A block of mass 100kg is kept on a horizontal rough surface. A man of mass 60kg is trying to pull the block with the help of ideal inextensible string. Find the minimum tension ( approximately ) in string to pull the block so that man remains at rest . Coefficient of friction between block and horizontal surface is 0.2 and that between man an horizontal surface is 0.5 ( g `= 10 m//s^(2)` )

Options

A. 100N
B. 166N
C. 180N
D. 196N

Answer

To solve the problem step by step, we need to analyze 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 (\( F_{f, man} \)) that can act on the man is given by the formula:
\[ F_{f, man} = \mu_{man} \times N_{man} \]
where:
- \( \mu_{man} = 0.5 \) (coefficient of friction between the man and the surface)
- \( N_{man} = m_{man} \times g = 60 \, \text{kg} \times 10 \, \text{m/s}^2 = 600 \, \text{N} \) (normal force acting on the man)

Now, substituting the values:
\[ F_{f, man} = 0.5 \times 600 \, \text{N} = 300 \, \text{N} \]

### Step 2: Calculate the maximum friction force acting on the block.
The maximum friction force (\( F_{f, block} \)) that can act on the block is given by:
\[ F_{f, block} = \mu_{block} \times N_{block} \]
where:
- \( \mu_{block} = 0.2 \) (coefficient of friction between the block and the surface)
- \( N_{block} = m_{block} \times g = 100 \, \text{kg} \times 10 \, \text{m/s}^2 = 1000 \, \text{N} \) (normal force acting on the block)

Now, substituting the values:
\[ F_{f, block} = 0.2 \times 1000 \, \text{N} = 200 \, \text{N} \]

### Step 3: Determine the minimum tension in the string.
For the man to remain at rest while pulling the block, the tension in the string (\( T \)) must not exceed the maximum friction force acting on the man. Therefore, we have:
\[ T \leq F_{f, man} = 300 \, \text{N} \]

However, to move the block, the tension must also overcome the friction acting on the block. Thus, the minimum tension required to start moving the block is equal to the maximum friction force acting on the block:
\[ T \geq F_{f, block} = 200 \, \text{N} \]

### Conclusion:
The minimum tension in the string to pull the block while ensuring the man remains at rest is:
\[ T = 200 \, \text{N} \]