A barometer kept in an elevator reads 76 cm when it is at rest. If the elevator goes up with increasing speed, the reading will be

Options

A. zero
B. 76cm
C. `lt`76cm
D. `gt`76cm

Answer

To solve the problem, we need to analyze the situation of a barometer in an elevator that is accelerating upwards. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Initial Condition
When the elevator is at rest, the barometer reads 76 cm of mercury. This reading corresponds to the atmospheric pressure at that location.

### Step 2: Identify the Change in Condition
When the elevator accelerates upwards, it experiences an effective gravitational acceleration that is different from when it is at rest. The gravitational acceleration \( g \) acts downwards, while the elevator's acceleration \( a \) acts upwards.

### Step 3: Calculate the Effective Acceleration
The effective gravitational acceleration \( g_e \) experienced by an observer inside the accelerating elevator can be expressed as:
\[ g_e = g + a \]
where \( g \) is the gravitational acceleration and \( a \) is the upward acceleration of the elevator.

### Step 4: Relate Barometer Reading to Effective Acceleration
The height of the mercury column in the barometer is related to the atmospheric pressure \( P \) and the effective acceleration \( g_e \) by the formula:
\[ P = \rho g_e h \]
where \( \rho \) is the density of mercury and \( h \) is the height of the mercury column.

### Step 5: Analyze the Relationship
Since the atmospheric pressure \( P \) is constant, we can express the height of the mercury column when the elevator is accelerating as:
\[ h' = \frac{P}{\rho g_e} \]
When the elevator is at rest, the height \( h \) is given by:
\[ h = \frac{P}{\rho g} \]
Thus, we can write the ratio of the heights:
\[ \frac{h'}{h} = \frac{g}{g_e} \]
Substituting \( g_e = g + a \):
\[ \frac{h'}{h} = \frac{g}{g + a} \]

### Step 6: Determine the Effect on Height
Since \( g + a > g \), it follows that:
\[ \frac{g}{g + a} < 1 \]
This implies:
\[ h' < h \]
Given that \( h = 76 \) cm, we conclude that:
\[ h' < 76 \text{ cm} \]

### Conclusion
Thus, when the elevator goes up with increasing speed, the reading of the barometer will be less than 76 cm.

### Final Answer
The reading will be **less than 76 cm** (Option 3).

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