Consider the equations
`P=Lim_(/_\srarr0)F/(/_\S) and P_1-P_2=rhogz`
In an elevator acceleratin upward

Options

A. both the equations are valid
B. the first is valid but not the second
C. the second is valid but not the first
D. both are invalid

Answer

To solve the problem, we need to analyze the two equations given in the context of an elevator that is accelerating upward.

### Step 1: Understand the first equation
The first equation is given as:
\[ P = \lim_{\Delta S \to 0} \frac{F}{\Delta S} \]
This equation represents the definition of pressure \( P \) as the limit of force \( F \) applied over an area \( \Delta S \) as the area approaches zero. This definition is universally valid for pressure in any fluid, regardless of the conditions.

### Step 2: Analyze the second equation
The second equation is:
\[ P_1 - P_2 = \rho g z \]
This equation relates the pressure difference \( P_1 - P_2 \) between two points in a fluid to the density \( \rho \), gravitational acceleration \( g \), and the height difference \( z \) between those two points.

### Step 3: Consider the effect of upward acceleration
When the elevator is accelerating upward, the effective gravitational acceleration changes. The effective gravitational acceleration \( g' \) becomes:
\[ g' = g + a \]
where \( a \) is the upward acceleration of the elevator.

### Step 4: Modify the second equation for upward acceleration
Given that the effective gravitational acceleration is now \( g + a \), we can modify the second equation:
\[ P_1 - P_2 = \rho (g + a) z \]
This means that the pressure difference in the fluid must account for the additional acceleration due to the upward movement of the elevator.

### Step 5: Conclusion about the validity of the equations
1. The first equation \( P = \lim_{\Delta S \to 0} \frac{F}{\Delta S} \) is always valid.
2. The second equation \( P_1 - P_2 = \rho g z \) is not valid in the case of an upward accelerating elevator, as it does not account for the additional acceleration \( a \).

### Final Answer
Thus, the first equation is valid, while the second equation is not valid when the elevator is accelerating upward.

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