Chapter 13: Universal Gravitation

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 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. 13 that these orbits are circular.

• Ch. 13 fits THE COURSE THEME OF NEWTON'S LAWS OF MOTION because he used his Gravitation Law & his 2nd Law in his

analysis of planetary motion. • His prediction that planet 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,

this 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. • This also gave Newton his major “claim to fame”. After this, he was

considered a “major player” 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

• 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. 13.1: Newton’s Universal Law of Gravitation

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

This must be true from

Newton’s 3rd Law

• 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 and inversely proportional to the square of the distance between them:

F12 = -F21 [(m1m2)/r2]

Direction of this force:

Along the line joining the 2 masses

Must be true from Newton’s 3rd Law

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

G a constant, the Universal Gravitational Constant

G is measured & is the same for ALL objects. G must be small!

1 22g

mmF G

r

• 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 ends of a light horizontal rod. Two larger masses were placed near the smaller ones.

• The angle of rotation is measured.

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

Measurement

Apparatus

F = G[(m1m2)/r2]• G = the Universal Gravitational Constant

• Measurements Find, in SI Units:

G = 6.673 10-11 N∙m2/kg2

• The force given above is strictly valid only for:

– Very small masses m1 & m2 (point masses)

– Uniform spheres

• For other objects: 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?

1 212 122

ˆmmG

rF r

12F

21F

Comments

1 212 122

ˆmmG

rF r

12 21F F

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

More Comments

1 212 122

ˆmmG

rF r

• Gravitation 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

• 3 billiard balls, masses m1 = m2 = m3 = 0.3 kg are 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: The 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

Using components, Fx = F21x + F31x

Fy = F21y + F31y

Example 13.1: Billiards