The total mass of an elevator with a 80 kg man in it is 1000 kg. This elevator moving upward with a speed of 8 `m//sec,` is brought to rest over a distance of 16m. The tension T in the cables supporting the elevator and the force exerted on the man by the elevator floor will respectively be-

Options

A. 7800N, 624N
B. 624N, 7800N
C. 78 N, 624 N
D. 624N , 78 N

Answer

To solve the problem, we need to determine two quantities: the tension \( T \) in the cables supporting the elevator and the force \( N \) exerted on the man by the elevator floor.

### Step 1: Identify the given data
- Mass of the man \( m = 80 \, \text{kg} \)
- Total mass of the elevator and man \( M = 1000 \, \text{kg} \)
- Initial speed \( u = 8 \, \text{m/s} \)
- Final speed \( v = 0 \, \text{m/s} \)
- Distance \( s = 16 \, \text{m} \)

### Step 2: Calculate the acceleration of the elevator
Using the third equation of motion:
\[ v^2 = u^2 + 2as \]
Substituting the known values:
\[ 0 = (8)^2 + 2a(16) \]
\[ 0 = 64 + 32a \]
Rearranging gives:
\[ 32a = -64 \implies a = -2 \, \text{m/s}^2 \]
The negative sign indicates that the elevator is decelerating.

### Step 3: Calculate the tension \( T \) in the cables
Using the free body diagram of the elevator, we have:
\[ T - Mg = Ma \]
Where:
- \( M = 1000 \, \text{kg} \) (total mass)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( a = -2 \, \text{m/s}^2 \) (deceleration)

Substituting the values into the equation:
\[ T - 1000 \times 9.8 = 1000 \times (-2) \]
\[ T - 9800 = -2000 \]
\[ T = 9800 - 2000 = 7800 \, \text{N} \]

### Step 4: Calculate the normal force \( N \) exerted on the man
For the man standing in the elevator, the forces acting on him are:
- Weight \( mg \) acting downward
- Normal force \( N \) acting upward
- Pseudo force due to acceleration of the elevator acting downward, which is \( ma \)

Setting up the equation:
\[ N - mg - ma = 0 \]
Rearranging gives:
\[ N = mg + ma \]
Substituting the values:
\[ N = (80 \times 9.8) + (80 \times -2) \]
Calculating each term:
\[ N = 784 - 160 = 624 \, \text{N} \]

### Final Answers
- Tension \( T = 7800 \, \text{N} \)
- Normal force \( N = 624 \, \text{N} \)