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Chapter 3 Celestial Sphere Movie

Gravity and Motion

Projects Test Monday Preview • I moved due-date for Part 1 to 10/21 Ch 1 Ch 2 • I added a descriptive webpage about the Galileo Movie projects. Essay 1: Backyard Astronomy Ch. 3 (just beginning) Northern Hemisphere sky: Big Dipper Little Dipper Cassiopeia Orion

Gravity The Problem of Astronomical Motion • Gravity gives the Universe its structure • Astronomers of antiquity did – It is a universal force that not connect gravity and causes all objects to pull on astronomical motion all other objects • Galileo investigated this everywhere connection with experiments – It holds objects together using projectiles and balls – It is responsible for holding rolling down planks the in its around • He put science on a course to the , the Sun in its orbit determine laws of motion and around the Milky Way, and to develop the scientific the Milky Way in its path method within the Local Group

Inertia and Newton’s First Law Inertia • This concept was • Galileo established the idea of inertia incorporated in – A body at rest tends to remain at rest Newton’s First Law – A body in motion tends to remain in motion of Motion: – Through experiments with inclined planes, Galileo demonstrated the idea of inertia and the A body continues in a importance of forces (friction) state of rest or uniform motion in a straight line unless made to change that state by forces acting on it

Newton’s First Law Astronomical Motion

• As seen earlier, • Must there be a force at move along curved work? • Important ideas of (elliptical) paths, or Newton’s First Law – The law implies that if • Yes! an object is not moving . – Force: A push or a pull with constant velocity , • – The force referred to is Speed and direction is then a nonzero net changing a net force force must be present

Gravity is that force! Orbital Motion and Gravity

• Although not the first to propose gravity as being responsible for celestial motion, Newton was the first to: – Spell out the properties of gravity – Write the equations of gravity-induced motion • Newton deduced that: – The ’s motion could be explained by the existence of a force (to deviate the Moon from a straight inertial trajectory) and that such a force decreased with distance – Orbital motion could be understood as a projectile moving “parallel” to the Earth’s surface at such a speed that its gravitational deflection toward the surface is offset by the surface’s curvature away from the projectile

Orbital Motion Using Newton’s Orbital Motion Using Newton’s First Law First Law • At a sufficiently high speed, the cannonball travels so far that the ground curves out from under it. • The cannonball literally misses the ground! • A cannonball fired at • A cannonball fired at a slow speed experiences higher speed feels the • The ball, now in orbit, one force – gravity, same force, but goes still experiences the pull pulling it downward farther of gravity!

Newton’s Second Law: Motion Newton’s 2nd Law: • Motion – An object is said to be in uniform motion if its speed and direction remain unchanged – An object in uniform motion is said to have a constant velocity – A force will cause an • Acceleration object to have non- – An object increasing or – Acceleration is produced uniform motion, a decreasing in speed along a by a force and experiments changing velocity straight line is accelerating show the two are – Acceleration is defined – An object with constant speed proportional as a change in velocity moving is a circle is accelerating

Newton’s Second Law: Mass Newton’s Second Law of Motion • Mass – Mass is the amount of matter an object contains F = ma – Technically, mass is a measure of an object’s inertia • Equivalently, the amount of acceleration (a) – Mass is generally measured in kilograms that an object undergoes is proportional to – Mass should not be confused the force applied (F) and inversely with , which is a force proportional to the mass (m) of the object related to gravity – weight – This equation applies for any force, may change from place to gravitational or otherwise place, but mass does not

F = ma Newton’s Law of Universal Gravity

• Everything attracts everything else!!

Measuring an Object’s Mass Newton’s Third Law of Motion Using Orbital Motion • When two objects • Basic Setup of an Orbital Motion Problem interact, they create – Assume a small mass object orbits around a much more massive object equal and opposite – Massive object can be assumed at rest (very little acceleration) forces on each – Assume orbit shape of small mass is a circle centered on large other mass • • This is true for any Using Newton’s Second Law – Acceleration in a must be: two objects, a = v 2/r including the Sun where v is the constant and r is the radius of the and the Earth! orbit – The force is that of gravity

Measuring an Object’s Mass Using Orbital Motion Surface Gravity • Method of Solution – Equate F = mv 2/r to F = GMm/r 2 and solve for v: v = (GM/r) 1/2 • Surface gravity is the acceleration a mass – One can also solve for M: undergoes at the surface of a celestial object (e.g., an asteroid, , or ) M = (v 2r)/G – v can be expressed in terms of the (P) on the • Surface gravity: small mass and its orbital radius: – Determines the weight of a mass at a celestial object’s v = 2 πr/P surface – Combining these last two equations: – Influences the shape of celestial objects M = (4 π2r3)/(GP 2) – Influences whether or not a celestial object has an – This last equation in known as Kepler’s modified third law and is often used to calculate the mass of a large celestial object from the orbital period and radius of a much smaller mass

Surface Gravity Calculations • Surface gravity is determined from Newton’s Second Law and the Law of Gravity: • ma = GMm/R 2 To overcome a celestial object’s gravitational force and escape into space, a mass must obtain a where M and R are the mass and radius of the celestial object, and m is the mass of the object whose acceleration a critical speed called the escape velocity we wish to know • Escape velocity: • The surface gravity, denoted by g, is then: – Determines if a spacecraft can move from one planet to g = GM/R 2 another • Notice dependence of g on M and R, but not m – Influences whether or not a celestial object has an atmosphere • g = 9.8 m/s 2 Earth – Relates to the nature of black holes • gEarth /g Moon = 5.6 and g /g Earth = 3

Escape Velocity Escape Velocity Calculation

• The escape velocity, V esc , is determined from Newton’s laws of motion and the Law of Gravity and is given by:

1/2 Vesc = (2GM/R) where M and R are the mass and radius of the celestial object from which the mass wishes to escape

• Notice dependence of V esc on M and R, but not m

• Vesc,Earth = 11 km/s, V esc,Moon = 2.4 km/s

Escape Velocity