Hukum Newton tentang gaya tarik menarik
Hukum Newton tentang gaya tarik menarik gravitasi umum
Hukum tarik-menarik gravitasi Newton dalam bidang fisika berarti gaya tarik untuk saling
mendekat satu sama lain. Dalam bidang fisika tiap benda dengan massa m1 selalu mempunyai
gaya tarik menarik dengan benda lain (dengan massa m2 ). Misalnya partikel satu dengan
partikel lain selalu akan saling tarik-menarik. Contoh yang dikemukakan oleh Sir Isaac
Newton dalam bidang mekanika klasik bahwa benda apapun di atas atmosfer akan ditarik
oleh bumi, yang kemudian banyak dikenal sebagai fenomena benda jatuh.
Gaya tarik menarik gravitasi ini dinyatakan oleh Isaac Newton melalui tulisannya di
journal Philosophiæ Naturalis Principia Mathematica pada tanggal 5 Juli 1687 dalam bentuk
rumus sebagai berikut:
,
di mana:
F adalah besarnya gaya gravitasi antara dua massa tersebut,
G adalah konstante gravitasi,
m1 adalah massa dari benda pertama
m2 adalah massa dari benda kedua, dan
r adalah jarak antara dua massa tersebut.
Teori ini kemudian dikembangkan lebih jauh lagi bahwa setiap benda angkasa akan saling
tarik-menarik, dan ini bisa dijelaskan mengapa bumi harus berputar mengelilingi matahari
untuk mengimbangi gaya tarik-menarik gravitasi bumi-matahari. Dengan menggunakan
fenomena tarik menarik gravitasi ini juga, meteor yang mendekat ke bumi dalam
perjalanannya di ruang angkasa akan tertarik jatuh ke bumi.
Di dalam astronomi, tiga Hukum Gerakan Planet Kepler adalah:
Setiap planet bergerak dengan lintasan elips, Matahari berada di salah satu fokusnya.
Luas daerah yang disapu pada selang waktu yang sama akan selalu sama.
Perioda kuadrat suatu planet berbanding dengan pangkat tiga jarak rata-ratanya dari
Matahari.
Ketiga hukum di atas ditemukan oleh ahli matematika dan astronomi Jerman: Johannes
Kepler (1571–1630), yang menjelaskan gerakan planet di dalam tata surya. Hukum di atas
menjabarkan gerakan dua benda yang saling mengorbit.
Karya Kepler didasari oleh data pengamatan Tycho Brahe, yang diterbitkannya sebagai
'Rudolphine tables'. Sekitar tahun 1605, Kepler menyimpulkan bahwa data posisi planet hasil
pengamatan Brahe mengikuti rumusan matematika cukup sederhana yang tercantum di atas.
Hukum Kepler mempertanyakan kebenaran astronomi dan fisika warisan zaman Aristoteles
dan Ptolemaeus. Ungkapan Kepler bahwa Bumi beredar sekeliling, berbentuk elips dan
bukannya epicycle, dan membuktikan bahwa kecepatan gerak planet bervariasi, mengubah
astronomi dan fisika. Hampir seabad kemudian, Isaac Newton mendeduksi Hukum Kepler
dari rumusan hukum karyanya, hukum gerak dan hukum gravitasi Newton, dengan
menggunakan Euclidean geometri klasik.
Pada era modern, hukum Kepler digunakan untuk aproksimasi orbit satelit dan benda-benda
yang mengorbit Matahari, yang semuanya belum ditemukan pada saat Kepler hidup (contoh:
planet luar dan asteroid). Hukum ini kemudian diaplikasikan untuk semua benda kecil yang
mengorbit benda lain yang jauh lebih besar, walaupun beberapa aspek seperti gesekan
atmosfer (contoh: gerakan di orbit rendah), atau relativitas (contoh: prosesi preihelion
merkurius), dan keberadaan benda lainnya dapat membuat hasil hitungan tidak akurat dalam
berbagai keperluan.
Figure 1: Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses, with
focal points ƒ1 and ƒ2 for the first planet and ƒ1 and &>. (2) The two shaded sectors A1 and A2 have
the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover
segment A2. (3) The total orbit times for planet 1 and planet 2 have a ratio a13/2 : a23/2.
Gerak harmonik sederhana adalah gerak bolak – balik benda melalui suatu titik
keseimbangan tertentu dengan banyaknya getaran benda dalam setiap sekon selalu konstan
Periode (T) adalah waktu yang diperlukan benda untuk melakukan satu getaran. Satuan
periode detik
Frekuensi (f) adalah banyaknya getaran yang dilakukan oleh benda selama satu detik. Satuan
frekuensi hertz
Hubungan antara periode dan frekuensi :
atau
Hukum Hooke menyatatakan bahwa besar gaya pegas pemulih sebanding dengan
pertambahan panjang pegas. Secara matematis, dapat dituliskan sebagai :
F = – k Δx
dengan k = tetapan pegas (N/m)
Tanda (-) diberikan karena arah gaya pemulih pada pegas berlawanan dengan arah gerak
pegas tersebut.
Dari hukum II Newton didapatkan periode pegas
Persamaan Gerak Harmonik Sederhana
Y = A sin ωt
di mana :
Y = simpangan
A = simpangan maksimum (amplitudo)
ω = kecepatan sudut = 2πf
t = waktu
Kecepatan maksimumnya adalah vmax = A ω
Energi pada Gerak Harmonik Sederhana adalah E = ½ kA2
Hukum Kepler I, II dan III
1. Hukum I Kepler
Menjelaskan tentang bentuk lingkaran orbit planet. Bunyi hukum Keppler 1 berisi
sebagai berikut.
Lintasan setiap planet mengelilingi matahari merupakan sebuah elips dengan matahari
terletak pada salah satu titik fokusnya.
Nah setelah melihat hukum pertama Kepler dapat dilihat ilustrasi tersebut seperti pada
Gambar berikut ini.
Ilustrasi orbit planet sesuai hukum I Kepler
2. Hukum II Kepler
Menjelaskan tentang kecepatan orbit planet. Apa perbedaan dengan hukum kepler
pertama? Perhatikan penjelasan berikut, hukum Keppler 2 berisi sebagai berikut
Setiap planet bergerak sedemikian sehingga suatu garis khayal yang ditarik dari
matahari ke planet tersebut mencakup daerah dengan luas yang sama dalam waktu
yang sama.
Untuk lebih jelasnya silahkan amati gambar berikut
Ilustrasi orbit planet sesuai hukum ketiga Kepler
Keterangan :
Garis AM akan menyapu lurus sampai ke garis BM, luasnya sama dengan daerah
yang disapu garis Cm hingga DM. Jika tAB = tCD. Hukum kedua ini juga
menjelaskan bahwa dititik A dan B planet harus lebih cepat dibanding saat di titik C
dan D.
3. Hukum III Kepler
Menjelaskan tentang periode revolusi planet. Periode revolusi planet ini dikaitkan
dengan jari-jari orbit rata-ratanya. Perhatikan penjelasan berikut, hukum Keppler 3
berisi sebagai berikut
Kuadrat periode planet mengitari matahari sebanding dengan pangkat tiga rata-rata
planet dari matahari.
Nah, Hubungan di atas dapat dirumuskan secara matematis denga persamaan seperti
berikut ini.
THE GRAVITATIONAL FORCE BETWEEN A PARTICLE AND A SPHERICAL MASS
We have already stated that a large sphere attracts a particle outside it as if the total mass of
the sphere were concentrated at its center. We now describe the force acting on a particle
when the extended object is either a spherical shell or a solid sphere, and then apply these
facts to some interesting systems.
Spherical Shell
Case 1. If a particle of mass m is located outside a spherical shell of mass M at, for instance,
point P in Figure 14.21a, the shell attracts the particle as though the mass of the shell were
concentrated at its center. We can show this, as Newton did, with integral calculus. Thus, as
far as the gravitational force acting on a particle outside the shell is concerned, a spherical
shell acts no differently from the solid spherical distributions of mass we have seen.
Case 2. If the particle is located inside the shell (at point P in Fig. 14.21b), the gravitational
force acting on it can be shown to be zero.
We can express these two important results in the following way:
The gravitational force as a function of the distance r is plotted in Figure 14.21c.
Figure 14.21 (a) The nonradial components of the gravitational forces exerted on a particle of
mass m located at point P outside a spherical shell of mass M cancel out. (b) The spherical
shell can be broken into rings. Even though point P is closer to the top ring than to the bottom
ring, the bottom ring is larger, and the gravitational forces exerted on the particle at P by the
matter in the two rings cancel each other. Thus, for a particle located at any point P inside the
shell, there is no gravitational force exerted on the particle by the mass M of the shell. (c) The
magnitude of the gravitational force versus the radial distance r from the center of the shell.
The shell does not act as a gravitational shield, which means that a particle inside a shell may
experience forces exerted by bodies outside the shell.
Solid Sphere
Case 1. If a particle of mass m is located outside a homogeneous solid sphere of mass M (at
point P in Fig. 14.22), the sphere attracts the particle as though the mass of the sphere were
concentrated at its center. We have used this notion at several places in this chapter already,
and we can argue it from Equation 14.25a. A solid sphere can be considered to be a collection
of concentric spherical shells. The masses of all of the shells can be interpreted as being
concentrated at their common center, and the gravitational force is equivalent to that due to a
particle of mass M located at that center.
Case 2. If a particle of mass m is located inside a homogeneous solid sphere of mass M (at
point Q in Fig. 14.22), the gravitational force acting on it is due only to the mass M’
contained within the sphere of radius shown in Figure 14.22. In other words,
This also follows from spherical-shell Case 1 because the part of the sphere that is
Figure 14.22 The gravitational force acting on a particle when it is outside a uniform solid
sphere is GMm/r2 and is directed toward the center of the sphere. The gravitational force
acting on the particle when it is inside such a sphere is proportional to r and goes to zero at
the center.
farther from the center than Q can be treated as a series of concentric spherical shells that do
not exert a net force on the particle because the particle is inside them. Because the sphere is
assumed to have a uniform density, it follows that the ratio of masses M’/M is equal to the
ratio of volumes V’/V, where V is the total volume of the sphere and V’ is the volume within
the sphere of radius r only:
Solving this equation for M’ and substituting the value obtained into Equation 14.26b, we
have
This equation tells us that at the center of the solid sphere, where r = 0, the gravitational force
goes to zero, as we intuitively expect. The force as a function of r is plotted in Figure 14.22.
Case 3. If a particle is located inside a solid sphere having a density ρ that is spherically
symmetric but not uniform, then M’ in Equation 14.26b is given by an integral of the form M’
= ∫ρ dV, where the integration is taken over the volume contained within the sphere of radius
r in Figure 14.22. We can evaluate this integral if the radial variation of ρ is given. In this
case, we take the volume element dV as the volume of a spherical shell of radius r and
thickness dr, and thus dV = 4πr2 dr. For example, if ρ = Ar, where A is a constant, it is left to a
problem (Problem 63)
Areal velocity
From Wikipedia, the free encyclopedia
Areal velocity is the area (shown in green) swept out per unit time by a particle moving along a curve
(shown in blue).
In classical mechanics, areal velocity (also called sector velocity or sectorial velocity) is the
rate at which area is swept out by a particle as it moves along a curve. In the adjoining figure,
suppose that a particle moves along the blue curve. At a certain time t, the particle is located
at point B, and a short while later, at time t + Δt, the particle has moved to point C. The area
swept out by the particle is the green area in the figure, bounded by the line segments AB and
AC and the curve along which the particle moves. The areal velocity equals this area divided
by the time interval Δt in the limit that Δt becomes vanishingly small.
Illustration of Kepler's second law. The planet moves faster near the Sun, so the same area is swept
out in a given time as at larger distances, where the planet moves more slowly.
The concept of areal velocity is closely linked historically with the concept of angular
momentum. Kepler's second law states that the areal velocity of a planet, with the sun taken
as origin, is constant. Isaac Newton was the first scientist to recognize the dynamical
significance of Kepler's second law. With the aid of his laws of motion, he proved in 1684
that any planet that is attracted to a fixed center sweeps out equal areas in equal intervals of
time. By the middle of the 18th century, the principle of angular momentum was discovered
gradually by Daniel Bernoulli and Leonhard Euler and Patrick d'Arcy; d'Arcy's version of the
principle was phrased in terms of swept area. For this reason, the principle of angular
momentum was often referred to in the older literature in mechanics as "the principle of
areas." Since the concept of angular momentum includes more than just geometry, the
designation "principle of areas" has been dropped in modern works.
Connection with angular momentum
In the situation of the first figure, the area swept out during time period Δt by the particle is
approximately equal to the area of triangle ABC. As Δt approaches zero this near-equality
becomes exact as a limit.
Let the point D be the fourth corner of parallelogram ABDC shown in the figure, so that the
vectors AB and AC add up by the parallelogram rule to vector AD. Then the area of triangle
ABC is half the area of parallelogram ABDC, and the area of ABDC is equal to the magnitude
of the cross product of vectors AB and AC. This area can also be viewed as a vector with this
magnitude, pointing in a direction perpendicular to the parallelogram; this vector is the cross
product itself:
Hence
The areal velocity is this area divided by Δt in the limit that Δt becomes vanishingly small:
But, is the velocity vector of the moving particle, so that
On the other hand, the angular momentum of the particle is
and hence the angular momentum equals 2m times the areal velocity.
Conservation of areal velocity is a general property of central force motion.[1]
The Universal Law of Gravitation
Gravity is described from the point of view of a universal law. This implies that gravity is a
force that should behave in similar ways regardless of where you are in the universe.
Gravity
It's a force of attraction that exists between any two objects that have mass. The more
mass they have, the greater the force of attraction. The closer they are, the greater the
force of attraction.
For most objects you get near every day, the force of attraction is so incredibly small that you
would never notice the force. Gravity is a very weak force, so between common objects like
you and your pencil, the force of attraction is very small because your mass and the mass of
your pencil are small. We only get noticeable amounts of gravity when at least one object is
very massive... like a planet. The force of attraction between you and the planet Earth is a
noticeable force! We call the force of attraction between you and the Earth, your weight.
Weight is another name for the force of gravity pulling down on you or anything else.
G is the universal gravitational constant.
It is
basically a conversion factor to adjust the number
and units so they come out to the correct value.
This is a universal constant so it is true
everywhere.
m1 is the mass of one of the objects.
m2 is the mass of the other object.
r is the radius of separation between the center of
masses of each object.
FG is the force of attraction between the two
objects.
Important Concepts
The direction of the force is not given by this formula since there are actually two
forces equal in size but opposite in direction. This formula calculates them both.
The formula is an inverse square law for radius of separation (notice the r2 on the
bottom of the equation). This means that if you double the separation you quarter the
force, or if you cut the separation in half you quadruple the force of attraction.
If you double a single mass, you double the force. If you cut one of the masses in
half, you cut the force in half. But if you double both masses you would quadruple
the force.
Common Misconceptions
"There is no gravity in space." FALSE If there were no gravity in space, the space
shuttle would not be able to orbit the Earth, the moon would not orbit the Earth, and
the Earth would not orbit the Sun. The reason we tend to think of there being no
gravity in space is that we have seen movies of the astronauts being "weightless".
They aren't actually weightless, they are still being pulled down by gravity but they
and the space shuttle are in a constant state of freefall around the Earth. So they seem
to be weightless as a result of the falling - just as you would seem weightless if you
were in an elevator when the cable broke.
"G and g are the same thing." FALSE Big G is the universal gravitational
constant. Little g is the acceleration due to the force of gravity and its value of
9.8m/s/s down is only true on this planet. It is not a universal constant.
"g is gravity." FALSE Little g is the effect of the force of gravity, but is not
gravity. Gravity is a force, little g is an acceleration caused by gravity.
©1998 Science Joy Wagon
Hukum tarik-menarik gravitasi Newton dalam bidang fisika berarti gaya tarik untuk saling
mendekat satu sama lain. Dalam bidang fisika tiap benda dengan massa m1 selalu mempunyai
gaya tarik menarik dengan benda lain (dengan massa m2 ). Misalnya partikel satu dengan
partikel lain selalu akan saling tarik-menarik. Contoh yang dikemukakan oleh Sir Isaac
Newton dalam bidang mekanika klasik bahwa benda apapun di atas atmosfer akan ditarik
oleh bumi, yang kemudian banyak dikenal sebagai fenomena benda jatuh.
Gaya tarik menarik gravitasi ini dinyatakan oleh Isaac Newton melalui tulisannya di
journal Philosophiæ Naturalis Principia Mathematica pada tanggal 5 Juli 1687 dalam bentuk
rumus sebagai berikut:
,
di mana:
F adalah besarnya gaya gravitasi antara dua massa tersebut,
G adalah konstante gravitasi,
m1 adalah massa dari benda pertama
m2 adalah massa dari benda kedua, dan
r adalah jarak antara dua massa tersebut.
Teori ini kemudian dikembangkan lebih jauh lagi bahwa setiap benda angkasa akan saling
tarik-menarik, dan ini bisa dijelaskan mengapa bumi harus berputar mengelilingi matahari
untuk mengimbangi gaya tarik-menarik gravitasi bumi-matahari. Dengan menggunakan
fenomena tarik menarik gravitasi ini juga, meteor yang mendekat ke bumi dalam
perjalanannya di ruang angkasa akan tertarik jatuh ke bumi.
Di dalam astronomi, tiga Hukum Gerakan Planet Kepler adalah:
Setiap planet bergerak dengan lintasan elips, Matahari berada di salah satu fokusnya.
Luas daerah yang disapu pada selang waktu yang sama akan selalu sama.
Perioda kuadrat suatu planet berbanding dengan pangkat tiga jarak rata-ratanya dari
Matahari.
Ketiga hukum di atas ditemukan oleh ahli matematika dan astronomi Jerman: Johannes
Kepler (1571–1630), yang menjelaskan gerakan planet di dalam tata surya. Hukum di atas
menjabarkan gerakan dua benda yang saling mengorbit.
Karya Kepler didasari oleh data pengamatan Tycho Brahe, yang diterbitkannya sebagai
'Rudolphine tables'. Sekitar tahun 1605, Kepler menyimpulkan bahwa data posisi planet hasil
pengamatan Brahe mengikuti rumusan matematika cukup sederhana yang tercantum di atas.
Hukum Kepler mempertanyakan kebenaran astronomi dan fisika warisan zaman Aristoteles
dan Ptolemaeus. Ungkapan Kepler bahwa Bumi beredar sekeliling, berbentuk elips dan
bukannya epicycle, dan membuktikan bahwa kecepatan gerak planet bervariasi, mengubah
astronomi dan fisika. Hampir seabad kemudian, Isaac Newton mendeduksi Hukum Kepler
dari rumusan hukum karyanya, hukum gerak dan hukum gravitasi Newton, dengan
menggunakan Euclidean geometri klasik.
Pada era modern, hukum Kepler digunakan untuk aproksimasi orbit satelit dan benda-benda
yang mengorbit Matahari, yang semuanya belum ditemukan pada saat Kepler hidup (contoh:
planet luar dan asteroid). Hukum ini kemudian diaplikasikan untuk semua benda kecil yang
mengorbit benda lain yang jauh lebih besar, walaupun beberapa aspek seperti gesekan
atmosfer (contoh: gerakan di orbit rendah), atau relativitas (contoh: prosesi preihelion
merkurius), dan keberadaan benda lainnya dapat membuat hasil hitungan tidak akurat dalam
berbagai keperluan.
Figure 1: Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses, with
focal points ƒ1 and ƒ2 for the first planet and ƒ1 and &>. (2) The two shaded sectors A1 and A2 have
the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover
segment A2. (3) The total orbit times for planet 1 and planet 2 have a ratio a13/2 : a23/2.
Gerak harmonik sederhana adalah gerak bolak – balik benda melalui suatu titik
keseimbangan tertentu dengan banyaknya getaran benda dalam setiap sekon selalu konstan
Periode (T) adalah waktu yang diperlukan benda untuk melakukan satu getaran. Satuan
periode detik
Frekuensi (f) adalah banyaknya getaran yang dilakukan oleh benda selama satu detik. Satuan
frekuensi hertz
Hubungan antara periode dan frekuensi :
atau
Hukum Hooke menyatatakan bahwa besar gaya pegas pemulih sebanding dengan
pertambahan panjang pegas. Secara matematis, dapat dituliskan sebagai :
F = – k Δx
dengan k = tetapan pegas (N/m)
Tanda (-) diberikan karena arah gaya pemulih pada pegas berlawanan dengan arah gerak
pegas tersebut.
Dari hukum II Newton didapatkan periode pegas
Persamaan Gerak Harmonik Sederhana
Y = A sin ωt
di mana :
Y = simpangan
A = simpangan maksimum (amplitudo)
ω = kecepatan sudut = 2πf
t = waktu
Kecepatan maksimumnya adalah vmax = A ω
Energi pada Gerak Harmonik Sederhana adalah E = ½ kA2
Hukum Kepler I, II dan III
1. Hukum I Kepler
Menjelaskan tentang bentuk lingkaran orbit planet. Bunyi hukum Keppler 1 berisi
sebagai berikut.
Lintasan setiap planet mengelilingi matahari merupakan sebuah elips dengan matahari
terletak pada salah satu titik fokusnya.
Nah setelah melihat hukum pertama Kepler dapat dilihat ilustrasi tersebut seperti pada
Gambar berikut ini.
Ilustrasi orbit planet sesuai hukum I Kepler
2. Hukum II Kepler
Menjelaskan tentang kecepatan orbit planet. Apa perbedaan dengan hukum kepler
pertama? Perhatikan penjelasan berikut, hukum Keppler 2 berisi sebagai berikut
Setiap planet bergerak sedemikian sehingga suatu garis khayal yang ditarik dari
matahari ke planet tersebut mencakup daerah dengan luas yang sama dalam waktu
yang sama.
Untuk lebih jelasnya silahkan amati gambar berikut
Ilustrasi orbit planet sesuai hukum ketiga Kepler
Keterangan :
Garis AM akan menyapu lurus sampai ke garis BM, luasnya sama dengan daerah
yang disapu garis Cm hingga DM. Jika tAB = tCD. Hukum kedua ini juga
menjelaskan bahwa dititik A dan B planet harus lebih cepat dibanding saat di titik C
dan D.
3. Hukum III Kepler
Menjelaskan tentang periode revolusi planet. Periode revolusi planet ini dikaitkan
dengan jari-jari orbit rata-ratanya. Perhatikan penjelasan berikut, hukum Keppler 3
berisi sebagai berikut
Kuadrat periode planet mengitari matahari sebanding dengan pangkat tiga rata-rata
planet dari matahari.
Nah, Hubungan di atas dapat dirumuskan secara matematis denga persamaan seperti
berikut ini.
THE GRAVITATIONAL FORCE BETWEEN A PARTICLE AND A SPHERICAL MASS
We have already stated that a large sphere attracts a particle outside it as if the total mass of
the sphere were concentrated at its center. We now describe the force acting on a particle
when the extended object is either a spherical shell or a solid sphere, and then apply these
facts to some interesting systems.
Spherical Shell
Case 1. If a particle of mass m is located outside a spherical shell of mass M at, for instance,
point P in Figure 14.21a, the shell attracts the particle as though the mass of the shell were
concentrated at its center. We can show this, as Newton did, with integral calculus. Thus, as
far as the gravitational force acting on a particle outside the shell is concerned, a spherical
shell acts no differently from the solid spherical distributions of mass we have seen.
Case 2. If the particle is located inside the shell (at point P in Fig. 14.21b), the gravitational
force acting on it can be shown to be zero.
We can express these two important results in the following way:
The gravitational force as a function of the distance r is plotted in Figure 14.21c.
Figure 14.21 (a) The nonradial components of the gravitational forces exerted on a particle of
mass m located at point P outside a spherical shell of mass M cancel out. (b) The spherical
shell can be broken into rings. Even though point P is closer to the top ring than to the bottom
ring, the bottom ring is larger, and the gravitational forces exerted on the particle at P by the
matter in the two rings cancel each other. Thus, for a particle located at any point P inside the
shell, there is no gravitational force exerted on the particle by the mass M of the shell. (c) The
magnitude of the gravitational force versus the radial distance r from the center of the shell.
The shell does not act as a gravitational shield, which means that a particle inside a shell may
experience forces exerted by bodies outside the shell.
Solid Sphere
Case 1. If a particle of mass m is located outside a homogeneous solid sphere of mass M (at
point P in Fig. 14.22), the sphere attracts the particle as though the mass of the sphere were
concentrated at its center. We have used this notion at several places in this chapter already,
and we can argue it from Equation 14.25a. A solid sphere can be considered to be a collection
of concentric spherical shells. The masses of all of the shells can be interpreted as being
concentrated at their common center, and the gravitational force is equivalent to that due to a
particle of mass M located at that center.
Case 2. If a particle of mass m is located inside a homogeneous solid sphere of mass M (at
point Q in Fig. 14.22), the gravitational force acting on it is due only to the mass M’
contained within the sphere of radius shown in Figure 14.22. In other words,
This also follows from spherical-shell Case 1 because the part of the sphere that is
Figure 14.22 The gravitational force acting on a particle when it is outside a uniform solid
sphere is GMm/r2 and is directed toward the center of the sphere. The gravitational force
acting on the particle when it is inside such a sphere is proportional to r and goes to zero at
the center.
farther from the center than Q can be treated as a series of concentric spherical shells that do
not exert a net force on the particle because the particle is inside them. Because the sphere is
assumed to have a uniform density, it follows that the ratio of masses M’/M is equal to the
ratio of volumes V’/V, where V is the total volume of the sphere and V’ is the volume within
the sphere of radius r only:
Solving this equation for M’ and substituting the value obtained into Equation 14.26b, we
have
This equation tells us that at the center of the solid sphere, where r = 0, the gravitational force
goes to zero, as we intuitively expect. The force as a function of r is plotted in Figure 14.22.
Case 3. If a particle is located inside a solid sphere having a density ρ that is spherically
symmetric but not uniform, then M’ in Equation 14.26b is given by an integral of the form M’
= ∫ρ dV, where the integration is taken over the volume contained within the sphere of radius
r in Figure 14.22. We can evaluate this integral if the radial variation of ρ is given. In this
case, we take the volume element dV as the volume of a spherical shell of radius r and
thickness dr, and thus dV = 4πr2 dr. For example, if ρ = Ar, where A is a constant, it is left to a
problem (Problem 63)
Areal velocity
From Wikipedia, the free encyclopedia
Areal velocity is the area (shown in green) swept out per unit time by a particle moving along a curve
(shown in blue).
In classical mechanics, areal velocity (also called sector velocity or sectorial velocity) is the
rate at which area is swept out by a particle as it moves along a curve. In the adjoining figure,
suppose that a particle moves along the blue curve. At a certain time t, the particle is located
at point B, and a short while later, at time t + Δt, the particle has moved to point C. The area
swept out by the particle is the green area in the figure, bounded by the line segments AB and
AC and the curve along which the particle moves. The areal velocity equals this area divided
by the time interval Δt in the limit that Δt becomes vanishingly small.
Illustration of Kepler's second law. The planet moves faster near the Sun, so the same area is swept
out in a given time as at larger distances, where the planet moves more slowly.
The concept of areal velocity is closely linked historically with the concept of angular
momentum. Kepler's second law states that the areal velocity of a planet, with the sun taken
as origin, is constant. Isaac Newton was the first scientist to recognize the dynamical
significance of Kepler's second law. With the aid of his laws of motion, he proved in 1684
that any planet that is attracted to a fixed center sweeps out equal areas in equal intervals of
time. By the middle of the 18th century, the principle of angular momentum was discovered
gradually by Daniel Bernoulli and Leonhard Euler and Patrick d'Arcy; d'Arcy's version of the
principle was phrased in terms of swept area. For this reason, the principle of angular
momentum was often referred to in the older literature in mechanics as "the principle of
areas." Since the concept of angular momentum includes more than just geometry, the
designation "principle of areas" has been dropped in modern works.
Connection with angular momentum
In the situation of the first figure, the area swept out during time period Δt by the particle is
approximately equal to the area of triangle ABC. As Δt approaches zero this near-equality
becomes exact as a limit.
Let the point D be the fourth corner of parallelogram ABDC shown in the figure, so that the
vectors AB and AC add up by the parallelogram rule to vector AD. Then the area of triangle
ABC is half the area of parallelogram ABDC, and the area of ABDC is equal to the magnitude
of the cross product of vectors AB and AC. This area can also be viewed as a vector with this
magnitude, pointing in a direction perpendicular to the parallelogram; this vector is the cross
product itself:
Hence
The areal velocity is this area divided by Δt in the limit that Δt becomes vanishingly small:
But, is the velocity vector of the moving particle, so that
On the other hand, the angular momentum of the particle is
and hence the angular momentum equals 2m times the areal velocity.
Conservation of areal velocity is a general property of central force motion.[1]
The Universal Law of Gravitation
Gravity is described from the point of view of a universal law. This implies that gravity is a
force that should behave in similar ways regardless of where you are in the universe.
Gravity
It's a force of attraction that exists between any two objects that have mass. The more
mass they have, the greater the force of attraction. The closer they are, the greater the
force of attraction.
For most objects you get near every day, the force of attraction is so incredibly small that you
would never notice the force. Gravity is a very weak force, so between common objects like
you and your pencil, the force of attraction is very small because your mass and the mass of
your pencil are small. We only get noticeable amounts of gravity when at least one object is
very massive... like a planet. The force of attraction between you and the planet Earth is a
noticeable force! We call the force of attraction between you and the Earth, your weight.
Weight is another name for the force of gravity pulling down on you or anything else.
G is the universal gravitational constant.
It is
basically a conversion factor to adjust the number
and units so they come out to the correct value.
This is a universal constant so it is true
everywhere.
m1 is the mass of one of the objects.
m2 is the mass of the other object.
r is the radius of separation between the center of
masses of each object.
FG is the force of attraction between the two
objects.
Important Concepts
The direction of the force is not given by this formula since there are actually two
forces equal in size but opposite in direction. This formula calculates them both.
The formula is an inverse square law for radius of separation (notice the r2 on the
bottom of the equation). This means that if you double the separation you quarter the
force, or if you cut the separation in half you quadruple the force of attraction.
If you double a single mass, you double the force. If you cut one of the masses in
half, you cut the force in half. But if you double both masses you would quadruple
the force.
Common Misconceptions
"There is no gravity in space." FALSE If there were no gravity in space, the space
shuttle would not be able to orbit the Earth, the moon would not orbit the Earth, and
the Earth would not orbit the Sun. The reason we tend to think of there being no
gravity in space is that we have seen movies of the astronauts being "weightless".
They aren't actually weightless, they are still being pulled down by gravity but they
and the space shuttle are in a constant state of freefall around the Earth. So they seem
to be weightless as a result of the falling - just as you would seem weightless if you
were in an elevator when the cable broke.
"G and g are the same thing." FALSE Big G is the universal gravitational
constant. Little g is the acceleration due to the force of gravity and its value of
9.8m/s/s down is only true on this planet. It is not a universal constant.
"g is gravity." FALSE Little g is the effect of the force of gravity, but is not
gravity. Gravity is a force, little g is an acceleration caused by gravity.
©1998 Science Joy Wagon