A quarter (diameter =2.4 cm ) is held at arm's length (70.0...

A quarter (diameter $=2.4 \mathrm{~cm}$ ) is held at arm's length $(70.0 \mathrm{~cm})$. The sun has a diameter of $1.39 \times 10^{9} \mathrm{~m}$ and is $1.50 \times 10^{11} \mathrm{~m}$ from the earth. What is the ratio of the angular size of the quarter to that of the sun?

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Key Concepts

Angular Size

Angular size refers to the angle an object appears to span in the field of view of the observer. It depends on both the actual physical size of the object and its distance from the observer. When the distance increases, the angular size decreases even if the actual size remains the same.

Small Angle Approximation

The small angle approximation is used when dealing with objects that subtend small angles (typically measured in radians). Under this approximation, the angular size can be approximated as the ratio of the object’s diameter to its distance. This simplification makes it easier to compare angular sizes without complex trigonometry.

Ratio of Angular Sizes

When comparing the apparent sizes of two different objects, the ratio of their angular sizes can be determined by taking the ratio of their diameters divided by the ratio of their distances. This concept is crucial to problems that ask for the relative angular extent of objects as seen from a particular point, by making use of proportional relationships.

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