PHY1012F CIRCULAR MOTION

1. PHY1012FCIRCULAR MOTION Gregor Leighgregor.leigh@uct.ac.za

2. MOTION IN A CIRCLE • Apply kinematics and dynamics knowledge, skills and techniques to circular motion. • Manipulate angular quantities and formulae against the background of an angular (rtz-) coordinate system. Learning outcomes:At the end of this chapter you should be able to…

3. UNIFORM CIRCULAR MOTION Any particle travelling at constant speed around a circle is engaged in uniform circular motion. The magnitude of is constant, but since is everywhere tangent to the circle, its direction changes continuously. r r O r In the particle model the centre of the circle lies outside the particle, and we speak of orbital motion.Later we shall apply the same principles to the rotation or spin of extended objects about axes within themselves.

4. PERIOD • Hence The time taken for the particle to complete one revolution (rev) is called the period, T,of the motion. r O E.g. Calculate the speed of a point on the rim of a CD in a 50x drive…

5. y O • x ANGULAR POSITION • It will be more convenient to describe the position of an orbiting particle in terms of polar coordinates rather than xy-coordinates.  , called the angular position of the particle, … r s • is positive when measured counterclockwise (ccw) from the positive x-axis; • is conveniently measured in radians (SI unit), where 1 rad is the angle subtended at the centre by an arc lengths = r; • is the single time-dependent quantity of circular motion. 

6. ANGULAR POSITION Notes: • ands = r ( in rad). • The radian is a dimensionless unit (as is any unit of angle). • y r s  O • x

7. ANGULAR VELOCITY • Change in angular position is called angular displacement, . tf = ti + t • y r  Analogous to linear motion, the rate of change of angular position is called average angular velocity: ti f r O i • x Allowing t0, we get (instantaneous) angular velocity: Units: [rad/s] (SI), but also[°/s, rev/s, and rev/min  rpm]

8. ANGULAR VELOCITY Notes: • A particle moves with uniform circular motion if and only if its angular velocity  is constant. • , sign by inspection… • Angular velocity is positive for counterclockwise motion…. • …negative for clockwise motion. • The graphical relationships we developed for position s and velocity vsin linear motion apply equally well to angular position  and angular velocity …  > 0  < 0

9. (rad) 2 0 • t (s) 2 4 6 8 –2 –4  (rad/s)  0 2 4 6 8 • t (s) –2 POSITION GRAPHS  VELOCITY GRAPHS Angular velocity  is equivalent to the slope of a -vs-t graph. Eg: A particle moves around a circle… For the first 3 s the is  slope velocity t

10. POSITION GRAPHS  VELOCITY GRAPHS Angular velocity  is equivalent to the slope of a -vs-t graph. (rad) Eg: A particle moves around a circle… 2 0 Between 3 s and 4 s the is • t (s) 2 4 6 8 –2 slope velocity –4  (rad/s)  0 2 4 6 8 • t (s) –2

11. POSITION GRAPHS  VELOCITY GRAPHS Angular velocity  is equivalent to the slope of a -vs-t graph. (rad) Eg: A particle moves around a circle… 2 0 Between 4 s and 8 s the is • t (s) 2 4 6 8 –2 slope velocity  –4 t  (rad/s)  0 2 4 6 8 • t (s) –2

12.  (rad/s) 2  0 • t (s) 2 4 6 8 FINDING POSITION FROM VELOCITY • A body’s angular position after a time interval t can be determined from its angular velocity using . Graphically, the change in angular position ( =t) is given by the area “under” a -vs-t graph: During the time interval 2 s to 8 s the body’s angular displacement is  t

13. THE rtz-COORDINATE SYSTEM • To facilitate the resolution of angular quantities, we introduce the rtz-coordinate system (centred on the orbiting particle and travelling around with it) in which… • the r-axis (radial axis) points from the particle towards the centre of the circle; • the t-axis (tangential axis) is tangent to the circle, pointing in the anticlockwise direction; • the z-axis is perpendicular to the plane of motion. • z • z • t r r O • t

14. THE rtz-COORDINATE SYSTEM • Viewed from above (with the z-axis pointing out of the screen) the axes are shown travelling around with the particle… • z • t r

15. r • t  THE rtz-COORDINATE SYSTEM Notes: • As in xyz-coordinate system, the r-, t-, and z-axes are mutually perpendicular. • The rtz-coordinate system is used only to resolve vector quantities associated with circular motion into radial and tangential components. The measurement of these quantities must necessarily take place in otherreference frames. • Given some vector in the plane of motion, making an angle of  with the r-axis, • Ar = Acos • At = Asin Asin Acos

16. VELOCITY and ANGULAR VELOCITY • The velocity vector has only a tangential component, vt . • t vt vt s s = r Differentiating with respect to time…  r r O [ rad /s] Hence vt = r and vr = 0 vt = r vz = 0

17. ACCELERATION and ANGULAR VELOCITY • Although the magnitude of remains constant in uniform circular motion, its direction changes continuously, so the particle must be accelerating. Motion diagram analysis reveals that the acceleration is centripetal. Notes: • For uniform circular motion, since the lengths of successive ’s are all the same, the magnitude of is constant. • These are all average velocity vectors…

18. ACCELERATION and ANGULAR VELOCITY Q • The instantaneous velocity and acceleration vectors are everywhere at right angles to each other. Q'  P' During time interval t … r • the particle travels an arc length vt between P and P' (PP' vt); • both the angular position and turn through angles of  ; P  r O …so OPP' ||| P'QQ'

19. ACCELERATION and ANGULAR VELOCITY • In vector notation: • t vt And since v= r… ar = 2r ar r O Centripetal acceleration has only a radial component, ar … at = 0az = 0

20. DYNAMICS OF UNIFORM CIRCULAR MOTION • From Newton II… • z • t r O Note! As always, is simply the result of any number of forces being applied by identifiable agents.(It is NOT some new, disembodied force!)

21. DYNAMICS OF UNIFORM CIRCULAR MOTION • In terms of r-, t-, and z-components: • z • t r O Necessarily so!

22. z • t r • Determine the maximum speed at which a car can corner on an unbanked, dry tar road without skidding. r O v will be a maximum when fs reaches its maximum value: fs= fs max = sn Fz = n – w = 0  n = w = mg

23. z • t r • z r  • A highway curve is banked at an angle  to the horizontal. Determine the maximum speed at which a car can take this corner without the assistance of friction. r O nr = nsin Fz = nz– w = 0  nz= ncos = w = mg nz  nr

24. CIRCULAR ORBITS • The force which keeps satellites (including the Moon) moving in circular orbits around the Earth is nothing other than the gravitational force of the Earth on them. A near-Earth satellite will maintain its circular orbit only if its centripetal acceleration ar is equal to g. r vorbit I.e. if

25. In 1957 Earth’s first artificial satellite, Sputnik I, was put into orbit 300 km above the Earth’s surface by the USSR. How long did observers have to wait between sightings? Whereas the period of Earth’s natural satellite, the Moon (384 000 km away) is… ??! The problem lies in the fact that g is only a local constant which can be used only near the surface of the Earth…

26. NEWTON’S LAW OF UNIVERSAL GRAVITATION • Any two particles in the universe exert a mutually attractive force on each other which is proportional to the product of their masses and inversely proportional to the square of the distance of their separation. Notes: • G is the universalgravitation constant. • G =6.67  10–11 Nm2/kg2. • The equation holds for extended spherical masses (e.g. planets) provided r, the distance between their centres, is large compared to their sizes.

27. NEWTON’S LAW OF UNIVERSAL GRAVITATION For a body of mass m at the surface of the Earth: PHY1012F But this is the body’s weight, w = mg… Hence, for Earth, (Why not 9.80 m/s2?) The value of the gravitational constant on the surface of any planet, gplanet, is thus a direct consequence of the size and mass of that planet. 27

28. VARIATION OF g WITH HEIGHT ABOVE GROUND 0 m Sea level 9.83 5 900 m Kilimanjaro 9.81 10 000 m Jet airliner 9.80 350 000 m International Space Station 8.85 35 900 000 m Geosynchronous satellite 0.22 For a satellite a distance h above the earth’s surface:

29. CIRCULAR ORBITS vorbit • We can now derive more universal formulae for any satellite: m FM on m r M And, since ,

30. CIRCULAR ORBITS So the correct period of the Moon is… T = 2.37  106 s = 27.4 days

31. INERTIAL and GRAVITATIONAL MASS • The connection between inertial mass (found by meas-uring a body’s acceleration a in response to a force F) and the gravitational mass which causes two bodies to attract each other is not immediately apparent… To date, however, the most precise experiments have been unable to determine any measurable difference between the two. Einstein’s general theory of relativity explains this principle of equivalence (minert = mgrav) as a fundamental property of space-time.

32. NON-UNIFORM CIRCULAR MOTION If the speed of an orbiting body varies, the body is exhibiting non-uniform circular motion. • t r • t PHY1012F at ar  In such cases, in addition to centripetal acceleration, the body also has non-zero tangential acceleration: ar  at The net acceleration vector, , is given by , where and . 32

33. t r DYNAMICS OF NON-UNIFORM CIRCULAR MOTION • The resultant force (the sum of any number of individual forces) acting on an orbiting particle can always be resolved into tangential and radial components if required… (Fnet)r  (Fnet)t

34. VERTICAL CIRCLES Motion in a vertical circle is NOT uniform. PHY1012F As a result of gravity, on its way down, the body speeds up; on the way up it slows down... • going down, at& vt are parallel; • going up, at& vt are antiparallel. at = 0 ar ar at at ar ar at at ar ar at = 0 at at Only at the top and bottom is at = 0. Elsewhere, the net acceleration is given by . The magnitude and direction of this net acceleration change continuously in a complex way… 34

35. VERTICAL CIRCLES PHY1012F The net force, , which produces this acceleration, is made up of the body’s weight and the tension force provided by the string. Like the acceleration, varies around the circle… …but at the top and bottom, where at = 0, is centripetal. Note: • At the top of the circle is the sumof and . • At the bottom, is given by the differenceof the two. 35

36. r • t VERTICAL CIRCLES • Apparent weight is actually a sensation arising from the contact forces which support you, rather than an awareness of the gravitational force of the Earth which acts simultaneously on every part of you. At the bottom of a vertical circle… vbot The extra force required to achieve this is what “adds to your g’s” in a bottom turn.

37. t r VERTICAL CIRCLES vtop At the top of a vertical circle… If (because of lack of speed) this term becomes too small (i.e. <w), n disappears, the body “comes unstuck” and goes into free fall. The speed at which n = 0is called the critical speed,vc: vc

38. ACCELERATION DUE TO GRAVITY • We are now in a position to understand why the measured value of g= 9.80 m/s2 is less than the value we calculated from Newton’s law of universal gravitation (g= 9.83 m/s2). Objects on the rotating Earth are in circular motion, so there must be a net force towards the centre. Thus wapp = mgapp = n < Fgrav. At mid-latitudes the reduction is about 0.03 m/s2, hence the measured value gapp= 9.80 m/s2.

39. MOTION IN A CIRCLE • Apply kinematics and dynamics knowledge, skills and techniques to circular motion. • Manipulate angular quantities and formulae against the background of an angular (rtz-) coordinate system. Learning outcomes:At the end of this chapter you should be able to…

40. NEWTON’S LAWS • The goals of Part I, Newton’s Laws, were to… • Learn how to describe motion both qualitatively and quantitatively so that, ultimately, we could analyse it mathematically. • Develop a “Newtonian intuition” for the explanation of motion: the connection between force and acceleration.