Für den Radius \(R\) und die Dichte \(\rho\) einer homogenen Kugel wurden die Werte \(\bar{R}=12,2 \mathrm{~cm}\) und \(\bar{\rho}=2,50 \mathrm{~g} / \mathrm{cm}^{3}\) ermittelt. Eine Schätzung der zugehörigen MeBunsicherheiten ergab \(\Delta R=0,15 \mathrm{~cm}\) und \(\Delta \rho=0,11 \mathrm{~g} / \mathrm{cm}^{3} .\) Welche Masse \(m\) besitzt die Kugel, mit welchem Maximalwert für die Me\betaunsicherheit von \(m\) muB man dabei rechnen?

In physical measurements, taking the uncertainty of the measured values into account is crucial. For the radius measurement of the sphere, we are given an average radius \( \bar{R} = 12.2 \text{ cm} \) and a corresponding uncertainty \( \Delta R = 0.15 \text{ cm} \). To understand how inaccurate our radius measurement might be, we calculate the relative uncertainty of the radius. This is done by dividing the uncertainty by the average value of the radius: \ \ \[ \text{Relative uncertainty of radius} = \frac{\Delta R}{\bar{R}} = \frac{0.15 \text{ cm}}{12.2 \text{ cm}} \approx 0.0123 \]
This means our radius measurement could be off by about 1.23%. Understanding this is important as the precision of the radius directly impacts other calculations, such as the volume of the sphere.

When measuring physical properties like density, similar precautions apply. The given average density for the sphere is \( \bar{\rho} = 2.50 \text{ g/cm}^3 \) with an associated uncertainty of \( \Delta \rho = 0.11 \text{ g/cm}^3 \). The relative uncertainty for the density can be calculated by dividing the uncertainty by the average density: \ \ \[ \text{Relative uncertainty of density} = \frac{\Delta \rho}{\bar{\rho}} = \frac{0.11 \text{ g/cm}^3}{2.50 \text{ g/cm}^3} \approx 0.044 \]
This means our density measurement has an uncertainty of about 4.4%. Keeping track of this helps us understand the potential error introduced when we use this density in further calculations, such as finding the mass of the sphere.

The mass of the sphere is calculated using its volume and density. First, we calculate the volume using the formula for the volume of a sphere: \ \ \[ V = \frac{4}{3} \pi \bar{R}^3 = \frac{4}{3} \pi (12.2 \text{ cm})^3 \ \]
Next, we use this volume to find the mass by multiplying with the density:
\[ m = V \cdot \bar{\rho} \]
Imagine the volume comes out to be a certain value, let's call it \( V \). Then the mass calculation will be: \ \ \[ m = V \cdot 2.50 \text{ g/cm}^3 \]
Thus, understanding both the measurement of volume and how to use it with the density is essential for calculating the mass accurately.

To find the overall uncertainty in the calculated mass, we need to consider the relative uncertainties of both the radius and the density. The formula for calculating the total relative uncertainty in the mass includes the individual relative uncertainties. For volume, because it depends on the cube of the radius, its relative uncertainty is three times the relative uncertainty of the radius:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{\Delta R}{\bar{R}} \approx 3 \cdot 0.0123 \approx 0.037 \]
Adding the relative uncertainties for volume and density gives us the total relative uncertainty for the mass:
\[ \frac{\Delta m}{m} = \frac{\Delta V}{V} + \frac{\Delta \rho}{\bar{\rho}} = 0.037 + 0.044 = 0.081 \]
Now, to find the maximum mass uncertainty, we multiply the total relative uncertainty by the calculated mass:
\[ \Delta m = \frac{\Delta m}{m} \cdot m \]
This calculation will provide the range within which the true mass may lie, reflecting the combined uncertainties that impacted our measurements.