3 ObjectivesStudents will be able to demonstrate an understanding in problem applications as well as in conceptual situations for:Uniform Circular MotionCentripetal forceBanked and unbanked highway curvesNewton’s law of universal gravitationUniversal Gravitation constantInverse square dependence on the distancePoint & spherical objectsCavendish experiment
4 ObjectivesStudents will be able to demonstrate an understanding in problem applications as well as in conceptual situations for:Kepler’s laws of orbital motionLaw of orbitsLaw of areasLaw of periods
5 Units of Chapter 5 Kinematics of Uniform Circular Motion Dynamics of Uniform Circular MotionHighway Curves, Banked and UnbankedNonuniform Circular MotionCentrifugationNewton’s Law of Universal GravitationGravity Near the Earth’s Surface; Geophysical ApplicationsSatellites and “Weightlessness”Kepler’s Laws and Newton’s SynthesisTypes of Forces in Nature
6 5-1 Kinematics of Uniform Circular Motion Uniform circular motion: motion in a circle of constant radius at constant speedInstantaneous velocity is always tangent to circle.
7 5-1 Kinematics of Uniform Circular Motion Looking at the change in velocity in the limit that the time interval becomes infinitesimally small, we see that(5-1)
8 5-1 Kinematics of Uniform Circular Motion Looking at the change in velocity in the limit that the time interval becomes infinitesimally small, we see that(5-1)
9 5-1 Kinematics of Uniform Circular Motion This acceleration is called the centripetal, or radial, acceleration, and it points towards the center of the circle.
10 5-1 Kinematics of Uniform Circular Motion Frequency (f) – the number of revolutions per secondPeriod (T) – the time required for one complete revolutionT = 1 / fV = distance / timesoV = 2πr / T = 2πrf
11 5-2 Dynamics of Uniform Circular Motion For an object to be in uniform circular motion, there must be a net force acting on it.We already know the acceleration, so can immediately write the force:(5-1)
12 5-2 Dynamics of Uniform Circular Motion We can see that the force must be inward by thinking about a ball on a string:Centrifugal (“center-fleeing”) force does NOT exist!
13 5-2 Dynamics of Uniform Circular Motion There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome.If the centripetal force vanishes, the object flies off tangent to the circle.
14 5-3 Highway Curves, Banked and Unbanked When a car goes around a curve, there must be a net force towards the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction.
15 5-3 Highway Curves, Banked and Unbanked If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show.
16 5-3 Highway Curves, Banked and Unbanked As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways:The kinetic frictional force is smaller than the static.The static frictional force can point towards the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve.
17 5-3 Highway Curves, Banked and Unbanked Banking the curve can help keep cars from skidding. In fact, for every banked curve, there is one speed where the entire centripetal force is supplied by thehorizontal component of the normal force, and no friction is required. This occurs when:
18 5-4 Nonuniform Circular Motion If an object is moving in a circular path but at varying speeds, it must have a tangential component to its acceleration as well as the radial one.
19 5-4 Nonuniform Circular Motion If an object is moving in a circular path but at varying speeds, it must have a tangential component to its acceleration as well as the radial one.
20 5-4 Nonuniform Circular Motion This concept can be used for an object moving along any curved path, as a small segment of the path will be approximately circular.
21 5-5 CentrifugationCentrifugal force – a force that tends to move the particles of a spinning object away from the spin axis.Centrifuge – a device for separating substances by spinning them at high speeds.
22 5-5 CentrifugationA centrifuge works by spinning very fast. This means there must be a very large centripetal force. The object at A would go in a straight line but for this force; as it is, it winds up at B.
23 Frame of Reference – a system for describing the location of objects (used to specify the positions and relative motions of objects).Inertial Frame of Reference – Newton’s laws will explain the behavior of objects; the observer is OUTSIDE the accelerating system.Non-Inertial Frame of Reference – Newton’s laws do NOT explain the behavior of objects; the observer is INSIDE the accelerating system.
24 Circular Motion – the motion of a body about an external axis. Examples: earth rotating around the sun; valve stem of tire rotating around axle.Rotary Motion – the motion of a body about an internal axis.Examples: earth spinning about its own axis; tire turning around axle.
25 Dynamics of Rotary Motion Constant Rotary Motion – the object must spin about a fixed axis at a steady rate.Example: the hands of a clockVariable rotary Motion – occurs if either the direction of the axis or the rate of spin varies.Example: either changing speed or direction while driving (tires rotate faster or slower and/or tires turn to new direction)
26 Dynamics of Rotary Motion An angle of one radian (a radian is a pure number) is the angle that, when placed with its vertex at the center of a circle, subtends (is opposite to) on the circumference an arc equal in length to the radius of the circle.rΔl = rΔlθr
28 Angular Velocity – the time rate of angular displacement Dynamics of Rotary MotionAngular Velocity – the time rate of angular displacement(units: radians/second)
29 Dynamics of Rotary Motion Angular velocity is a vector quantity represented by a vector along the axis of rotation. The length of the vector indicates the magnitude of the angular velocity. The direction of the vector is the direction in which the thumb of the right hand points when the fingers of the right hand points when the fingers of the right hand encircle the vector in the direction in which the body is rotating. This is known as the Right Hand Rule of Thumb.
30 Dynamics of Rotary Motion Angular acceleration (measured in radians /s2) is any change in either the rate of rotation or the direction of the axis changes the angular velocity.α = Δω / ΔtUse the linear kinematics equations but substitute:Θ for xω = vα for a
32 Dynamics of Rotary Motion Rotational Inertia (kg m2) – the resistance of a rotating object to changes in its angular velocity – includes both the mass and shape of a rotating object.Torque = rotational Inertia x angular accelerationTorque = IRαIR is measured in kg m2α is measured in rad/s2Use the Right Hand Rule of Thumb to determine the direction of the torque vector.
33 Dynamics of Rotary Motion Rotational Inertia Equations: Thin ring: IR = mr2Disk spinning about CG: IR = ½ mr2Thick ring: IR = ½ m(r12 + r22)Solid ball: IR = 2/5 mr2Thin rod spinning about CG: IR = 1/12 ml2Thin rod spinning about one end: IR = ⅓ ml2Disk spinning about a diameter: IR = ¼ mr2
34 Precession – occurs when an object rotates about two different axes Dynamics of Rotary MotionPrecession – occurs when an object rotates about two different axes
35 Dynamics of Rotary Motion The earth precesses by spinning about an axis that is tilted with respect to its plane of revolution around the sun. The rotation of the earth produces an equatorial bulge. The gravitational pull of the sun on the part of the bulge closest to the sun is stronger than the pull on the part of the bulge on the other side of the earth. Because of the resulting torque, the earth’s axis undergoes precessional motion and does not point constantly at the same place in space.The earth’s axis completes a single processional cycle in 26,000 years, thus changing the north star.
36 5-6 Newton’s Law of Universal Gravitation If the force of gravity is being exerted on objects on Earth, what is the origin of that force?Newton’s realization was that the force must come from the Earth.He further realized that this force must be what keeps the Moon in its orbit.
37 5-6 Newton’s Law of Universal Gravitation The gravitational force on you is one-half of a Third Law pair: the Earth exerts a downward force on you, and you exert an upward force on the Earth.When there is such a disparity in masses, the reaction force is undetectable, but for bodies more equal in mass it can be significant.
38 5-6 Newton’s Law of Universal Gravitation Therefore, the gravitational force must be proportional to both masses.By observing planetary orbits, Newton also concluded that the gravitational force must decrease as the inverse of the square of the distance between the masses.In its final form, the Law of Universal Gravitation reads:where(5-4)
39 5-6 Newton’s Law of Universal Gravitation The magnitude of the gravitational constant G can be measured in the laboratory.This is the Cavendish experiment.
41 5-7 Gravity Near the Earth’s Surface; Geophysical Applications Now we can relate the gravitational constant to the local acceleration of gravity. We know that, on the surface of the Earth:Solving for g gives:Now, knowing g and the radius of the Earth, the mass of the Earth can be calculated:(5-5)
42 5-7 Gravity Near the Earth’s Surface; Geophysical Applications The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical.
43 5-7 Gravity Near the Earth’s Surface; Geophysical Applications Factors which affect the value of gThe earth is not a perfect sphereThe earth has mountains and valleys and bulges at the equatorThe Earth’s mass is not uniformly distributedThe Earth’s rotation affects the value of g
44 5-8 Satellites and “Weightlessness” Satellites are routinely put into orbit around the Earth. The tangential speed must be high enough so that the satellite does not return to Earth, but not so high that it escapes Earth’s gravity altogether.
45 5-8 Satellites and “Weightlessness” The satellite is kept in orbit by its speed – it is continually falling, but the Earth curves from underneath it.
46 5-8 Satellites and “Weightlessness” Geosynchronous satellitestays above the same point on the EarthPossible only if it is above a point on the EquatorUsed forTV and radio transmissionWeather forecastingCommunication relays
47 5-8 Satellites and “Weightlessness” Objects in orbit are said to experience weightlessness. They do have a gravitational force acting on them, though!The satellite and all its contents are in free fall, so there is no normal force. This is what leads to the experience of weightlessness.
48 5-8 Satellites and “Weightlessness” More properly, this effect is called apparent weightlessness, because the gravitational force still exists. It can be experienced on Earth as well, but only briefly:
49 A Brief History of Astronomy Ptolemy – Greek philosopher who believed the sun and all planets and stars orbit the Earth.Ptolemaic or Geocentric frame of reference:The earth is a stationary reference point – the center of the universe.
50 A Brief History of Astronomy Copernicus – observed that the movements of the planets and stars did not agree with an Earth centered (Geocentric) view of the universe. He showed that the motion of planets is more easily explained by assuming the Earth and all planets orbit the sun.
51 A Brief History of Astronomy Copernican or Heliocentric frame of reference:The sun is stationary reference point – the center of the universe.This frame of reference is used most often in calculations because it simplifies the mathematical descriptions.
52 A Brief History of Astronomy Tyco Brahe – Danish nobleman and astronomerGranted funding to build a research institute where he built astronomical instruments and made careful measurements for more than 20 years.Made the most accurate astronomical observations of his time.No one before him had attempted to make so many redundant observations.Catalogued the planets and stars with enough accuracy so as to determine whether the Ptolemaic or Copernican system was more valid in describing the heavens.
53 A Brief History of Astronomy KeplerKepler was a student of Brahe who used Brahe’s careful measurements to develop his own laws of planetary motion.Kepler mathematically analyzed all of Brahe’s data.Kepler discovered the laws that describe the motions of every planet and satellite.
54 A Brief History of Astronomy GalileoMade improvements to the telescope“Father of modern observational astronomy”Discovered the four largest satellites of Jupiter (named the Galilean moons in his honor).Observed and analyzed sunspots.Championed the Copernican view of the universe for which he was placed under house arrest for the last years of his life by the Inquisition.
55 A Brief History of Astronomy Isaac NewtonNewton based his work on Universal Gravitation on Kepler’s idea that planets travel in elliptical paths.CavendishThe value of G was first measured by Cavendish about 100 years after Newton mathematically predicted the value of G.
56 A Brief History of Astronomy Halleyshowed that comets were merely one of a large number of objects that circled the sun based on Newton’s Universal Law of Gravitation.EinsteinIn his Theory of Gravity, Einstein proposed that gravity is NOT a force, but an effect of space itself (Mass causes space to be curved, and other bodies are accelerated because of the way they follow this curved space.)
57 5-9 Kepler’s Laws and Newton's Synthesis Kepler’s laws describe planetary motion.The orbit of each planet is an ellipse, with the Sun at one focus.
58 5-9 Kepler’s Laws and Newton's Synthesis 2. An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times.
59 5-9 Kepler’s Laws and Newton's Synthesis The ratio of the square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun.
60 5-9 Kepler’s Laws and Newton's Synthesis The ratio of the square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun.
61 5-9 Kepler’s Laws and Newton's Synthesis Kepler’s laws can be derived from Newton’s laws. Irregularities in planetary motion led to the discovery of Neptune, and irregularities in stellar motion have led to the discovery of many planets outside our Solar System.
62 5-9 Kepler’s Laws and Newton's Synthesis Kepler’s laws can be derived from Newton’s laws. Irregularities in planetary motion led to the discovery of Neptune, and irregularities in stellar motion have led to the discovery of many planets outside our Solar System.
63 5-10 Types of Forces in Nature Modern physics now recognizes four fundamental forces:GravityElectromagnetismWeak nuclear force (responsible for some types of radioactive decay)Strong nuclear force (binds protons and neutrons together in the nucleus)
64 5-10 Types of Forces in Nature So, what about friction, the normal force, tension, and so on?Except for gravity, the forces we experience every day are due to electromagnetic forces acting at the atomic level.
65 5-10 Types of Forces in Nature So, what about friction, the normal force, tension, and so on?Except for gravity, the forces we experience every day are due to electromagnetic forces acting at the atomic level.
66 Summary of Chapter 5An object moving in a circle at constant speed is in uniform circular motion.It has a centripetal accelerationThere is a centripetal force given byThe centripetal force may be provided by friction, gravity, tension, the normal force, or others.
67 Summary of Chapter 5 Newton’s law of universal gravitation: Satellites are able to stay in Earth orbit because of their large tangential speed.
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