A man is standing on a weighing machine kept on the floor of an elevator. The ratio of measured weight of the man when elevator is at rest and when it is moving with downward acceleration a is 3 : 2. The value of a is :

Options

A. `(g)/(3)`
B. `(3g)/(2)`
C. g
D. `(2g)/(3)`

Explanation

To solve the problem step-by-step, we will analyze the situation of a man standing on a weighing machine in an elevator that is either at rest or moving downward with acceleration.

### Step 1: Understand the Forces Acting on the Man
When the elevator is at rest, the only force acting on the man is his weight, which is given by:
\[ W_1 = mg \]
where \( m \) is the mass of the man and \( g \) is the acceleration due to gravity.

### Step 2: Analyze the Situation When the Elevator is Moving Downward
When the elevator is moving downward with an acceleration \( a \), the effective weight of the man (the reading on the weighing machine) will be:
\[ W_2 = mg - ma \]
This is because the downward acceleration of the elevator reduces the normal force acting on the man.

### Step 3: Set Up the Ratio of Weights
According to the problem, the ratio of the measured weight of the man when the elevator is at rest to when it is moving downward is given as:
\[ \frac{W_1}{W_2} = \frac{3}{2} \]
Substituting the expressions for \( W_1 \) and \( W_2 \):
\[ \frac{mg}{mg - ma} = \frac{3}{2} \]

### Step 4: Cross Multiply to Solve for \( a \)
Cross-multiplying gives us:
\[ 2mg = 3(mg - ma) \]

### Step 5: Distribute and Rearrange the Equation
Expanding the right side:
\[ 2mg = 3mg - 3ma \]
Rearranging this equation to isolate terms involving \( a \):
\[ 2mg - 3mg = -3ma \]
\[ -mg = -3ma \]
Dividing both sides by \( -m \) (assuming \( m \neq 0 \)):
\[ g = 3a \]

### Step 6: Solve for \( a \)
Now, we can solve for \( a \):
\[ a = \frac{g}{3} \]

### Final Answer
Thus, the value of \( a \) is:
\[ a = \frac{g}{3} \]