Speed, Velocity & Acceleration Understanding Motion, Energy & Gravity Chapter 4 speed: distance traveled per unit time (e.g., m/s, mph, km/ hr) velocity: speed & direction acceleration: change in velocity (speed and/or direction) with time constant velocity is not felt, but acceleration (speeding/ slowing, turning into curve) is Acceleration of Gravity Acceleration due to Gravity on earth is a constant = 9.8 m / s 2. I.e., objects fall faster by 9.8 m / s with each passing second Acceleration is independent of the mass of an object. This was first demonstrated by Galileo who dropped objects of different masses from the Leaning Tower of Pisa Acceleration of Gravity Acceleration due to Gravity on earth is a constant = 9.8 m / s 2. I.e., objects fall faster by 9.8 m / s with each passing second Acceleration is independent of the mass of an object. This was first demonstrated by Galileo who dropped object of different masses from the Leaning Tower of Pisa
Momentum & Force Momentum = mass x velocity (p = mv) a force must be applied to change the momentum. I.e., constant velocity = no net force. Change in momentum = mass x (velocity1 - velocity2) = mass x acceleration Magnitude of force & change in momentum: which hurts more, being hit by a bug traveling at 30 mph or a baseball traveling at 30 mph? Both transfer momentum to you (and you feel a force as a result) during the collision The mere presence of a force doesn t result in a change in momentum. E.g., a driver can achieve a constant velocity even though air resistance & friction with the road are present Acceleration & net force change in momentum = change in velocity acceleration = application of a non-zero net force I.e., we feel forces when we accelerate in a car (being pushed back) or drive on a curvy road (pushed to the side) planets accelerate as they orbit the Sun. Thus, something must be applying a force Angular Momentum angular momentum, or circular/turning momentum is the momentum an object has when spinning on its axis An ice skater has angular momentum when they are spinning in place the Earth has angular momentum due to its rotation (rotational angular momentum) and its orbit around the Sun (orbital angular momentum) Angular Momentum Angular momentum changes from the application of a torque, or twisting force E.g., pushing on the hinges of the door results in no net torque, but pushing on the end of the door does r planet Sun F
Weight vs Mass Mass: amount of matter in an object Weight: a force, e.g., a force that a scale measures when you stand on it. Weight = (Mass) x (acceleration due to gravity) Weight vs Mass Your measured weight can vary, e.g., when riding an elevator accelerating up = you weigh more due to the elevator s acceleration accelerating down = you weigh less moving at a constant speed = no weight change Your measured weight can vary, e.g., when riding an elevator if the cable breaks... you are in free fall (gravity accelerates you at the same rate as your surroundings)... and thus weightless. astronauts in the Space Shuttle experience weightlessness by falling around the Earth Weight vs Mass Isaac Newton (1642-1727 A.D.) Born just after Galileo's death Developed calculus Used math to develop scientific ideas Fundamental contributions: 1.Three laws of motion 2.Law of gravity 3.Applying the laws of physics to the heavens
Newton s 1st law of motion An object moves at constant velocity if there is no net force acting upon it. Objects at rest (velocity = 0) stay at rest Moving object stays in motion unless a force acts upon it. For a moving car, the forces acting upon it are air resistance, friction with the road (and maybe gravity if one is driving uphill) You don't feel motion when moving at a constant velocity (e.g, a smooth airplane flight) In physics, this is the concept of inertia Newton s 2nd law of motion Force = mass x acceleration Another way to think of it force is change in momentum One can throw a baseball farther than a bowling ball. Jupiter exerts a stronger gravitational force on passing comets than the Earth does 2nd law: circular motion What force is in play when one swings a ball on a rope? The inward force along the string keeps it in circular motion For a driver turning into a curve, the inward force is friction For a planet orbiting the Sun, the inward force is gravity Newton s 3rd law of motion For any force, there is always an equal and opposite reaction force. Standing on the ground: you push on the ground, it pushes back on you no net force. Better example: What happens when you jump in the air from a log floating on water This is what allows a rocket to lift off.
Conservation Laws Newton s laws arise from conservation principles: Conservation of momentum Conservation of angular momentum Conservation of energy Conservation of Momentum Total momentum of interacting objects does not change unless an external force is applied. An object can gain or lose momentum only if some other object s momentum changes by a precisely opposite amount E.g., Billiard balls. E.g., Rocket: forward momentum of rocket = backward momentum of gas ball at rest or in motion = 1st law. collision - transfer of momentum = 2nd and 3rd law. Conservation of Angular Momentum In the absence of an external torque, the total angular momentum of a set of interacting objects does not change. Angular momentum = mass x speed x radius of orbit Earth's orbit: Orbital angular momentum constant Orbit continues unless something takes away angular momentum. Orbital speed is greatest when closest to the Sun (i.e., Kepler s 2nd law) Conservation of Rotational Angular Momentum The Earth will keep rotating at the same speed, as long as it doesn t transfer angular momentum to another object An ice skater is a great example of conservation of angular momentum. Angular momentum = mass x speed x radius
Conservation of Energy Objects gain or lose energy only by exchanging it with other objects. 1 Joule = 1 kg m 2 / s 2 1 Calorie = 4184 Joules Energy type 1: Kinetic, or energy of motion Energy type 2: Radiative, or light energy Energy type 3: Potential, or stored energy Thermal Energy i.e., kinetic energy of random motion of atoms and molecules Temperature ~ average kinetic energy of the particles Thermal Energy i.e., kinetic energy of random motion of atoms and molecules Temperature ~ average kinetic energy of the particles normal hot
Temperature Scales Thermal Energy Thermal energy depends both on the number of particles (density) and their temperature. I.e., a boiling pot of water has more thermal energy than a hot oven. 25 Gravitational Potential Energy gravitational potential energy: depends on the mass of the object & and how far it can fall as a result of gravity E.g., = mass x acceleration of gravity x height above ground Mass-Energy Einstein showed that mass and energy are related: Potential energy stored in nuclei. Nuclear fusion can release some of this potential energy. 1 megaton H-bomb explosion results from converting 0.1 kg of mass into energy. Mass comes from energy too particle accelerators 11-megaton ROMEO Event at Bikini Atoll (NNSA) 27
Gravity Newton is rumored to have observed a falling apple Newton's law of gravity: The gravitational force between two objects is Force on mass 1 from 2 is the same as force on mass 2 from mass 1. Gravity affects anything with mass. Gravity follows an inverse square law with distance. Gravity What happens to the force of gravity if the distance between the objects doubles? If we were to replace the Sun (radius = 7 x 10 8 m) with a neutron star (radius = 10,000 m) with the same mass, would Earth's orbit change? Gravity Gravity Gravity keeps the planets in orbit around the Sun planet's velocity planet force of gravity orbital path Sun Weak between two people because the masses involved are small. Strong between a person & Earth because the Earth is massive & the distance is small. Strong between Sun & planets because the Sun is very massive.
Newton: Ellipses (& circles) are only one kind of orbit Ellipses (& circles) are bound orbits Parabolic & hyperbolic orbits are unbound Circles, Ellipses, parabolas, and hyperbolas are conic sections For all orbits, objects move faster when closer to the object being orbited Types of Orbits Kepler s 3rd Law Gravity = Centrifugal Force = mass x centripetal acceleration Circumference = Speed x Period So, Center of Mass Perception: E.g., the Earth orbits the Sun Reality: Both the Earth & Sun orbit their common center of mass For objects of equal mass, the center of mass is exactly halfway between them. For the Earth-Sun, the center of mass is below the surface of the Sun Center of Mass Kepler s third law (modified)
Orbital Energy A planet s total orbital energy always stays the same Orbits cannot change spontaneously Gravitational Encounters The Voyager Missions However, orbits can change through the exchange of orbital energy with other objects E.g., comets passing close to Jupiter can lose so much orbital energy that the comet becomes bound to Jupiter Spacecrafts can use the exchange of energy with planets to boost their speed. I.e., these probes made use of the gravity of several planets to redirect & accelerate them
Atmospheric Drag Satellites can lose orbital energy via drag with the upper atmosphere of the Earth Orbital energy is converted to thermal energy Result: falling satellite burns up in the atmosphere Why do all objects fall at the same rate? It turns out that the mass of the falling object cancels out Gravity = mass x acceleration of falling object Pieces of Russian Space Station MIR during its reentry on March 23, 2001 42 Escape Velocity Escape Velocity Escape velocity: i.e., sufficient speed (i.e., kinetic energy) to escape the gravitational pull of a massive object
Tidal Force Tidal Force As a example, consider the Earth as being comprised of particles of mass, m. Consider 3 particles on the Earth - one at the near edge (1) to the Moon, in the center (2), and at the far edge (3). 3 2 1 ΔR01 ΔR02 Tidal Force: a differential gravitational force, ΔF, that tends to deform a body due to the tidal effects of its neighbor ΔR03 Tidal Force The gravitational force from the Moon felt by the particle on the near edge is stronger than the force felt by the particle in the center. The force from the Moon felt by the particle in the center is stronger than the force felt by the particle on the far edge. This results in a differential gravitational force, ΔF, between the particles. Tidal Force 3 2 1 3 2 1 ΔR01 ΔR01 ΔR03 ΔR02 ΔR03 ΔR02 ΔF ΔF The larger the distance between 2 objects relative to the size of each object, ΔR02» (ΔR01 - ΔR03), the weaker the tidal force. A strong tidal force can actually tear an object apart
the Sun also exerts tidal forces on the Earth 50% weaker than the Moon s tidal force because the Sun is so far away - thus the differential force is weaker New & full moon - Sun and Moon tidal forces on Earth are aligned 1st & 3rd quarter Moon - Sun and Moon tidal forces are perpendicular to each other Tidal Force Earth No rotation Tidal Force: long-term effects Moon Axis of greatest tidal stress No rotation: the tidal bulge is aligned with the Earth-Moon line Tidal Force: long-term effects Tidal Force: long-term effects Earth Moon Earth Moon Axis of greatest tidal stress Axis of greatest tidal stress Rotation: tidal bulge is pulled out of phase with the Earth-Moon line as the Earth attempts to pull the bulge around in the rotational direction. Rotation: the tidal bulge pulls the moon forward.
Tidal Force: long-term effects Tidal Force: long-term effects Earth Moon Axis of greatest tidal stress Rotation: the Moon s gravity tries to pull the tidal bulge back Result 1: the Earth s rotation is slowing down - i.e., the Earth is losing rotational angular momentum Result 2: the Moon is gaining angular momentum Result 3: If the Moon stays in orbit around the Earth, the Earth-Moon rotation will eventually be synchronous Tidal Stress Synchronous Rotation As the moon orbits the planet, different parts of the moon pass through the axis of greatest tidal stress The end result is the moon is heated by friction The heat is expelled through moonquakes and volcanism Dissipation of energy slows the moon s rotation Earth Moon orbital period = rotational period
Synchronous Rotation Tidal Forces: Effects on Moon & Earth Minimum frictional energy loss: when the orbital period of the Moon is equal to the rotation period I.e., the same parts of the moon are always aligned with the axis of maximum tidal stress Earth Moon The Earth s day was once much shorter The Moon was once much closer to the Earth, and it rotated much faster I.e., friction, and thus heat generation, is minimized 58 Tides and Other Objects Galilean moons exhibit synchronous rotation Pluto and Charon are both tidally locked Many binary star systems exhibit this. Mercury has a more complex configuration - 3 rotational periods = 2 orbital periods Tidal forces can drive geological activity - Oceans on Europa - Volcanoes on Io Comet Shoemaker-Levy 9 was broken apart due to tidal forces from Jupiter (Credit: Dr. Hal Weaver and T. Ed Smith (STScI), and NASA) Putting it all into context... Newton's laws - Explain Kepler's laws - Allow us to determine the mass of planets, stars in binary systems,... - Understanding of orbits and observations of transits allow us to get radius, density. Center of mass - Important for understanding how we detect extrasolar planets. Tides - Explain why the Moon shows the same face toward Earth - Important for understanding Jupiter's moons
Putting it all into context... Thermal energy - Important in understanding whether a planet can hold an atmosphere Angular momentum - Important for understanding the formation of the solar system