Problem 43 - Precalculus: Functions and Graphs

Question

Earth's rotation Earth rotates about its axis once every 23 hours, 56 minutes, and 4 seconds. Approximate the number of radians Earth rotates in one second.

Step-by-Step Solution

Answer

Earth rotates approximately 0.00007292115 radians per second.

Step 1: Convert rotational period to seconds

First, we need to convert Earth's rotational period into seconds. Earth rotates once every 23 hours, 56 minutes, and 4 seconds.Convert hours to seconds: \[ 23 \text{ hours} = 23 \times 3600 = 82800 \text{ seconds} \]Convert minutes to seconds:\[ 56 \text{ minutes} = 56 \times 60 = 3360 \text{ seconds} \]Add the seconds:\[ 4 \text{ seconds} \]Thus, total seconds for one full rotation is:\[ 82800 + 3360 + 4 = 86164 \text{ seconds} \]

Step 2: Calculate radians per rotation

The Earth rotates 360 degrees per full rotation. Convert 360 degrees to radians:Radians per rotation:\[ 360^{\circ} = 2\pi \text{ radians} \]

Step 3: Calculate radians per second

Knowing the full rotation in radians and the rotation period in seconds, calculate the radians the Earth rotates in one second:\[ \text{Radians per second} = \frac{2\pi}{86164} \]Simplify the formula:\[ \frac{2\pi}{86164} \approx 0.00007292115 \text{ radians per second} \]

Key Concepts

Radians

Radians are a unit of angular measure used in mathematics and physics to express angles. Unlike degrees, where a circle is divided into 360 parts, radians divide the circle into parts based on the circle's radius. One radian corresponds to the angle made when the arc length is equal to the radius of the circle. The full rotation of a circle is equal to 2π radians.

• Radians provide a natural way of describing angles in terms of lengths.
• There are exactly 2π radians in a complete circle.
• Conversion between degrees and radians is essential in many calculations, particularly in physics and engineering.

Knowing that 360 degrees is equivalent to 2π radians helps in converting rotations of the Earth and other celestial bodies into radians.
The use of radians is particularly useful in calculus, as many mathematical functions involving angles behave more naturally when their angles are expressed in radians.

Rotational period

The rotational period of a body is the time it takes to complete one full rotation about its axis. For Earth, this period is approximately 23 hours, 56 minutes, and 4 seconds. This time measurement tells us how long it takes for a single rotation relative to the distant stars, which is known as a sidereal day.

• The rotational period is crucial in understanding the movement of celestial bodies.
• Earth's rotational speed has a significant impact on the length of a day.
• Accurate measurement of the rotational period helps in predicting various phenomena, like the cycle of day and night.

Understanding Earth's rotational period is important in fields ranging from astronomy to navigation and timekeeping. This period is what defines the length of a day as we conventionally understand it.

Conversion to seconds

To solve problems involving time, converting different units into a single unit is crucial for clarity and consistency. When dealing with Earth's rotation, converting hours and minutes into seconds makes calculations simpler and more straightforward.

• There are 3600 seconds in an hour. Multiply the number of hours by 3600 to convert to seconds.
• There are 60 seconds in a minute. Multiply the number of minutes by 60 to convert to seconds.
• Adding these values together gives the total number of seconds.

In the case of Earth's rotation, converting 23 hours, 56 minutes, and 4 seconds into seconds is necessary for calculating how many radians the Earth rotates in one second. By aligning all time units, the formulas used become simpler and more accurate.