chapter 13: universal gravitation this cartoon mixes two legends: 1. the legend of newton, the apple...
TRANSCRIPT
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