3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a...

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3- Newton's law of gravity ة ل ا ق ث ل ل ن ت و ي ن ون ن ا ق

Transcript of 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a...

Page 1: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

3- Newton's law of gravity

للثقالة نيوتن قانون

Page 2: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

Galileo Galilei(1564-1641)

Using a telescope he Using a telescope he made, Galileo made, Galileo observed:observed:

Moons of Jupiter.Moons of Jupiter.Phases of Venus.Phases of Venus.

His findings supported His findings supported a Copernican model.a Copernican model.He spent the end of He spent the end of his life under “house his life under “house arrest” for his beliefs.arrest” for his beliefs.

Page 3: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

Johannes Kepler German astronomer (1571–1630)

Kepler has try to deduce a mathematical model for the motion of the planets.

Page 4: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

Isaac Newton (1642-1727).

Page 5: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• " Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them ".

Page 6: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

يجذب • الكون في جسيم أيتتناسب بقوة آخر جسيم أيكتلتيهما مضروب مع & طرديا

مربع مع عكسيا وتتناسب. بينهما فيما المسافة

Page 7: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

If the particles have masses m1 and m2 and are separated by a distance r, the magnitude of this gravitational force is:

الجسيمان كتلة كانت و m1إذاm2 مسافة يفصالهما ، rوكان

مقدار التثاقل فإن : يكون قوة

2

21

g r

mm GF

Page 8: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• where G is a universal constant called the universal gravitational constant which has been measured experimentally.

• The value of G depends on the system of units used, its value in SI units is:

يسمى Gحيث عام العام ثابت التثاقل ثابت . قيمة تعتمد & معمليا مقاس نظام Gوهو على

النظام في وقيمته المستخدمة الوحدات:الدولي

G = 6.672 x 10-11 N. m2 / kg2

Page 9: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• The force law is an:

inverse-square law

العكسي التربيع قانون

because the magnitude of the force varies as the inverse square of the separation of the particles.

Page 10: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• We can express this force in vector form اتجاهي by شكلdefining a unit vector

r12 الوحدة متجه

Page 11: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• Because this unit vector is in the direction of the displacement vector r12 directed from m1 to m2, the force exerted on m2, by m1 is :

F21 = - (G ( m1 m2 ) / r122 ) r12

Page 12: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• Likewise, by Newton's third law the force exerted on m1 by m2, designated F12, is equal in magnitude to F21 and in the opposite direction.

Page 13: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• That is these forces form an action-reaction pair ورد الفعل قوى من زوجالفعل

F12 = F21

Page 14: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• the gravitational force exerted by a finite-size, spherically symmetric mass distribution on a particle outside the sphere is the same as if the entire mass of the sphere were concentrated at its center

كتلة • أن لو كما القوى تعملمركزها الكرة في .مركزة

Page 15: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• For example, the force exerted by the Earth on a particle of mass m at the Earth's surface has the magnitude

Fg = G ( mE m ) / RE2

mE is the Earth's mass األرض and كتلةRE is the Earth's radius قطر نصف األرض

Page 16: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• This force is directed toward the center of the Earth

األرض مركز نحو موجهة

Page 17: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• At points inside the earth:

We would find that the force decreases as we approach the center.

Exactly at the center the gravitational force on a body would be zero.

Page 18: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

4-Measurement of the gravitational constantالعام قياس التثاقل ثابت

The universal gravitational constant, G, was measured by Henry Cavendish in 1798

Page 19: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

5- Weight and gravitational forceالتثاقل وقوة الوزن

If g is the magnitude of the free-fall acceleration, and since the force on a freely falling body of mass m near the surface of the Earth is given by

F = m g, we can equate

m g = G ( mE m / RE2 )

Page 20: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

2

E

E

R

mGg

Page 21: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.
Page 22: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

Using the facts that g = 9.80 m/s2 at the Earth's surface and RE = 6.38 x 10)6( m, we find that

mE = 5.98 x 10)24( kg.

Page 23: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• From this result, the average density of the Earth is calculated to be :

• ρE = mE / VE = mE / ( 4/3 π RE3 )

• = 5.98 x 10 24 / ( 4/3 π 6.38 x 106 m )3

= 5500 kg/m3 = 5.5 g/cm3

Page 24: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• Since this value is higher than the density of most rocks at the Earth's surface (density of granite =

3 g/cm3), • we conclude that the inner core

of the Earth has a density much higher than the average value.

من • أعلى قيمتها الكثافة تلك أن بماعلى الصخرية المواد معظم كثافة

أن ذلك من نستنتج فإننا األرضكثافة له لألرض الداخلي القلب

لكثافة المتوسطة القيمة من أعلى األرض.

Page 25: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• The magnitude of the gravitational force acting on this mass is:

• Fg = G ( ME m / r2 )

= G ( ME m / ( RE + h )2 )

Page 26: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• If the body is in free-fall, then

Fg = mg' and we see that g', the free-fall acceleration experienced by an object at the altitude h, is

g' = G mE / r2

= G mE / ( RE + h ) 2

Page 27: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• Thus, it follows that g' decreases with increasing altitude الجاذبية تقل عجلة

األرض سطح عن ارتفعنا . كلما

• Since the true weight of an object is mg , we see that as r →∞, the true weight approaches zero.

Page 28: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

6- The Gravitational Field مجال التثاقل

• When a particle of mass m is placed at a point where the field is the vector g, the particle experiences a force Fg = m g.

Page 29: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• the gravitational field is defined by:

g = Fg / m

Page 30: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• consider an object of mass m near the Earth's surface. The gravitational force on the object is directed toward the center of the Earth and has a magnitude (m g).

Page 31: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• Since the gravitational force on the object has a magnitude :

(G mE m) / r2

Page 32: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

field g is

rr

mG

m

Fg

2

Eg

where r is a unit vector pointing radially outward from the Earth, and the minus sign indicates that the field points toward the center of the Earth and is always opposite to r

Page 33: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• We have used the same symbol الرمز g for gravitational field نفس

magnitude that we used earlier for the acceleration of free fall.

The units of the two quantities are the same نفس لهما الكميتان. الوحدات

Page 34: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.
Page 35: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

Example• A ring-shaped body with radius a

has total mass M. Find the gravitational field at point p, at a distance x from the center of the ring, along the line through the center and perpendicular to the plane of the ring.

Page 36: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.
Page 37: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• We imagine the ring as being divided into small segments Δs, each with mass ΔM. At point P each segment produces a gravitational field Δg with magnitude.

Page 38: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• Δg = (G ΔM) / r2 = (G ΔM) / (x2 + a2)• The component of this field along the x-

axis is given by : Δgx = - Δg cosφ =• - G ΔM . x • x2 + a2 (x2 + a2) ½

• = - G ΔM x• (x2 + a2)3/2

Page 39: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• we simply sum all the ΔM 's. This sum is equal to the total mass M.

Page 40: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

23

22x

)ax(

xMGg

Page 41: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

7-Gravitatiuonal Potential energy الوضع للجاذبية طاقة

• we know that the earth's gravitational force on a body of mass m, at any point outside the earth, is given by

• w = fg = (G m mE ) / r2

Page 42: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

We compute the work Wgrav done by the gravitational force when r changes

from r1 to r2

2

1

r

r

rgravdrFW

Page 43: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

Thus Wgrav is given by:

1

E

2

E

r

r

2Egrav r

mmG

r

mmG

r

drmmGW

2

1

Page 44: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• Wgrav = U1 - U2 where U1 and U2 are the potential energies of positions 1 and 2 . So Comparing this with the eq. of Wgrav gives:

Page 45: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

r

mmGU E

Page 46: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

8- Kepler's laws كبلر قوانين

•The complete analysis is summarized in three statements, known as Kepler's laws:

Page 47: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• l. All planets move in elliptical orbits with the Sun at one of the focal points. شكل على مدارات في تتحرك الكواكب كل

بؤرتيه إحدى في الشمس تقع ناقص .قطع

Page 48: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• 2. The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals.

و • الشمس بين ما المخطوط القطري المتجهأزمنة في متساوية مساحات يمسح ما كوكب

.متساوية

Page 49: 3- Newton's law of gravity قانون نيوتن للثقالة Galileo Galilei (1564-1641) Using a telescope he made, Galileo observed: Moons of Jupiter. Phases of Venus.

• 3. The square of the orbital period of any planet is proportional the cube of the semi major axis of the elliptical orbit.

مع • يتناسب كوكب ألي الدوري الزمن ربعشكل على الذي للمدار األفقي المحور مكعب

ناقص . قطع