1 / 27

Apples and Planets

Apples and Planets. PTYS206-2 28 Feb 2008. List of Symbols. F, force a, acceleration (not semi-major axis in this lecture) v, velocity M, mass of Sun m, mass of planet d, general distance r,radius of circle, semi-major axis of orbit R, radius of Earth. Newton’s Laws.

oksana
Download Presentation

Apples and Planets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Apples and Planets PTYS206-2 28 Feb 2008

  2. List of Symbols • F, force • a, acceleration (not semi-major axis in this lecture) • v, velocity • M, mass of Sun • m, mass of planet • d, general distance • r,radius of circle, semi-major axis of orbit • R, radius of Earth

  3. Newton’s Laws • Newton devised a uniform and systematic method for describing motion, which we today refer to as the Science of Mechanics. It remains the basic description of motion, requiring correction only at very high velocities and very small distances. • Newton summarized his theory in 3 laws: • An object remains at rest or continues in uniform motion unless acted upon by a force. • Force is equal to mass x acceleration (F=ma) • For every action there is an equal and opposite reaction.

  4. Newton and Gravity Cambridge was closed because of the Plague. As the story goes, Newton was sitting under the apple tree outside his farmhouse (shown right) and while watching the apples fall he realized that the force that made the apples fall also made the planets orbit the sun. Using his newly invented Calculus, Newton was able to show that Kepler’s 3 laws of planetary motion followed directly from this hypothesis. Link for animation

  5. Falling Apples and Orbiting Planets Splat What do these have in common?

  6. Newton’s cannonball From Principia

  7. Apples and Planets We will know analyze the motion of terrestrial falling bodies and orbiting planets in more detail. We will analyze both phenomenon in the same way and show that Newton’s theory explains both. The plan is to combine Newton’s second law with Newton’s law of gravitation to determine the acceleration. The interesting thing here is that we are applying laws determined for motion on Earth to the motion of heavenly bodies. What an audacious idea!

  8. Gravitational Force: Units According to Newton’s 2nd law, Force=mass x acceleration The units must also match. Units of mass = kilograms Units of acceleration = meters/sec2 Unit of force must be kilograms-meters/sec2 = kg m s-2 (shorthand) We define a new unit to make notation more simple. Let’s call it a Newton. From the definition we can see that 1 Newton = 1 kg m s-2 From now on we measure force in Newtons.

  9. What are the units of G? Newton’s law of gravitation F = GMm/d2 Let’s solve for G (multiply by d2, divide by Mm) G = Fd2/Mm Examine the units Fd2/Mm has units of N m2/kg2 or N m2 kg-2 Or, expressing Newtons in kg, m, and s (1 N = 1 kg m s-2) Fd2/Mm has units of N m2 kg-2 = (kg ms-2)m2 kg-2= m3 s-2 kg-1 G has units of m3 s-2 kg-1 Numerically, G = 6.6710-11m3 s-2 kg-1

  10. Newton’s Second Law Newton’s Law of Gravity Force = mass x acceleration F = ma • All bodies exert a gravitational force on each other. • The force is proportional to the product of their masses and inversely proportional to the square of their separation. F = GMm/d2 where m is mass of one object, M is the mass of the other, and d is their separation. • G is known as the constant of universal gravitation.

  11. Falling Apples: Gravity on Earth F = m a = G m M/ R2 F = m a = G m M/ R2 (cancel the m’s) a = G M/ R2 where: G = 6.67x10-11 m3kg-1s-2 M= 5.97x1024 kg On Earth’s surface: R = 6371 km Thus: a = G M/ R2 = 9.82 m s-210 m s-2 a on Earth is sometimes called g. • The separation, d, is the distance between the centers of the objects.

  12. Newton Explains Galileo The acceleration does not depend on m! Bodies fall at the same rate regardless of mass.

  13. Planetary motion is more complicated, but governed by the same laws.First, we need to consider the acceleration of orbiting bodies

  14. Circular Acceleration Acceleration is any change in speed or directionof motion. Circular motion is accelerated motion because direction is changing. For circular motion: a = v2/r

  15. Real Life ExampleA Circular Race Track Acceleration

  16. Orbiting Planets Continued So, orbiting planets are accelerating. This must be caused by a force. Let’s assume that the force is gravity. We should be able to calculate the force and acceleration using Newton’s second law and Newton’s law of gravity.

  17. Orbits come in a varietyof shapes (eccentricities).In order to keep the mathsimple, we will considerin this lecture only circularorbits. All of our results also apply to ellipticalorbits, but we will not derive them that way.

  18. Step 1: Calculate the Velocity We take as given that acceleration and velocity in circular motion are related by a = v2/r According to Newton’s 2nd law F = ma = mv2/r According to Newton’s law of gravity F = GMm/r2 Equating the expressions for force we have mv2/r = GMm/r2 Solving for v2 gives v2 = GM/r

  19. Step 2: The Velocity is related to the semi-major axis and period The velocity is related to the semi-major axis and the period in a simple way: velocity = distance/time distance = 2r, where r=semi-major axis, radius of circle time = Period, P v = 2r/P = distance/time

  20. Step 3: Relate the Period to the Orbital Radius We have v2 =GM/r And v = 2r/P So it follows that (2r/P)2 = GM/r Or 42r2/P2 = GM/r

  21. How Does This Relate to Kepler’s Third Law? We have 42r2/P2 = GM/r Multiply both sides by r 42r3/P2 = GM Multiply both sides by P2 42r3 = GM P2 Divide both sides by 42 r3 = (GM/42) P2

  22. Newton’s form of Kepler’s Third Law We have r3 = (GM/42) P2 Kepler’s third law was a3=P2, where a=semi-major axis (not acceleration). Since today we are using r=semi-major axis, this equation is the same as Kepler’s 3rd if (GM/42) = 1 AU3/year2 Let’s check

  23. Do Newton and Kepler Agree? We want to know if (GM/42) = 1 AU3/year2 Plug in G = 6.710-11m3 s-2 kg-1, M=2.01030 kg (GM/42) = 3.41018m3 s-2 Recall 1 AU = 1.51011m and 1 year = 3.1107s So 1 AU3/year2 = (1.51011m)3/(3.1107s)2 1 AU3/year2 = 3.41018m3 s-2 Wow!!!

  24. Using Newton’s Form of Kepler’s Third Law: Example 1 Planet Gabrielle orbits star Xena. The semi major axis of Gabrielle's orbit is 1 AU. The period of its orbit is 6 months. What is the mass of Xena relative to the Sun?

  25. Using Newton’s Form of Kepler’s Third Law: Example 2 Planet Linus orbits star Lucy. The mass of Lucy is twice the mass of the Sun. The semi-major axis of Linus' orbit is 8 AU. How long is 1 year on Linus?

  26. Using Newton’s Form of Kepler’s Third Law: Example 3 Jupiter's satellite (moon) Io has an orbital period of 1.8 days and a semi-major axis of 421,700 km. What is the mass of Jupiter?

  27. Using Newton’s Form of Kepler’s Third Law: Example 4 The moon has an orbit with a semi-major axis of 384,400 km and a period of 27.32 days. What is the mass of the Earth?

More Related