Determine the mass of an astronomical body from free-fall acceleration at its surface
Describe how the value of
varies due to location and Earth’s rotation
In this section, we observe how Newton’s law of gravitation applies at the surface of a planet and how it connects with what we learned earlier about free fall. We also examine the gravitational effects within spherical bodies.
Weight
Recall that the acceleration of a free-falling object near Earth’s surface is approximately
. The force causing this acceleration is called the weight of the object, and from Newton’s second law, it has the value
mg . This weight is present regardless of whether the object is in free fall. We now know that this force is the gravitational force between the object and Earth. If we substitute
mg for the magnitude of
in Newton’s law of universal gravitation,
m for
, and
for
, we obtain the scalar equation
where
r is the distance between the centers of mass of the object and Earth. The average radius of Earth is about 6370 km. Hence, for objects within a few kilometers of Earth’s surface, we can take
(
[link] ). The mass
m of the object cancels, leaving
This explains why all masses free fall with the same acceleration. We have ignored the fact that Earth also accelerates toward the falling object, but that is acceptable as long as the mass of Earth is much larger than that of the object.
Masses of earth and moon
Have you ever wondered how we know the mass of Earth? We certainly can’t place it on a scale. The values of
g and the radius of Earth were measured with reasonable accuracy centuries ago.
Use the standard values of
g ,
, and
[link] to find the mass of Earth.
Estimate the value of
g on the Moon. Use the fact that the Moon has a radius of about 1700 km (a value of this accuracy was determined many centuries ago) and assume it has the same average density as Earth,
.
Strategy
With the known values of
g and
, we can use
[link] to find
. For the Moon, we use the assumption of equal average density to determine the mass from a ratio of the volumes of Earth and the Moon.
As soon as Cavendish determined the value of
G in 1798, the mass of Earth could be calculated. (In fact, that was the ultimate purpose of Cavendish’s experiment in the first place.) The value we calculated for
g of the Moon is incorrect. The average density of the Moon is actually only
and
at the surface. Newton attempted to measure the mass of the Moon by comparing the effect of the Sun on Earth’s ocean tides compared to that of the Moon. His value was a factor of two too small. The most accurate values for
g and the mass of the Moon come from tracking the motion of spacecraft that have orbited the Moon. But the mass of the Moon can actually be determined accurately without going to the Moon. Earth and the Moon orbit about a common center of mass, and careful astronomical measurements can determine that location. The ratio of the Moon’s mass to Earth’s is the ratio of [the distance from the common center of mass to the Moon’s center] to [the distance from the common center of mass to Earth’s center].
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
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Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature