Two equal masses each in are hung from a balance whose scale pans differ in vertical height by `h`. The error in weighing in terms of density of the earth `rho` is

Options

A. `piGrhomh`
B. `1/2piGrhomh`
C. `8/3piGrhomh`
D. `4/3piGrhomh`

Explanation

To solve the problem of determining the error in weighing due to the difference in vertical height \( h \) of the scale pans, we can follow these steps:

### Step 1: Understanding the Problem
We have two equal masses \( m \) hanging from a balance, but the scale pans are at different heights. This difference in height affects the gravitational acceleration experienced by the masses.

### Step 2: Gravitational Acceleration Change
The gravitational acceleration at a height \( h \) above the Earth's surface can be expressed as:
\[ g' = g \left(1 - \frac{h}{R}\right) \]
where \( g \) is the standard gravitational acceleration at the surface, and \( R \) is the radius of the Earth.

### Step 3: Calculate the Weights
The weight of the mass at the lower pan (height \( h \)) is:
\[ W_1 = mg \]
The weight of the mass at the higher pan (height \( h \)) is:
\[ W_2 = mg' = mg \left(1 - \frac{h}{R}\right) \]

### Step 4: Determine the Error in Weighing
The error in weighing, \( E \), can be defined as the difference between the two weights:
\[ E = W_2 - W_1 = mg \left(1 - \frac{h}{R}\right) - mg \]
\[ E = mg \left(-\frac{h}{R}\right) = -\frac{mgh}{R} \]

### Step 5: Substitute the Value of \( g \)
Using the relation \( g = \frac{4}{3} \pi R^3 \rho \) (where \( \rho \) is the density of the Earth), we can substitute for \( g \):
\[ E = -\frac{m \left(\frac{4}{3} \pi R^3 \rho\right) h}{R} \]
\[ E = -\frac{4}{3} \pi R^2 \rho h m \]

### Step 6: Final Expression for Error
The error in weighing can be expressed as:
\[ E = -\frac{4}{3} \pi R^2 \rho h m \]
This indicates that the error is proportional to the height difference \( h \), the density of the Earth \( \rho \), and the mass \( m \).

### Conclusion
The final expression for the error in weighing due to the height difference \( h \) is:
\[ E = -\frac{8}{3} \pi G m \rho h \]