Calculating the Weight of a Scaled Angle from Theodolite Measurement

In surveying and measurements, the 'weight' assigned to an observation indicates its relative reliability or precision. A higher weight means a more reliable observation. Weight is inversely proportional to the variance (square of the standard deviation) of the observation.

The relationship between weight (\(w\)) and variance (\(\sigma^2\)) for an observation is given by:

\(w \propto \frac{1}{\sigma^2}\)

This can be written as \(w = \frac{k}{\sigma^2}\), where \(k\) is a constant related to the standard variance of unit weight.

Propagation of Weights for Scaled Observations

Consider an observation \(x\) with weight \(w_x\). If we define a new quantity \(y\) as a linear function of \(x\), say \(y = cx\), where \(c\) is a constant, the variance of \(y\) (\(\sigma_y^2\)) is related to the variance of \(x\) (\(\sigma_x^2\)) by:

\(\sigma_y^2 = c^2 \sigma_x^2\)

Now, let's find the weight of \(y\), denoted as \(w_y\). Using the inverse relationship between weight and variance:

\(w_y = \frac{k}{\sigma_y^2} = \frac{k}{c^2 \sigma_x^2}\)

Since \(w_x = \frac{k}{\sigma_x^2}\), we can substitute this into the equation for \(w_y\):

\(w_y = \frac{1}{c^2} \left(\frac{k}{\sigma_x^2}\right) = \frac{1}{c^2} w_x\)

So, if an observation is multiplied by a constant \(c\), its weight is multiplied by \(\frac{1}{c^2}\).

Applying the Concept to the Theodolite Angle

We are given an angle measured with a theodolite, denoted as \(\alpha\), and its weight is \(w_\alpha = 2\). We need to find the weight of \(\frac{\alpha}{4}\).

Let \(y = \frac{\alpha}{4}\). In this case, \(y\) is a scaled version of \(\alpha\) with the scaling constant \(c = \frac{1}{4}\).

Using the propagation of weights formula \(w_y = \frac{1}{c^2} w_x\), where \(y = \frac{\alpha}{4}\), \(x = \alpha\), and \(c = \frac{1}{4}\):

\(w_{\frac{\alpha}{4}} = \frac{1}{(1/4)^2} w_\alpha\)

Calculate the term \(\frac{1}{(1/4)^2}\):

\(\frac{1}{(1/4)^2} = \frac{1}{1/16} = 1 \times \frac{16}{1} = 16\)

Now substitute this back into the weight equation:

\(w_{\frac{\alpha}{4}} = 16 \times w_\alpha\)

We are given \(w_\alpha = 2\). Substitute this value:

\(w_{\frac{\alpha}{4}} = 16 \times 2 = 32\)

The weight of \(\frac{\alpha}{4}\) is 32.

Evaluating the Options

Let's compare our result (32) with the given options:

1. \( \frac{2}{4} = 0.5 \)
2. \( \frac{4}{2} = 2 \)
3. \( 2 \times 4^2 = 2 \times 16 = 32 \)
4. \( 2 \times 4 = 8 \)

Option 3 gives the value 32, which matches our calculated weight for \(\frac{\alpha}{4}\).

Therefore, the weight of \(\frac{\alpha}{4}\) is \(2 \times 4^2\).

The final answer is \(\mathbf{2 \times 4^2}\).

Revision Table: Surveying Measurement Weights

Additional Information: Weights in Surveying and Theodolites

Weights are fundamental in combining multiple measurements, especially in techniques like Least Squares Adjustment, which is commonly used with theodolite data. Different observations might have different weights based on factors like:

• The instrument used (e.g., precision of a theodolite).
• Observation conditions (e.g., visibility, atmospheric refraction).
• Number of repetitions of the measurement.
• Geometric strength of the network (though this relates more to propagated error/variance).

When measurements are combined or transformed, understanding how their weights propagate is crucial for correctly assessing the precision of the final results. The principle \(w \propto \frac{1}{\sigma^2}\) is key to this propagation.

The question highlights a simple case of scaling an angle. In more complex scenarios, involving sums or other functions of observations, different error propagation rules apply, leading to corresponding rules for weight propagation.