Ch. 6: Gravitation & Newton’s Synthesis

This cartoon mixes two legends: 1. The legend of Newton, the apple & gravity which led to the

Universal Law of Gravitation.

2. The legend of William Tell & the apple.

• It was very SIGNIFICANT & PROFOUND in the 1600's when Sir Isaac Newton first wrote

Newton's Universal Law of Gravitation! This was done at the young age of about 30. It was this, more than any of his other achievements,

which caused him to be well-known in the world scientific community of the late 1600's.

• He used this law, along with Newton's 2nd Law (his 2nd Law!) plus Calculus, which he

also (co-) invented, to PROVE that the orbits of the planets around the sun must be ellipses. – For simplicity, we will assume in Ch. 6 that these orbits are circular.

• Ch. 6 fits THE COURSE THEME OF NEWTON'S LAWS OF MOTION because Newton used his Gravitation Law & his 2nd Law in his analysis of planetary motion.

• His prediction that planetary orbits are elliptical is in excellent agreement with Kepler's analysis of observational data & with Kepler's empirical laws of planetary motion.

• When Newton first wrote the

Universal Law of Gravitation, it was the first time, anyone had EVER written a theoretical expression (physics in math form) & used it to PREDICT something that is in agreement with observations! For this reason,

Newton's Formulation of his Universal Gravitation Law is considered

THE BEGINNING OF THEORETICAL PHYSICS. • It also gave Newton his major “claim to fame”. After this, he was considered to

be a “major leader” in science & math among his peers.

• In modern times, this, plus the many other things he did, have led to the consensus that Sir Isaac Newton was the

GREATEST SCIENTIST WHO EVER LIVED

Newton’s Grave & a

Monumentto him are in

Westminster Abbeyin

London, England.

Inscription on Newton’s Gravestone:“Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colors thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25th December, 1642, and died on 20th March 1727.

Newton’s Monument in Westminster Abbey.

• This is an EXPERIMENTAL LAW describing the gravitational force of attraction between 2 objects.

• Newton’s reasoning:

the Gravitational force of attraction between 2 large objects (Earth - Moon, etc.) is the SAME force as the attraction of objects to the Earth.

• Apple story: This is likely not a true historical account, but the reasoning discussed there is correct. This story is probably legend rather than fact.

Sect. 6-1: Newton’s Universal Law of Gravitation

If the force of gravity is being exerted on objects on Earth, What is the Origin of that Force?

Newton’s realization was that the force must come from the Earth itself! He further realized that this same force must

be what keeps the Moon in its orbit!

The gravitational force on you is half of a Newton’s 3rd Law pair: Earth exerts a downward force on you, & you exert an upward force on Earth. When there is such a large difference in the 2 masses, the reaction force (the force you exert on the Earth) is undetectable, but for 2 objects with masses closer in size to each other, it can be significant.

This must be true from

Newton’s 3rd Law!

The Force of Attraction between 2 small masses is the same as the force between Earth & Moon, Earth & Sun, etc.

By observing planetary orbits, Newton also concluded that the gravitational force decreases as the inverse of the square of the distance r between the masses.

Newton’s Universal Law of Gravitation: “Every particle in the Universe attracts every other particle in the Universe with a force that is proportional to the product of their masses &

inversely proportional to the square of the distance between them:

F12 = -F21 [(m1m2)/r2]

The direction of this force: Along the line joining the 2 masses

This must be true from

Newton’s 3rd Law!

Newton’s Universal Gravitation Law• This force is written as:

G a constant,

the Universal Gravitational Constant

G is measured & is the same for ALL objects. G must be small! • The measurement of G in the lab is tedious &

sensitive because it is so small. – First done by Cavendish in 1789.

• Modern version of Cavendish experiment: Two small masses are fixed at the ends of a light horizontal rod. Two larger masses are placed near the smaller ones.

• The angle of rotation is measured.

• Use Newton’s 2nd Law to get the vector force between the masses. Relate to angle of rotation & can extract G.

Cavendish Measurement

Apparatus

• G = the Universal Gravitational Constant

• Measurements Find, in SI Units:

• The force given above is strictly valid only for:

– Very small masses m1 & m2 (point masses)

– Uniform spheres

• For other objects: We need integral calculus!

• The Universal Law of Gravitation is an example of an Inverse Square Law– The magnitude of the force varies as the inverse square of the

separation of the particles

• The law can also be expressed in vector form

The negative sign means it’s an attractive force• Aren’t we glad it’s not repulsive?

Comments

Force exerted by particle 1 on particle 2

Force exerted by particle 2 on particle 1

This tells us that the forces form a Newton’s 3rd Law

action-reaction pair, as expected.

The negative sign in the above vector equation tells us thatparticle 2 is attracted toward particle 1

21

F21 = - F12

More Comments

• Gravity is a “field force” that always exists between two masses, regardless of the medium between them.

• The gravitational force decreases rapidly as the distance between the two masses increases– This is an obvious consequence of the

inverse square law

A 50-kg person & a 70-kg person are sitting on a bench close to each other. Estimate the magnitude of the gravitational force each exerts on the other.

Example 6-1: Gravitational Force Between 2 People

Example 6-2: Spacecraft at 2rE

m

ME

• Spacecraft at twice the Earth radius

Earth Radius: rE = 6320 km

Earth Mass: ME = 5.98 1024 kg

Example 6-2: Spacecraft at 2rE

m

ME

• Spacecraft at twice the Earth radius

Earth Radius: rE = 6320 km

Earth Mass: ME = 5.98 1024 kg

FG = G(mME/r2)

• At surface (r = rE)

FG = weight

= mg = G[mME/(rE)2]

• At r = 2rE

FG = G[mME/(2rE)2]

= (¼)mg = 4900 N

Example 6-3: Force on the MoonFind the net force on the

Moon due to the gravitational

attraction of both the Earth &

the Sun, assuming they are at

right angles to each other.

ME= 5.99 1024kg

MM=7.35 1022kg

MS = 1.99 1030 kg

rME = 3.85 108 m

rMS = 1.5 1011 m

F = FME + FMS

(vector sum!)

F = FME + FMS

(vector sum!)

FME = G [(MMME)/(rME)2]

= 1.99 1020 N

FMS = G [(MMMS)/(rMS)2]

= 4.34 1020 N

F =[ (FME)2 + (FMS)2]

= 4.77 1020 N

tan(θ) = 1.99/4.34

θ = 24.6º

Gravitational Force Due to a Mass Distribution• In can be shown, with integral calculus, that:

The gravitational force exerted by a

SPHERICALLY SYMMETRICmass distribution of uniform density on a particle outside the distribution is

the same as if the entire mass of the distribution were concentrated at the center.• So, assuming that the Earth is such a sphere, the gravitational force exerted by the Earth on a particle of

mass m on or near the Earth’s surface is

FG = G[(mME)/r2]; ME Earth Mass, rE Earth Radius

• Similarly, to treat the gravitational force due to large spherical shaped objects, it can be shown with calculus, that: 1) If a (point) particle is outside a thin spherical shell, the gravitational force on the particle is the same as if all the mass of the sphere were at center of shell. 2) If a (point) particle is inside a thin spherical shell, the gravitational force on the particle is zero. So, we can model a sphere as a series of thin shells. For a mass outside any large spherically symmetric mass, the gravitational force acts as though all the mass of the sphere is at the sphere’s center.

In vector form,

The figure gives the directions of the displacement & force vectors.

If there are many particles, the total force is the vector sum of the individual forces:

Sect. 6-2: Vector Form of Universal Gravitation Law

• 3 billiard (pool) balls, masses m1 = m2 = m3 = 0.3 kg on a table as in the figure. Triangle sides: a = 0.4 m, b = 0.3 m,

c = 0.5 m. Calculate the magnitude & direction of the total gravitational force F on m1 due to m2 & m3.

Note: Gravitational force is a vector, so we have to add the vectors F21 & F31 to get the vector F (using the vector addition methods of

earlier). F = F21 + F31

Example: Billiards (Pool)

Using components: Fx = F21x + F31x = 0 + 6.67 10-11 N Fy = F21y + F31y = 3.75 10-11 N + 0

So, F = [(Fx)2 + (Fy)2]½ = 3.75 10-11 Ntanθ = 0.562, θ = 29.3º