Problem 77 - Physics

Question

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?

Step-by-Step Solution

Answer

The ratio of the angular size of the quarter to the sun is approximately 3.70.

Step 1: Determine Angular Size of the Quarter

The formula for angular size is given by \( \theta = \frac{d}{D} \), where \( d \) is the diameter of the object and \( D \) is the distance from the object. Using the given diameter and distance for the quarter, the angular size of the quarter is calculated as follows:\[ \theta_{quarter} = \frac{2.4 \mathrm{~cm}}{70.0 \mathrm{~cm}} = \frac{2.4}{70.0} \] \[ \theta_{quarter} \approx 0.0343 \text{ radians} \]

Step 2: Determine Angular Size of the Sun

Using the same formula \( \theta = \frac{d}{D} \), now calculate the angular size of the sun using its given diameter and distance from Earth:\[ \theta_{sun} = \frac{1.39 \times 10^9 \mathrm{~m}}{1.50 \times 10^{11} \mathrm{~m}} \] \[ \theta_{sun} \approx 0.00927 \text{ radians} \]

Step 3: Calculate the Ratio of Angular Sizes

To find the ratio of the angular size of the quarter to that of the sun, divide the angular size of the quarter by the angular size of the sun:\[ \text{Ratio} = \frac{\theta_{quarter}}{\theta_{sun}} = \frac{0.0343}{0.00927} \] \[ \text{Ratio} \approx 3.70 \]

Key Concepts

Diameter

When we talk about the diameter of an object, we're referring to a straight line that passes from one side of the circle to the other, through the center. This is a crucial measurement as it directly affects the perceived size of the object when viewed from a distance.
The diameter of the quarter in this exercise is 2.4 cm. Likewise, the diameter of the sun is immense, measuring at 1.39 x 10^9 meters. These diameters are essential to compute the angular size of the objects.
Angular size gives us insight into how large an object appears from a particular viewpoint. So, next time you hear the term "diameter," remember it helps us understand how bulky or slender an object is from side to side!

Distance

Distance is the space between two points, which in our case, refers to how far the observer is from the object being observed.
In this exercise, the quarter is held 70 cm from our eye, while the sun is a staggering 1.50 x 10^11 meters away. The distance from the observer determines the apparent size of the object, which is calculated in the form of angular size.
When comparing distances, notice how larger distances dramatically reduce the angular size despite the object's real size. A remarkable demonstration of this is how the enormous sun can appear smaller than a quarter in our sky due to its vast distance from the Earth.

Ratio of Angular Sizes

The ratio of angular sizes is a comparative measure of how two objects appear relative to each other.
In this scenario, we compute the ratio between the quarter and the sun's angular sizes. Calculating this helps us understand how they stack up visually against each other.
To find this ratio, we divide the angular size of the quarter by the angular size of the sun. This exercise yields an approximate result of 3.70.
This ratio indicates that the quarter appears about 3.70 times larger than the sun when viewed from their respective distances.

Radians

Radians are a way of measuring angles more commonly used in mathematics than degrees. An angle in radians measures the length of the arc of a unit circle that the angle subtends. This makes them a convenient unit when dealing with angular sizes because they provide a simple, dimensionless measure of size.
In the given exercise, both the quarter and the sun's angular sizes are determined in radians, allowing for easy comparison. For the quarter, it is approximately 0.0343 radians, and for the sun, about 0.00927 radians.

• Using radians simplifies mathematical calculations, especially in relation to geometry and trigonometry.
• It provides a universal way of comparing the apparent sizes of objects regardless of their real size!

Understanding radians can make evaluating angular sizes much easier, and it is a skill that can be particularly useful in physics and astronomy.