An elevator and its load weigh a total of 1600 kg. Find the tension T in the supporting cable when the elevator, originally moving downward at 20 m/s is brought to rest with constant acceleration in a distance of 50 m.

Options

A. `2024xx10^(2)N`
B. `1024xx10^(4)N`
C. `2024xx10^(4)N`
D. `2024xx10^(-4)N`

Answer

To solve the problem of finding the tension \( T \) in the supporting cable of an elevator that is brought to rest, we can follow these steps:

### Step 1: Identify the Given Data
- Total weight of the elevator and load, \( W = 1600 \, \text{kg} \)
- Initial velocity, \( u = 20 \, \text{m/s} \) (downward)
- Final velocity, \( v = 0 \, \text{m/s} \) (at rest)
- Displacement, \( s = 50 \, \text{m} \)

### Step 2: Use the Third Equation of Motion
We can use the third equation of motion to find the acceleration \( a \):
\[ v^2 = u^2 + 2as \]
Substituting the known values:
\[ 0 = (20)^2 + 2a(50) \]
\[ 0 = 400 + 100a \]
Rearranging gives:
\[ 100a = -400 \]
\[ a = -4 \, \text{m/s}^2 \]

### Step 3: Identify the Direction of Acceleration
Since the elevator is moving downward and is being brought to rest, the acceleration \( a \) is upward. Thus, the effective acceleration acting on the elevator is:
\[ a = 4 \, \text{m/s}^2 \, \text{(upward)} \]

### Step 4: Calculate the Weight of the Elevator
The weight \( W \) of the elevator is given by:
\[ W = mg = 1600 \times 9.8 = 15680 \, \text{N} \]

### Step 5: Apply Newton's Second Law
According to Newton's second law, the net force acting on the elevator can be expressed as:
\[ T - W = ma \]
Where:
- \( T \) is the tension in the cable,
- \( W \) is the weight of the elevator,
- \( m \) is the mass of the elevator,
- \( a \) is the acceleration.

Substituting the values we have:
\[ T - 15680 = 1600 \times 4 \]
\[ T - 15680 = 6400 \]

### Step 6: Solve for Tension \( T \)
Rearranging the equation to find \( T \):
\[ T = 15680 + 6400 \]
\[ T = 22080 \, \text{N} \]

### Final Answer
The tension in the supporting cable when the elevator is brought to rest is:
\[ T = 22080 \, \text{N} \]

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