To solve the problem step by step, we need to analyze the forces acting on the body and apply Newton's laws of motion.
### Step 1: Identify the forces acting on the body
The forces acting on the body are:
- The applied force \( P \)
- The static friction force \( f_s \) (which is at its maximum when the body is just about to move)
- The weight of the body \( W = mg \)
Given:
- Mass \( m = 40 \, \text{kg} \)
- Coefficient of static friction \( \mu_s = 0.5 \)
- Coefficient of kinetic friction \( \mu_k = 0.4 \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)
### Step 2: Calculate the maximum static friction force
The maximum static friction force can be calculated using the formula:
\[
f_s = \mu_s \cdot N
\]
where \( N \) is the normal force. On a horizontal surface, the normal force \( N \) is equal to the weight of the body:
\[
N = mg = 40 \, \text{kg} \times 10 \, \text{m/s}^2 = 400 \, \text{N}
\]
Now, substituting the values:
\[
f_s = \mu_s \cdot mg = 0.5 \cdot 400 \, \text{N} = 200 \, \text{N}
\]
### Step 3: Determine the force \( P \) when the body is just about to move
When the body is just about to start moving, the applied force \( P \) is equal to the maximum static friction force:
\[
P = f_s = 200 \, \text{N}
\]
### Step 4: Calculate the kinetic friction force once the body starts moving
Once the body starts moving, the frictional force acting on it is the kinetic friction force \( f_k \):
\[
f_k = \mu_k \cdot N = \mu_k \cdot mg = 0.4 \cdot 400 \, \text{N} = 160 \, \text{N}
\]
### Step 5: Apply Newton's second law to find the acceleration
According to Newton's second law:
\[
F_{\text{net}} = P - f_k = ma
\]
Substituting the known values:
\[
200 \, \text{N} - 160 \, \text{N} = 40 \, \text{kg} \cdot a
\]
\[
40 \, \text{N} = 40 \, \text{kg} \cdot a
\]
Dividing both sides by 40 kg:
\[
a = 1 \, \text{m/s}^2
\]
### Final Answer
The acceleration of the body is \( 1 \, \text{m/s}^2 \).
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