Problem 37 - College Physics

Question

How many minutes of arc does the Earth rotate in 1 min of time?

Step-by-Step Solution

Answer

The Earth rotates 15 minutes of arc in 1 minute of time.

Step 1: Understanding Angular Movement

The Earth makes a complete rotation of 360 degrees in 24 hours. To find the angular movement for 1 minute of time, we need to determine how many degrees the Earth rotates per minute.

Step 2: Calculating Degrees Per Hour

First, calculate how many degrees the Earth rotates in 1 hour. Since there are 360 degrees in a full rotation and 24 hours in a day, you divide 360 degrees by 24 hours.\[\frac{360}{24} = 15 \text{ degrees per hour}\]

Step 3: Converting Hours to Minutes

Since there are 60 minutes in an hour, you need to calculate how many degrees the Earth rotates in 1 minute of time. Divide the degrees per hour by 60.\[\frac{15}{60} = 0.25 \text{ degrees per minute}\]

Step 4: Converting Degrees to Minutes of Arc

Remember that 1 degree is equal to 60 minutes of arc. To find the rotation in minutes of arc for 0.25 degrees, multiply 0.25 by 60.\[0.25 \times 60 = 15 \text{ minutes of arc}\]

Key Concepts

Angular Movement

The concept of angular movement describes how an object rotates around a point or an axis. In the context of the Earth, angular movement refers to its rotation around its own axis. This rotation is what causes the phenomenon of day and night. A full rotation of the Earth corresponds to an angular movement of 360 degrees, which happens approximately every 24 hours.
This constant rotation means we can predictably divide the Earth's movement into smaller increments. For example, if the Earth rotates 360 degrees in 24 hours, we can calculate its angular movement over shorter periods, such as hours or minutes, by breaking down the full rotation into these smaller units. This is particularly useful in various fields such as astronomy and navigation.

Minutes of Arc

A minute of arc is a unit of angular measurement that allows for more precise calculations. Just as there are 60 minutes in an hour, there are 60 minutes of arc in one degree. This can sometimes be denoted as 60 arcminutes, and it is important for detailing angles in degrees more finely.
Using minutes of arc is beneficial when dealing with astronomy, geolocation, or even mapping where accurate measurements are necessary. For example, in the exercise, when we calculated the Earth's rotation in terms of minutes of arc, we noted that 0.25 degrees corresponds to 15 minutes of arc. This detail is crucial for understanding how small changes in degrees translate into this more precise unit of measurement.

Degrees per Minute

Degrees per minute measures the Earth's angular movement in its simplest form. From the problem, we learn that the Earth rotates 0.25 degrees every minute. This understanding helps to make calculations about the Earth's rotation more manageable and practical.
By simplifying the Earth's rotation to 15 degrees per hour (or 0.25 degrees per minute), we can easily perform quick calculations for real-world applications. For instance, if you wanted to know how many degrees the Earth rotates in 5 minutes, you would simply multiply 0.25 by 5, resulting in a rotation of 1.25 degrees.

Conversion of Units

Converting between different units of angular measurement is a key skill. In this exercise, we saw how easily you can switch from degrees to minutes of arc to make calculations more precise.
For example, when converting 0.25 degrees to minutes of arc, you multiply by 60, which gives 15 minutes of arc. Understanding this conversion is crucial in fields where precise measurements are necessary, such as in navigation or astronomy.

• 1 degree = 60 minutes of arc
• 1 hour = 60 minutes

This ability to convert units helps in many practical situations where different parts of the world or different technologies might use different standard measurements.