Name: how earth’s shape affects apparent weight
Uploaded: 2023-04-30T11:06:22.000Z
Duration: 1 min 20 s
Channel: JoVE

K12 Standards

Virginia  Standards of Learning

HIgh School

Earth Science

ES.3

ES.3.b

variation-in-acceleration-due-to-gravity-near-the-earth-s-surface

Variation in Acceleration due to Gravity near the Earth's Surface

Rozważmy obiekt o masie m zawieszony na skali sprężynowej przymocowanej do sztywnego wspornika. Na równiku Ziemi odczyt skali, który jest wielkością pozornej masy obiektu, jest równy wielkości jego rzeczywistej masy pomniejszonej o wielkość siły dośrodkowej.
Jeśli jednak obiekt zawieszony na skali znajduje się na innej szerokości geograficznej λ, wówczas siła dośrodkowa działająca na obiekt jest skierowana w stronę punktu P na osi obrotu, z dala od środka Ziemi.
Stąd pozorna masa obiektu jest równa ciężarowi rzeczywistemu pomniejszonej o składową cosinusową siły dośrodkowej skierowanej w stronę środka Ziemi.
Ogólnie rzecz biorąc, dzieląc równanie przez m, przyspieszenie netto g' obiektu od biegunów jest mniejsze niż przyspieszenie grawitacyjne o współczynnik równy przyspieszeniu dośrodkowemu.
Ziemia jest spłaszczoną kulą o promieniu równikowym większym niż jej promień biegunowy. W ten sposób jego zmienna gęstość przyczynia się również do zmiany przyspieszenia pod wpływem grawitacji.

zmiany przyspieszenia spowodowane grawitacją w pobliżu powierzchni ziemi

Pozorna waga obiektu to jego ciężar zmierzony za pomocą wagi sprężynowej w jego miejscu. Różni się od jego prawdziwego ciężaru, czyli siły, z jaką Ziemia go przyciąga, w wyniku obrotu Ziemi. Matematycznie pozorna waga obiektu jest równa jego prawdziwej masie pomniejszonej o siłę dośrodkową, która utrzymuje go w ruchu kołowym wzdłuż powierzchni Ziemi co 24 godziny.

Różnica między ciężarami rzeczywistymi i pozornymi jest proporcjonalna do kwadratu prędkości kątowej Ziemi. Ponieważ prędkość kątowa Ziemi jest stosunkowo mała, a jej kwadrat pojawia się w różnicy, wynik netto obrotu Ziemi wynosi tylko 0,34% masy obiektu.

Jednak efekt ten płynnie maleje w miarę przesuwania się od równika do biegunów. Wariację formułuje się, odwołując się do prostej geometrii; ponieważ każdy obiekt obraca się po okręgu pokrywającym szerokość geograficzną, na której się znajduje, różnica między wagą rzeczywistą i pozorną zależy od szerokości geograficznej obiektu. Maksymalny efekt występuje na równiku, a minimalny na biegunach. Na biegunach ciężar pozorny jest równy ciężarowi rzeczywistemu, ponieważ bieguny Ziemi nie obracają się.

Promień Ziemi również ma wpływ na różnicę między wagą rzeczywistą a pozorną. Ziemia nie jest idealną kulą; jego promień jest o około 30 km większy na równiku w porównaniu do biegunów, ponieważ wnętrze jest częściowo płynne, co zwiększa wybrzuszenie Ziemi na równiku w wyniku jej obrotu. Zatem wybrzuszenie Ziemi powoduje, że pozorna masa obiektu jest mniejsza od rzeczywistej wagi o dodatkowy czynnik. Można wykazać, że współczynnik ten jest porównywalny ze współczynnikiem związanym z obrotem Ziemi.

Ten tekst jest adaptacją <a href="https://openstax.org/books/university-physics-volume-1/pages/13-2-gravitation-near-earths- powierzchni">Openstax, University Physics, tom 1, sekcja 13.2: Grawitacja w pobliżu powierzchni Ziemi.</a>

Zmiany przyspieszenia spowodowane grawitacją w pobliżu powierzchni Ziemi

Pozorna masa obiektu to jego waga mierzona za pomocą wagi sprężynowej w jego miejscu. Różni się od jego prawdziwej wagi, siły, z jaką ...

Education

JoVE Core

Physics

Fizyka

Unit 1

Jednostka 1

Gravitation

Grawitacja

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

how earth’s shape affects apparent weight

Apparent weight changes slightly on Earth because of rotation and the planet’s shape. Apparent weight is the weight measured by a spring balance at a location. It is different from true weight, which is the force Earth pulls on an object with.

Earth’s rotation reduces apparent weight by adding a centripetal force. Centripetal force is the inward force that keeps an object moving in a circle. The apparent weight equals the true weight minus this force. The difference depends on the square of Earth’s angular speed, so the effect is small. In fact, rotation changes weight by only about 0.34%.

This change also depends on latitude. Latitude is a position north or south of the equator. The effect is largest at the equator and decreases smoothly toward the poles. At the poles, the apparent weight equals the true weight because Earth’s poles do not rotate in the same way.

Earth’s radius adds another small change. Earth is not a perfect sphere. Its radius is about 30 km larger at the equator than at the poles because the planet bulges outward. This bulge comes from Earth’s partly liquid interior and its rotation. The bulging makes apparent weight smaller than true weight by an additional factor, and this factor is comparable to the one caused by rotation.

How Earth’s Shape Affects Apparent Weight

how-earth-s-shape-affects-apparent-weight

gravitation

Newton’s Law Behind Planet Motion

Newton’s law of gravitation links falling objects, the Moon’s orbit, and planetary motion. Before Newton, many people thought the sky and Earth followed different laws. Kepler later described the three laws of planetary motion, but their connection to a deeper force was still unclear. Isaac Newton made that connection in 1665 to 1666. He showed that the same phenomenon explains why objects fall to Earth and why the Moon moves around Earth. He also found that the Moon’s centripetal acceleration...

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newton's law of gravitation

Newton's Gravity and the Motion of Bodies

Newton's gravity links the fall of objects on Earth with the motion of the Moon and planets. People once thought the downward pull seen near Earth was unique to our planet. Later, Nicolaus Copernicus proposed the heliocentric theory, which placed Earth and the other planets around the Sun, with the Moon orbiting Earth. Isaac Newton connected these motions in the 17th century. He reasoned that the same attraction that keeps the Moon on its path around Earth also makes an apple fall from a tree.

gravitation between spherically symmetric masses

gravitation-between-spherically-symmetric-masses

Gravitational Potential Inside a Shell

Gravitational potential energy for spherically symmetric bodies depends on their masses and the distance between them. The idea is to treat each body as if its mass were concentrated at its center of mass. This makes the calculation easier for high school physics. A spherically symmetric mass can be thought of as many concentric spherical shells. Now place a point mass at a distance r from the center of one shell. Every particle on a given spherical ring is the same distance from that point...

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gravity between spherical bodies

Point-Mass Approximation in Spherical Gravity

Newton’s law of gravitation gives the force between two point masses. For large spherical objects, such as the Earth, the Moon, and other planets, the same law can be used by treating each body as if its mass were concentrated at its center. This idea can be proven by comparing gravitational potential energy. The transcript shows that a hollow sphere of mass M and a point mass m give the same energy result as two extended solid spherical objects. To set up the proof, a tiny ring on the shell...

reduced mass coordinates: isolated two-body problem

reduced-mass-coordinates-isolated-two-body-problem

Center of Mass in the Two-Body Problem

Reduced mass coordinates help simplify the classical two-body problem. In classical mechanics, this problem describes the motion of two interacting bodies under gravity or another central force. The key idea is that the system can be treated like a single-body problem when reduced mass is used. For this description, one body is imagined to move with the reduced mass while the other body is fixed in place with infinite mass. The reduced mass is a property of the two-body system and is defined...

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acceleration due to gravity on earth

Gravity on Earth: Mass, Direction, and Value

Gravity near Earth gives objects a downward acceleration called acceleration due to gravity. It comes from Earth’s gravitational pull and points toward the center of the Earth. Newton’s law of gravitation shows that the gravitational force depends on mass, while Newton’s second law shows that the acceleration produced by a force depends on how that force acts on the object. Because of this relationship, the acceleration due to gravity is independent of an object’s mass. An object and the Earth...

acceleration due to gravity on other planets

acceleration-due-to-gravity-on-other-planets

Measuring Gravity on Distant Planets

Gravity near Earth’s surface is measured as acceleration due to gravity. On Earth, students can test it with simple experiments. On other planets, that kind of direct experiment is not possible. Instead, astronomers use observations to find a planet’s gravity. They study how the planet affects nearby objects, especially its satellite. The key idea is that the acceleration depends on the planet’s gravity, not on the object’s mass. To make the calculation, scientists track a satellite’s path...

apparent weight and the earth's rotation

Earth's Spin and Apparent Weight

Earth’s spin changes how weight is measured on the surface. Because the Earth turns once every 24 hours, objects on the surface move in a circle. That motion requires a net centripetal force, which points toward the center of the circle. The north and south poles are the only exception. At the equator, the force that provides this inward pull comes from Earth’s gravity. It is the difference between an object’s true weight and the normal force from the ground. The normal force is the support...

variation in acceleration due to gravity near the earth's surface

Apparent weight changes slightly on Earth because of rotation and the planet’s shape. Apparent weight is the weight measured by a spring balance at a location. It is different from true weight, which is the force Earth pulls on an object with. Earth’s rotation reduces apparent weight by adding a centripetal force. Centripetal force is the inward force that keeps an object moving in a circle. The apparent weight equals the true weight minus this force. The difference depends on the square of...

potential energy due to gravitation

Gravitational Binding and Potential Energy

Gravitational potential energy describes how gravity stores energy between two masses. Because gravity is a conservative force, the work needed to move an object between two points depends on the starting and ending positions, not on the path taken. That is why a gravitational potential energy function can be written in terms of spatial coordinates. For a mass bound by gravity to another object, the potential energy is negative near the attracting body and rises toward zero as the distance...

the principle of superposition and the gravitational field

the-principle-of-superposition-and-the-gravitational-field

Adding Gravitational Forces with Superposition

Gravitational forces can be added using the principle of superposition. For objects that are sufficiently far apart, the net gravitational force on a point object is the vector sum of the forces from each mass. This means each force is found first, and then the forces are added as vectors. Superposition is useful because gravity from more than one object can combine in ways that are not always easy to see. One force may be increased or reduced by another force, or the final effect may be a...

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escape velocity

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circular orbits and critical velocity for satellites

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Orbital Speed and Period in Circular Paths

Orbital speed and period in circular paths explain how the Moon, planets, and satellites stay in motion around larger bodies. The Moon orbits Earth. Earth and the other planets orbit the Sun. The space above our atmosphere also contains many artificial satellites. A circular orbit is the simplest orbit to study. It helps show how an object’s speed and its period, or the time for one full orbit, depend on where it is and what it circles. Nicolaus Copernicus first suggested that Earth and the...

energy of a satellite in a circular orbit

energy-of-a-satellite-in-a-circular-orbit

Satellite Orbits and Energy in Space

Satellite orbits and energy in space depend on altitude, orbit shape, and gravity. Thousands of artificial satellites circle Earth every day at many different distances. Satellites below 1,600 km are in low-Earth orbit, or LEO. Research satellites and Earth observation satellites are usually placed in LEO. They mostly move in elliptical orbits, which are oval-shaped paths. Navigation satellites are placed in medium-Earth orbit, or MEO, which ranges from 2,000 km to 36,000 km above Earth’s...

kepler's first law of planetary motion

Planet Orbits as Ellipses

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kepler's second law of planetary motion

Kepler's Equal-Area Planet Motion

Kepler's second law explains how a planet moves faster and slower in its orbit around the Sun. Johannes Kepler, a German astronomer and mathematician in the early 17th century, developed three laws for planetary motion. His first law says that planets travel in elliptical orbits, with the Sun at one focus of the ellipse. Because the planet-Sun distance changes during the orbit, the planet does not move at the same speed all the time. In an elliptical orbit, the total energy of the planet is...

kepler's third law of planetary motion

Kepler’s Planet Distance Law

Kepler’s third law links a planet’s average distance from the Sun to the time it takes to orbit the Sun. This rule gives a clear mathematical relationship between orbital distance and orbital period, which is the time for one full revolution. Johannes Kepler developed this law in the early 17th century after building on the observations of Nicolaus Copernicus and Tycho Brahe. Kepler first formed his first two laws from those earlier observations. He later published the third law of planetary...

tidal forces

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Earth’s ocean tides are driven by gravitational tidal forces. Scientists have studied the origin of tides for more than 2000 years, and Newton’s work marked the start of a proper understanding. These same tidal forces also act on other astronomical bodies. They can produce internal heat that leads to volcanic activity on Io, one of Jupiter’s moons, and they can tear apart stars that pass too close to black holes. Newton’s law of gravitation says that gravitational force is inversely...

schwarzschild radius and event horizon

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Escape velocity helps explain how black holes and event horizons form. It is the minimum speed a projectile needs to leave the surface of an object and eventually escape its gravitational field. In this context, escape velocity does not depend on the mass of the object being launched. The key idea is that no object with finite mass can move faster than the speed of light in a vacuum. When escape velocity is compared with that speed limit, the Schwarzschild radius becomes important. If an...

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Black holes are found through indirect astronomical evidence. They were first proposed in the 1920s, but they stayed outside observational astronomy until the 1970s. Their closest known cousins are neutron stars, which are made mostly of packed neutrons and are extremely dense. A neutron star can have the mass of the Sun while its diameter is only a few kilometers. Because of that, the escape velocity from its surface is close to the speed of light. Interest in black holes rose again in the...

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space-time curvature and the general theory of relativity

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Einstein’s Gravity and Mercury’s Orbit

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key terms and concepts

Learn why apparent weight changes with latitude on Earth. Explore the effects of rotation, centripetal force, and the planet’s equatorial bulge.

How Earth’s Shape Affects Apparent Weight | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

PH.3

PH.3.b

Florida Department of Education

High School

Earth and Space Science

SC.912.E.5

SC.912.E.5.10

SC.912.E.5.11

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.H

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

Earth and Space Science (ESS)

Earth’s Place in the Universe

E-ESS1-6.

new-york-state-science-learning-standards

New York State Science Learning Standards

Physical Science

HS. Forces and Interactions

HS-PS2-4.

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ES.3.a

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Apparent Weight and the Earth's Rotation

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An object&#39;s apparent weight is its weight measured by a spring balance at its location. It is different from its true weight, the force with which the ...

The student will investigate and understand that Earth is unique in our solar system. Key ideas include the dynamics of the sun-Earth-moon system cause seasons, tides, and eclipses.

How Earth’s Shape Affects Apparent Weight

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