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
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product