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 2 nd 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 2 nd 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 3 rd 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: F 12 = -F 21  [(m 1 m 2 )/r 2 ] Direction of this force:  Along the line joining the 2 masses  Must be true from Newton’s 3 rd 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! Measurement of G in the lab is tedious & sensitive because it is so small. –First done by Cavendish in 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 2 nd Law to get vector force between the masses. Relate to angle of rotation & can extract G.  Measurement Apparatus

F = G[(m 1 m 2 )/r 2 ] G = the Universal Gravitational Constant Measurements Find, in SI Units: G =  N∙m 2 /kg 2 The force given above is strictly valid only for: –Very small masses m 1 & m 2 (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?

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 3 rd Law action-reaction pair, as expected. The negative sign in the above vector equation tells us that particle 2 is attracted toward particle 1

More Comments 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 m 1 = m 2 = m 3 = 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 m 1 due to m 2 & m 3. Note: The gravitational force is a vector, so we have to add the vectors F 21 & F 31 to get the vector F (using the vector addition methods of earlier). F = F 21 + F 31 Using components, F x = F 21x + F 31x F y = F 21y + F 31y Example 13.1: Billiards