To solve the problem, we need to find the minimum tension in the string that allows the man to pull the block without moving himself. We will consider the forces acting on both the man and the block.
### Step 1: Calculate the maximum friction force acting on the man
The maximum friction force that can act on the man is given by:
\[ F_{\text{friction, man}} = \mu_{\text{man}} \cdot N_{\text{man}} \]
Where:
- \( \mu_{\text{man}} = 0.5 \) (coefficient of friction between the man and the surface)
- \( N_{\text{man}} = m_{\text{man}} \cdot g \) (normal force on the man)
Calculating \( N_{\text{man}} \):
\[ N_{\text{man}} = 60 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 600 \, \text{N} \]
Now, substituting this into the friction force equation:
\[ F_{\text{friction, man}} = 0.5 \cdot 600 \, \text{N} = 300 \, \text{N} \]
### Step 2: Calculate the maximum friction force acting on the block
The maximum friction force that can act on the block is given by:
\[ F_{\text{friction, block}} = \mu_{\text{block}} \cdot N_{\text{block}} \]
Where:
- \( \mu_{\text{block}} = 0.2 \) (coefficient of friction between the block and the surface)
- \( N_{\text{block}} = m_{\text{block}} \cdot g \) (normal force on the block)
Calculating \( N_{\text{block}} \):
\[ N_{\text{block}} = 100 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 1000 \, \text{N} \]
Now, substituting this into the friction force equation:
\[ F_{\text{friction, block}} = 0.2 \cdot 1000 \, \text{N} = 200 \, \text{N} \]
### Step 3: Determine the minimum tension in the string
For the man to remain at rest, the tension \( T \) in the string must be equal to the maximum friction force acting on the man. Therefore, we have:
\[ T = F_{\text{friction, man}} \]
Since the block must also remain at rest, the tension must not exceed the maximum friction force acting on the block. Thus, we have:
\[ T \leq F_{\text{friction, block}} \]
From our calculations:
- Maximum friction force on the man: \( 300 \, \text{N} \)
- Maximum friction force on the block: \( 200 \, \text{N} \)
The minimum tension required to pull the block while keeping the man at rest is determined by the limiting friction of the block:
\[ T = 200 \, \text{N} \]
### Conclusion
Thus, the minimum tension in the string required to pull the block so that the man remains at rest is approximately \( 200 \, \text{N} \).
