text{kg}} = \frac{6}{6} = 1 \, \text{m/s}^2 \]
### Step 6: Determine if the upper block will slip
Now we need to check if the upper block will slip or not. The maximum static friction force (f2) between the two blocks is given by:
\[ f_2 = \mu_2 \cdot N = 0.4 \cdot 20 \, \text{N} = 8 \, \text{N} \]
### Step 7: Compare forces to determine motion
The force required to accelerate the upper block (m2) at the same acceleration (a = 1 m/s²) is:
\[ F_{\text{required}} = m_2 \cdot a = 2 \, \text{kg} \cdot 1 \, \text{m/s}^2 = 2 \, \text{N} \]
Since the maximum static friction force (8 N) is greater than the force required (2 N), the upper block will not slip.
### Conclusion
Both blocks will move together with the same acceleration. Therefore, the acceleration of the upper block is:
\[ \text{Acceleration of the upper block} = 1 \, \text{m/s}^2 \]
