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Chapter 5 Uniform Circular Motion ac=v2/r Uniform circular motion: Motion in a circular path with constant speed v s r 1) Speed and period • Period, T: time for one revolution • Speed is related to period: – Path for one revolution: s = 2r – Speed: v = s/T = 2r/T 2) Centripetal Acceleration v s r • Directed toward centre 2 v • Magnitude: a c r ac 3) Centripetal force v s ac r Fc • Toward centre (parallel to acceleration) since F = ma • Magnitude: mv 2 Fc mac r 4) Driving around circular curves a) Friction only: vmax s gr b) Banked curve, no friction: v2 tan rg 4) Driving around circular curves c) Banked curve with friction fs(ii) fs(i) Net force F is vector sum of gravity, the normal force, and friction: r r r F FN mg fs Since the ideal (zero friction) angle is given by v2 tan rg i) If v2 > gr tan friction prevents sliding up ii) If v2 < gr tan friction prevents sliding down 4) Driving around circular curves c) Banked curve with friction Example: Find maximum velocity for µ = 0.80, = 47º, and r = 60 m FN fs(ii) y fs(i) x Fc r r r Fc FN mg fs fs y-motion x-motion 0 FN cos mg fs sin When v vmax , fs FN so mg FN cos sin Fc FN sin fs cos mg mv 2 When v vmax , fs FN , and using Fc = , r mvmax 2 FN r(sin cos ) 4) Driving around circular curves c) Banked curve with friction Example: Find maximum velocity for µ = 0.80, = 47º, and r = 60 m FN fs(ii) y x fs(i) Fc r r r Fc FN mg fs fs mg Eliminate FN , and solve for vmax : vmax 2 sin cos gr cos sin vmax 88m/s 317km/h 5) Satellites in circular orbits a) Speed and radius For a distance r, the gravitational force is mM FG 2E r For circular motion with radius r, the centripetal force is mv 2 Fc r • Radius of orbit determines speed (independent of mass) For circular motion around the earth, F Fc , so • Accelerations of all objects at the same radius are equal (no acceleration relative to each other) mM E mv 2 G 2 r r GM E v r or ==> No apparent force between them: apparent weightlessness in orbiting satellite 5) Satellites in circular orbits b) Period, Synchronous orbits (i) Period and radius Time for one revolution: T Speed is constant, so v For earth orbit, v 2 r T GM E or r 3 2 r 2 T GM E 2 r T GM E , so r 5) Satellites in circular orbits b) Period, Synchronous orbits (ii) Radius of synchronous orbit: Definition: satellite is stationary above earth’s surface Conditions: • T = 1 sidereal day • above equator Radius is given by r 3 2 T GM E 2 Using T 24h 4 min = 86160 s gives r 4.22 10 7 m = 6.61RE 6) Centrifugal force* and artificial gravity a) Artificial gravity A rotating cylinder exerts a centripetal force on objects on the inside surface: e.g. if r = 1700 m, and Fc = mg (usual force of gravity), then mg = mv2/r, giving v gr 130 m/s = 468 km/h Centripetal force acts toward the centre (up, like the normal force on earth) but gravity acts down. What has taken the place of gravity, from the perspective of the cylinder-dwellers? Centrifugal force… does not exist is a fictitious, pseudo, virtual force is an inertial force * The concept of centrifugal force is not required material. 6) Centrifugal force and artificial gravity a) Artificial gravity A rotating cylinder exerts a centripetal force on objects on the inside surface: e.g. if r = 1700 m, and Fc = mg (usual force of gravity), then mg = mv2/r, giving v gr 130 m/s = 468 km/h Centripetal force acts toward the centre (up, like the normal force on earth) but gravity acts down. What has taken the place of gravity, from the perspective of the cylinder-dwellers? Centrifugal force… does not exist (in inertial frames) is a fictitious, pseudo, virtual force is an inertial force 6) Centrifugal force and artificial gravity This episode of Quirks & Quarks on CBC radio illustrates the confusion that can exist about artificial gravity. http://www.cbc.ca/quirks/archives/01-02/mp3/qq131001f.mp3 The question was: “What happens when you jump inside a rotating cylinder like the one on the movie 2001: A Space Odysey” Contrary to the answer given, you do not float to the other side and hit your head. The simulated gravity does not require continuous contact with the surface. From your perspective (the jumper’s), you will be pulled back to the surface just as you would in a gravitational field, apart from a small deviation depending on the radius of the cylinder. To understand this, we will consider a simpler inertial force first. 6) Centrifugal force and artificial gravity b) Inertial force -- Apparent force resulting from an accelerating reference frame. e.g. An accelerating spaceship (far from planets) m FN a From an inertial frame: FN ma ma0 a0 6) Centrifugal force and artificial gravity b) Inertial force -- Apparent force resulting from an acceleration reference frame. e.g. An accelerating spaceship (far from planets) m ma0 FN F 0 a0 FN ma0 0 From an inertial frame: FN ma ma0 Inside the spaceship, acceleration is zero, so net force should be zero. Same equation, new interpretation: Inside the ship an inertial force equal to ma0 is acting toward the floor. 6) Centrifugal force and artificial gravity b) Inertial force -- Apparent force resulting from an acceleration reference frame. e.g. An accelerating spaceship (far from planets) a0 From an inertial frame: FN ma ma0 Inside the spaceship, acceleration is zero, so net force should be zero. Same equation, new interpretation: Inside the ship an inertial force equal to ma0 is acting toward the floor. A dropped ball falls to the floor just as it would if a constant force acted on it. 6) Centrifugal force and artificial gravity b) Inertial force -- Apparent force resulting from an acceleration reference frame. e.g. An accelerating spaceship (far from planets) a0 From an inertial frame: FN ma ma0 Inside the spaceship, acceleration is zero, so net force should be zero. Same equation, new interpretation: Inside the ship an inertial force equal to ma0 is acting toward the floor. A dropped ball falls to the floor just as it would if a constant force acted on it. 6) Centrifugal force and artificial gravity b) Inertial force -- Apparent force resulting from an acceleration reference frame. e.g. An accelerating spaceship (far from planets) a0 From an inertial frame: FN ma ma0 Inside the spaceship, acceleration is zero, so net force should be zero. Same equation, new interpretation: Inside the ship an inertial force equal to ma0 is acting toward the floor. A dropped ball falls to the floor just as it would if a constant force acted on it. 6) Centrifugal force and artificial gravity c) Centrifugal force: the inertial force in a rotating reference frame Consider a cup of water swung around in a vertical circle. What keeps the water in the cup when it is upside-down? In an inertial frame, only the normal force and gravity act to give centripetal force which produces centripetal acceleration (no such thing as centrifugal force) Answer: Inertia. The cup pulls the water down faster than its natural falling rate. Gravity plus the normal force provide centripetal force that produces circular motion: mv 2 FN mg r mg FN 6) Centrifugal force and artificial gravity c) Centrifugal force: the inertial force in a rotating reference frame F=mg+FN=mv2/r Inside the cup, the water is at rest, as is an observer in a boat inside the cup; i.e. they do not accelerate in this reference frame. What keeps water in the cup from the perspective of the cup-dweller? mg Answer: Invent centrifugal force to balance real forces and ensure zero acceleration. So F 0 mv 2 FN mg 0 r FN 6) Centrifugal force and artificial gravity c) Centrifugal force: the inertial force in a rotating reference frame F=mg+FN What happens if a ball is dropped? If the force holding it Fh is removed, it should fall radially (down to the cupdweller), but in an inertial frame it moves tangentially. How can both be true? Fh mg FN 6) Centrifugal force and artificial gravity This animation shows what happens when a ball is dropped inside a rotating cylinder. 6) Centrifugal force and artificial gravity In the rotating reference frame it moves radially outward (down to the observer), apart from a small curvature which decreases for larger cylinder radii. An object released moves with constant velocity on the tangent following to Newton’s first law. 6) Centrifugal force and artificial gravity An object that leaves the floor with a small radial velocity (like a jumping person) will have velocity with both tangential and radial components and will move at a small angle to the the tangent. It will leave the floor and then return to it again very close the point of departure. To an observer inside the cylinder, it moves up and down (except for small displacement due to finite radius of the cylinder). 6) Centrifugal force and artificial gravity c) Centrifugal force: the inertial force in a rotating reference frame An accelerating reference frame can duplicate the effect of gravity (not only when in contact). Centrifugal force is introduced within the reference frame to account for the observed acceleration. However, when analyzing the situation from an inertial frame (as we do in this course), inertial forces are not present. “Centrifugal force does not exist.” 6) Centrifugal force and artificial gravity d) Equivalence principle - Inertial mass (F = ma) is the same as gravitational mass (F = GmM/r2) - It is, however, clear that science is fully justified in assigning such a numerical equality only after this numerical equality is reduced to an equality of the real nature of the two concepts.” -- Einstein - The happiest thought of my life… The gravitational field has only a relative existence... Because for an observer freely falling from the roof of a house - at least in his immediate surroundings - there exists no gravitational field. -- Einstein A uniform gravitational field is completely equivalent to a uniformly accelerated reference frame. • No local experiment can distinguish them • Gravity is geometrical 7) Vertical circular motion • Not usually uniform motion since the speed is changing • Net force not always toward the centre • Component of acceleration toward the centre (the centripetal component) is still v2/r, so mv 2 Fc r 7) Vertical circular motion mv32 FN 3 mg r FN 2 FN 4 mv4 2 r mv12 FN1 mg r mv2 2 r 7) Vertical circular motion mv32 FN 3 mg r Minimum speed at top: FN 3 0 mv32 mg r v3 rg For r = 6 m, v = 7.6 m/s or 27 km/h