/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 The Earth's rate of rotation is ... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

The Earth's rate of rotation is constantly decreasing, causing the day to increase in duration. In the year 2006 the Earth took about 0.840 s longer to complete 365 revolutions than it did in the year \(1906 .\) What was the average angular acceleration of the Earth during this time? Give your answer in \(\mathrm{rad} / \mathrm{s}^{2}\).

Short Answer

Expert verified
The average angular acceleration of the Earth was approximately \(5.31 \times 10^{-22}\) rad/s².

Step by step solution

01

Understand the Problem

The Earth's rotation is slowing down, meaning each day is slightly longer. Between 1906 and 2006, the Earth took an additional 0.840 seconds to complete 365 rotations. We need to find the average angular acceleration over this 100-year period.
02

Determine Time Interval

The time interval given is from 1906 to 2006, which is 100 years. To convert this into seconds, use the conversion: 1 year = 365.25 days (to account for leap years), thus \[ 100 \text{ years} \times 365.25 \text{ days/year} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour} \approx 3.15576 \times 10^9 \text{ seconds}.\]
03

Understand the Change in Angular Displacement

The additional 0.840 seconds occurs over 365 rotations. Since each complete rotation corresponds to an angular displacement of \(2\pi\) radians, the total angular displacement is \[ \Delta \theta = 365 \times 2\pi \approx 2290 \times \pi \text{ radians}.\]
04

Calculate Angular Velocity Change

The change in time for these rotations accounts for 0.840 seconds more over the 100 years. The change in angular velocity \(\Delta \omega\) can be expressed as:\[ \Delta \omega = \frac{0.840 \text{ seconds}}{3.15576 \times 10^9 \text{ seconds}} \times \frac{2\pi \text{ radians}}{1\text{ rotation}}.\]\[ \Delta \omega \approx 1.67553 \times 10^{-12} \text{ rad/s}.\]
05

Compute Angular Acceleration

Angular acceleration \(\alpha\) is defined as the change in angular velocity \(\Delta \omega\) over a given time interval \(\Delta t\). Therefore:\[\alpha = \frac{\Delta \omega}{\Delta t} \approx \frac{1.67553 \times 10^{-12} \text{ rad/s}}{3.15576 \times 10^9 \text{ s}} \approx 5.31 \times 10^{-22} \text{ rad/s}^2.\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Earth's rotation
The planet Earth rotates around its axis, causing the cycle of day and night. This rotation is what gives us the apparent motion of the Sun moving across the sky each day. However, Earth's rotation is not constant. Over long periods, the rate of rotation can change due to various factors, such as gravitational interactions with the Moon. These interactions are known as tidal forces.
- The rotational period of Earth currently takes approximately 24 hours to complete one full turn relative to the Sun.
- However, historical data shows that this period is gradually increasing, indicating Earth's rotation is slowing down.
This deceleration isn't something we typically notice on a daily basis, but over decades and centuries, the cumulative effect becomes significant. Scientists study these changes in rotation speed to understand not only Earth's physical dynamics but also its interactions with the surrounding celestial bodies.
Angular velocity
Angular velocity is a measure of how quickly an object rotates or revolves relative to another point. It shows the rate of change of an angle over time.
- In the context of Earth's rotation, angular velocity describes how quickly the planet spins around its axis.
- It is typically represented in radians per second. A radian is a way to measure angles using the radius of a circle. When talking about Earth's rotation, it's often simplified by stating Earth's angular velocity is about 0.0000727 rad/s. This small number highlights the immense size and slow rotation of the Earth.
Changes in angular velocity, like those measured from 1906 to 2006, reflect how Earth's rotation rate evolves, which can be calculated using the difference in rotation period over time.
Time conversion
Time conversion is an essential skill in physics, especially when dealing with large timescales or precise measurements. When calculating physical phenomena such as Earth's rotation, scientists often need to convert time from one unit to another, like years to seconds.
- For instance, in calculations regarding Earth's rotation slowing down, converting 100 years into seconds is necessary for finding changes over long periods. - This is typically done by converting years into days, days into hours, and then hours into seconds.
Considering the average year has about 365.25 days (accounting for leap years), performing these conversions accurately ensures precise calculations. This meticulous conversion process is vital for understanding the tiny changes in Earth's angular velocity and acceleration, as small miscalculations can lead to major differences when interpreting long-term trends.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

When astronauts return from prolonged space flights, they often suffer from bone loss, resulting in brittle bones that may take weeks for their bodies to rebuild. One solution may be to expose astronauts to periods of substantial "g forces" in a centrifuge carried aboard their spaceship. To test this approach, NASA conducted a study in which four people spent 22 hours each in a compartment attached to the end of a 28 -foot arm that rotated with an angular speed of \(10.0 \mathrm{rpm}\). (a) What centripetal acceleration did these volunteers experience? Express your answer in terms of \(g .\) (b) What was their linear speed?

At 3: 00 the hour hand and the minute hand of a clock point in directions that are \(90.0^{\circ}\) apart. What is the first time after 3: 00 that the angle between the two hands has decreased by half to \(45.0^{\circ} ?\)

The following angles are given in radians. Convert them to degrees: \(\pi / 6,0.70 \pi, 1.5 \pi, 5 \pi\).

\(A\) bicycle coasts downhill and accelerates from rest to a linear speed of \(8.90 \mathrm{m} / \mathrm{s}\) in \(12.2 \mathrm{s}\). (a) If the bicycle's tires have a radius of \(36.0 \mathrm{cm},\) what is their angular acceleration? (b) If the radius of the tires had been smaller, would their angular acceleration be greater than or less than the result found in part (a)?

A centrifuge is a common laboratory instrument that separates components of differing densities in solution. This is accomplished by spinning a sample around in a circle with a large angular speed. Suppose that after a centrifuge in a medical laboratory is turned off, it continues to rotate with a constant angular deceleration for 10.2 s before coming to rest. (a) If its initial angular speed was 3850 rpm, what is the magnitude of its angular deceleration? (b) How many revolutions did the centrifuge complete after being turned off?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.