A man of mass `85kg` stands on a lift of mass `30kg` When he pulls on the rope he exerts a force of `400N` on the floor of the lift Calculate acceleration of the lift Given `g =10 m//s^(2)` .

Explanation

To solve the problem, we need to analyze the forces acting on the lift and the man. Here are the steps to calculate the acceleration of the lift:

### Step 1: Identify the forces acting on the system
The forces acting on the man and the lift are:
1. The gravitational force acting on the man (weight) = \( W_m = m_m \cdot g = 85 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 850 \, \text{N} \)
2. The gravitational force acting on the lift (weight) = \( W_l = m_l \cdot g = 30 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 300 \, \text{N} \)
3. The force exerted by the man on the floor of the lift = \( F = 400 \, \text{N} \)

### Step 2: Calculate the total weight of the system
The total weight of the system (man + lift) is given by:
\[ W_{total} = W_m + W_l = 850 \, \text{N} + 300 \, \text{N} = 1150 \, \text{N} \]

### Step 3: Determine the net force acting on the lift
When the man pulls on the rope, he exerts a force of \( 400 \, \text{N} \) downward on the lift. The net force acting on the lift can be calculated as:
\[ F_{net} = F - W_{total} = 400 \, \text{N} - 1150 \, \text{N} = -750 \, \text{N} \]
(Note: The negative sign indicates that the net force is directed downward.)

### Step 4: Calculate the total mass of the system
The total mass of the system (man + lift) is:
\[ m_{total} = m_m + m_l = 85 \, \text{kg} + 30 \, \text{kg} = 115 \, \text{kg} \]

### Step 5: Apply Newton's second law to find acceleration
According to Newton's second law, the net force is equal to the mass times acceleration:
\[ F_{net} = m_{total} \cdot a \]
Substituting the values we have:
\[ -750 \, \text{N} = 115 \, \text{kg} \cdot a \]

### Step 6: Solve for acceleration
Rearranging the equation to solve for \( a \):
\[ a = \frac{-750 \, \text{N}}{115 \, \text{kg}} \approx -6.52 \, \text{m/s}^2 \]

### Conclusion
The acceleration of the lift is approximately \( -6.52 \, \text{m/s}^2 \). The negative sign indicates that the lift is accelerating downward.

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