Isaac Newton AST101 Lecture 4 – Feb 3, 2003

Isaac Newton was born in 1642, the year Galileo died and 12 years after Kepler’s death. From age 27, Newton was Lucasian Professor of Mathematics at Cambridge University. Newton was a shy and solitary person who shunned controversy – his major discoveries went unpublished for nearly 20 years.

Newton’s achievement span a huge array of topics and include:

Discovering the binomial theorem

Inventing calculus

Experimenting with light and developing the theory of optics

The Law of Gravitation

Enunciating the laws of mechanics governing all motion

Inventing the first reflecting telescope (mirrors instead of lenses)

Experiments in alchemy (probably getting mercury poisoning, and leading to episodes of insanity)

No one would argue with a claim for Newton as one of the most productive and influential scientists ever.

Many would nominate him as Man of the Millenium in Science

1. The planets move in ellipses, with the Sun at one focus.

2. The line from the Sun to the moving planet sweeps out equal areas in equal times.

3. The square of the planet’s orbital period (P) is proportional to the cube of the semi-major axis of the ellipse.

Newton knew of course of Galileo’s and Kepler’s work, and understood that what was lacking was understanding of the cause behind the observed motions of the planets. Kepler’s suggested that the Sun somehow reached out with invisible paddles to guide the planets around their orbits.

Galileo thought that the natural state of motion was for bodies to travel in circles.

Newton sought a more comprehensive theory that could explain planets’ orbits, falling apples and cannonball trajectories from a single set of principles. This theory was worked out in 1665 – 1667 while Cambridge classes were suspended due to the bubonic plague. But only in 1687 were his results published in Philosophiae Naturalis Principia Mathematica – the cornerstone of physics and astronomy ever since.

In this work, he introduced the essential ingredient of using mathematics to describe the physical world, and he provided the explanation for Kepler’s Laws:

Summary of Newton’s laws of motion and law of gravitation:1. Vectors and scalars

Many quantities we use to describe the world are simple numbers – the size of the number matters, but there is no sense of direction associated.

Examples are Mass (or weight), Time, Temperature, or Energy (for example heat, kinetic energy of motion, potential energy due to position)

These quantities are called SCALARS

Other quantities called VECTORS have an essential component of direction. For example Displacement refers to the location of an object relative to some origin or starting place.

**

The direction of a displacement is essential to know – if you travel for 10 miles north from campus, you have a quite different (wetter) experience than if you travel 10 miles west!

East

North

The location of the tree relative tothe origin (x=0, y=0) is given by the red DISPLACEMENT vector. Both the magnitude or distance (200 m) and the direction (30O north of east) are essential to telling where to find the tree.

200 m

30O

For VECTORS, both magnitude and direction must be specified.

Examples of vectors:

Displacement – locating an object relative to another, or relative to the origin of our coordinate system. Often denote by r . The → sign is used to tell us the quantity is a vector.

Velocity – the rate at which the displacement changes with time. So, if the change in r is r over a time interval t, we define the velocity as

v = r/tposition at time t1

position at time t2

difference in position = r . The time difference is t = t1-t2. The velocity is v = r/t

Case (a) is like the car accelerating from the stop light

Case (b) is what planets in circular orbit do

(A more complex combination of the two ways is possible)

Define ‘speed’ as magnitude of velocity (speed is a scalar)

What happens if velocity changes? For example when your car starts from v = 0 at a stop light and increases as you hit the gas pedal. You ACCELERATE .

Acceleration = rate of change of velocity

But velocity is a vector, so acceleration must be vector too:

acceleration = a = v/t

There are two basic ways to get acceleration:

a) change the magnitude of velocity, but keep velocity in same direction

b) change the direction of velocity, but keep magnitude the same

Acceleration:

R

R

v1

v2

v1v2

v = change in velocity

Even though magnitude of velocity does not change here, the direction does. v is in direction toward center of circle. Can show with geometry that the magnitude of the acceleration is:

a = v2/R (Centripetal acceleration)

A planet or moon in circular orbit is accelerating (continually ‘falling’ toward center) !

Motion in a circle with constant speed (for example, moon around Earth)

Velocity at time t1 =

v1; Velocity at time t2

= v2

Radius of circle = R

Newton’s Laws of Motion

1. If no force is applied to a body, its velocity does not change. (If originally at rest, stays at rest; if moving it continues at same magnitude and direction of velocity.)

2. If a force is applied to a body of mass m, it is accelerated with

F = m a

3. For any pair of objects, the force exerted on the second by the first is equal in magnitude but opposite in direction to the force exerted by the first on the second.

1. If no force is applied to a body, its velocity does not change. (If originally at rest, stays at rest; if moving it continues at same velocity.)

Newton understood what Aristotle and Galileo did not – that the natural state of motion is to continue in a straight path if there are no external influences (forces). Not to come to rest (Aristotle), and not to move in circles (Galileo). The first law is sometimes called the law of inertia.

2. If a force is applied to a body of mass m, it is accelerated with

F = m a .

In the 2nd law, we should consider a Force to be the cause; an acceleration to be the effect or result; and the mass the body experiencing the force to determine the size of the effect.

Mass: a property inherent in every object (including planets, stars and us). For objects on earth, the mass is related to the weight. It basically determines the resistance of a body to acceleration.

Force: a push or pull, just as in common language usage. A force is a vector since it matters whether the push is to the right or left. There are different types of forces (a push with your arm, friction, electrical, gravitational etc.). Although some forces occur when two objects are in contact, others like gravity or the electric force act even when the objects are at a distance from each other.

For centripetal acceleration of a planet in orbit, the force must be toward the center of the circle. The planet is in continual ‘free fall’ around the sun.

The 2nd law says that the direction of an acceleration is always in the direction of the force. Thus a forward force on a car due to the action of the engine on the wheels produces a forward acceleration.

For a fixed mass M, a force F2 = 2 F1 will produce twice the acceleration as that of F1

F1

a1

F2 = 2F1

a2 = 2 a1

Mass M Mass M

F

a1

Mass M Mass 2M

F

a2 = ½ a1

For a fixed force F, a mass 2M will receive half the acceleration as a mass M

F = m a Consequences of the Second Law

Show that the second law implies the first (remember that zero acceleration means the velocity is not changing)

3. For any pair of objects, the force exerted on the second by the first is equal in magnitude but opposite in direction to the force exerted by the first on the second.

F1 F2

F1 is the force that the triangle exerts on the star

F2 is the force that the star exerts on the triangle

Newton’s 3rd law tells us that the magnitudes of F1 and F2 are the same, but that they are opposite (directed along the same line but in opposite directions).

Thus, while the Sun attracts the Earth, the Earth also attracts the Sun with an equal but opposite force.

In astronomy, the gravitational force is of primary importance.

Newton also determined the character of the GRAVITATIONAL FORCE.

The gravitational force, Fgrav , between two bodies of masses M1 and M2 depends also on the distance, r, separating them (actually the distances between their centers).

G is a constant that is determined only by experiment. It is called Newton’s constant of Gravitation. Its value is found to be G = 6.67 x 10-11 Nm2/kg2 but you will not need to remember it for this course. (It is a small number.)

The direction of the gravitational force is attractive, directed along the line separating the two masses.

Force on M1 due to M2

Force on M2 due to M1

M1 M2

Forces on M1 and on M2 are equal in magnitude and opposite in direction (Newton’s 3rd law)

Fgrav = G M1M2 /r2

r

No contact of the two masses is needed; gravity force operates at a distance.

The strength of the force is proportional to the product of the masses of the two bodies.

Fgrav The strength of the force diminishes as the separation increases (falls off like 1/r2).

The Newton constant G is small, so to get appreciable force, we need very large masses (e.g. the Sun, the Earth etc.). The force between two people standing 1 m apart is only 3 x 10-10 times the force exerted on either person by the Earth (their weight).

Example:

A stone falling toward Earth’s surface is responding to the Gravitational force between Earth and stone.

F (on stone) = G MEMS /R2

ME is mass of earth; MS is mass of stone; R is the distance from the center of Earth to the stone. For heights not far above the surface of the earth, R is approximately the radius of the Earth. (The radius of Earth is 6.4 x 106 m, so raising a stone to 6 m is a negligible distance compared to radius.

For any such stone, G, ME and R are constants and we can lump them into one constant called g .

F = MS g , with g = G ME /R2

Compare this with Newton’s 2nd law F = MSa and we learn that g is just the acceleration experienced by any falling body near the surface of the Earth.

Measuring g in the lab, knowing G, and knowing R from mapping the Earth – we can turn the equation for g around to get:

ME = g R2/G Thus experiments on falling bodies tell us the mass of the Earth!

Example: Derive Kepler’s 3rd Law

Consider a planet in orbit around the sun (approximate orbit as circle for simplicity). Call the mass of the Sun M1, and mass of planet M2.

r

M2

M1

F = GM1M2/r2 = M2 a = M2 v2/r

Don’t measure v directly, but v = (orbit circumference) / period: v = 2r/P. Substitute:

GM1M2/r2 = M2 (2r)2 / (P2 r).

Solve this equation for P2 to get:

P2 = r3 G M1

This looks like Kepler’s 3rd law since r for a circle is the semi-major axis.

Newton made one essential improvement: In fact, by Newton’s 3rd law, planet also exerts force on Sun, so the Sun also moves. Both planet and Sun orbit a common point called the center of mass. This leads to the change in Kepler’s 3rd law by Newton:

P2 = r3 G(M1 +M2)

For earth – Sun, the earth’s mass is negligible and Kepler’s form is OK. But if the ‘planet’ were as massive as Sun, need Newton’s fix.

Do the algebra to show this →

Can simplify by choosing appropriate units for period P, masses M and semi-major axis a:

For Earth ( ) around Sun ( )

period P = 1 year

semi-major axis a = 1 AU

M1 + M2 = M

So using units with P in years, a in AU and Masses in solar masses,

P2 = a3 /(M1+M2)

Newton says that (units are seconds, meters and kilograms)

or P2 ~ a3 /(M1+M2) in any set of units.

P2 = r3 G(M1 +M2)

.

Shorthand symbols for Earth and Sun

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Newton further showed that the Gravitational force requires two orbiting objects to both move in ELLIPSES of same eccentricies, with a COMMON focus located at their fixed center of mass (c.m.) (Circle is special case)

Center of mass like the fulcrum of a see-saw: M

1

M2

d1 d2

M1/M2 = d2/d1

Larger mass is closer to c.m. For Sun and planet, c.m. is inside the Sun. Motion on elliptical orbits is faster for the lighter object on the larger ellipse.

The relative orbit (e.g. the Earth’s orbit seen from the Sun) is also an ellipse with same eccentricity as either absolute orbit. It is this relative orbit for which P2 = a3 /(M1+M2)