When an empty lift is moving down with an acceleration of `g/4 ms^(-2)` the tension in the cable is 9000N. When the lift is moving up with an acceleration of `g/3 ms^(-2)` the tension in the cable is

Options

A. 16,000 N
B. 18,000 N
C. 12,000 N
D. 15,000 N

Answer

To solve the problem step by step, we will analyze the two scenarios given: when the lift is moving down and when it is moving up.

### Step 1: Analyze the first scenario (Lift moving down)
When the lift is moving down with an acceleration of \( \frac{g}{4} \, \text{m/s}^2 \):
- Let the mass of the lift be \( m \).
- The weight of the lift is \( mg \).
- The tension in the cable is \( T \).
- According to Newton's second law, we can write the equation of motion:

\[ mg - T = m \left(\frac{g}{4}\right) \]

### Step 2: Rearranging the equation
Rearranging the equation from Step 1 gives us:

\[ T = mg - m \left(\frac{g}{4}\right) \]

Factoring out \( m \):

\[ T = m \left(g - \frac{g}{4}\right) = m \left(\frac{4g}{4} - \frac{g}{4}\right) = m \left(\frac{3g}{4}\right) \]

### Step 3: Substitute the tension value
We know from the problem that \( T = 9000 \, \text{N} \):

\[ 9000 = \frac{3mg}{4} \]

### Step 4: Solve for \( mg \)
To find \( mg \):

\[ 3mg = 9000 \times 4 \]
\[ 3mg = 36000 \]
\[ mg = \frac{36000}{3} = 12000 \, \text{N} \]

### Step 5: Analyze the second scenario (Lift moving up)
Now, when the lift is moving up with an acceleration of \( \frac{g}{3} \, \text{m/s}^2 \):
- The equation of motion now becomes:

\[ T - mg = m \left(\frac{g}{3}\right) \]

### Step 6: Rearranging the equation for upward motion
Rearranging gives:

\[ T = mg + m \left(\frac{g}{3}\right) \]

Factoring out \( m \):

\[ T = m \left(1 + \frac{1}{3}\right)g = m \left(\frac{4g}{3}\right) \]

### Step 7: Substitute \( mg \) value
Now substitute \( mg = 12000 \, \text{N} \):

\[ T = \frac{4}{3} \times 12000 \]
\[ T = 16000 \, \text{N} \]

### Final Answer
The tension in the cable when the lift is moving up with an acceleration of \( \frac{g}{3} \, \text{m/s}^2 \) is **16000 N**.

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