Cicula and Rotational Motion Sections 5.-5.5 and Chapte 10
Basic Definitions
Unifom Cicula Motion Unifom cicula motion efes to the motion of a paticle in a cicula path at constant speed. The instantaneous velocity is always tangent to the cicle.
Centipetal Acceleation Looking at the change in velocity in the limit that the time inteval becomes infinitesimally small, we see that the coesponding acceleation is given by: v a R
Centipetal Acceleation This acceleation is called the centipetal, o adial, acceleation, and it points towad the cente of the cicle.
Centipetal Foce If an object is being acceleated then a net foce must be acting on it. We aleady know the acceleation, so can immediately wite the foce: v F R ma R m
If we think about a ball on a sting, we can see that the foce must point inwad, towads the cente of the cicula path. Stings only pull; they neve push. Centipetal Foce
In the game of tetheball, the stuck ball whils aound a pole. In what diection does the net foce on the ball point? ConcepTest 5.6 Tetheball 1) towad the top of the pole ) towad the gound 3) along the hoizontal component of the tension foce 4) along the vetical component of the tension foce 5) tangential to the cicle T W
In the game of tetheball, the stuck ball whils aound a pole. In what diection does the net foce on the ball point? ConcepTest 5.6 Tetheball 1) towad the top of the pole ) towad the gound 3) along the hoizontal component of the tension foce 4) along the vetical component of the tension foce 5) tangential to the cicle The vetical component of the tension balances the weight. The hoizontal component of tension povides the centipetal foce that W T T points towad the cente of the cicle. W
ConcepTest 5.9 Ball and Sting Two equal-mass ocks tied to stings ae whiled in hoizontal cicles. The adius of cicle is twice that of cicle 1. If the peiod of motion is the same fo both ocks, what is the tension in cod compaed to cod 1? 1) T = ¼T 1 ) T = ½T 1 3) T = T 1 4) T = T 1 5) T = 4T 1
ConcepTest 5.9 Ball and Sting Two equal-mass ocks tied to stings ae whiled in hoizontal cicles. The adius of cicle is twice that of cicle 1. If the peiod of motion is the same fo both ocks, what is the tension in cod compaed to cod 1? 1) T = ¼T 1 ) T = ½T 1 3) T = T 1 4) T = T 1 5) T = 4T 1 The centipetal foce in this case is given by the tension, so T = mv /. Fo the same peiod, we find that v = v 1 (and this tem is squaed). Howeve, fo the denominato, we see that = 1 which gives us the elation T = T 1.
A Ping-Pong ball is shot into a cicula tube that is lying flat (hoizontal) on a tabletop. When the Ping-Pong ball leaves the tack, which path will it follow? ConcepTest 5.8 Missing Link
ConcepTest 5.8 Missing Link A Ping-Pong ball is shot into a cicula tube that is lying flat (hoizontal) on a tabletop. When the Ping-Pong ball leaves the tack, which path will it follow? Once the ball leaves the tube, thee is no longe a foce to keep it going in a cicle. Theefoe, it simply continues in a staight line, as Newton s Fist Law equies! Follow-up: What physical foce povides the centipetal foce?
Centifugal Foce Thee is no centifugal foce pointing outwad. If the centipetal foce vanishes, the object flies off at a tangent to the cicle, not outwads.
Pola Coodinates
Definitions ), ( y x ), ( j y i x ˆ sin ˆ cos sin cos sin cos y x y x x y tan
3D Coodinates Cylindical Coodinates Spheical Coodinates
Angula Velocity, w In unifom cicula motion: = constant = (t) Given a peiod T, and given that thee ae p adians in a full cicle, then: w p T
Position Function, (t) Any velocity is defined as the ate of change of the position with espect to time. Theefoe: d w dt Integating both sides with espect to t, we get: ( t) wt Notice that (t=0) = 0, theeby 0 is the initial angle. 0
Linea Velocity i x ˆ ) cos( ) ( 0 w t t What is the elationship between v and w? j y ˆ ) sin( ) ( 0 w t t Diffeentiating both sides with espect to t ) ( sin ) ( cos ) ( sin ) ( cos ' ' 0 0 0 0 w w w w w w w t t t t y x v j y ˆ ) cos( ) ( ' 0 w w t t Then, i x ˆ ) sin( ) ( ' 0 w w t t v w Theefoe,
Linea Velocity v
Centipetal Acceleation, a c Diffeentiating x(t) and y(t) again, we get x y ( ) t w cos( wt ) ˆi " 0 ( t) w sin( wt ) ˆj " 0 Using the Pythagoean theoem once again, we find that a c w Theefoe, v a c
Centipetal Acceleation, a c v a c
Highway Cuves
Banked and Unbanked Cuves When a ca goes aound a cuve, thee must be a net foce towad the cente of the cicle of which the cuve is an ac. If the oad is flat, that foce is supplied by fiction.
Banked and Unbanked Cuves If the fictional foce is insufficient, the ca will tend to move moe nealy in a staight line, as the skid maks show.
Banked and Unbanked Cuves As long as the ties do not slip, the fiction is static. If the ties do stat to slip, the fiction is kinetic, which is bad in two ways: 1. The kinetic fictional foce is smalle than the static.. The static fictional foce can point towad the cente of the cicle, but the kinetic fictional foce opposes the diection of motion, making it vey difficult to egain contol of the ca and continue aound the cuve.
Banked and Unbanked Cuves Banking the cuve can help keep cas fom skidding. In fact, fo evey banked cuve, thee is one speed at which the entie centipetal foce is supplied by the hoizontal component of the nomal foce, and no fiction is equied. This occus when: v F sin m N
ConcepTest 5.7a Aound the Cuve I You ae a passenge in a ca, not weaing a seat belt. The ca makes a shap left tun. Fom you pespective in the ca, what do you feel is happening to you? 1) you ae thown to the ight ) you feel no paticula change 3) you ae thown to the left 4) you ae thown to the ceiling 5) you ae thown to the floo
ConcepTest 5.7a Aound the Cuve I You ae a passenge in a ca, not weaing a seat belt. The ca makes a shap left tun. Fom you pespective in the ca, what do you feel is happening to you? 1) you ae thown to the ight ) you feel no paticula change 3) you ae thown to the left 4) you ae thown to the ceiling 5) you ae thown to the floo The passenge has the tendency to continue moving in a staight line. Fom you pespective in the ca, it feels like you ae being thown to the ight, hitting the passenge doo.
ConcepTest 5.7b Aound the Cuve II Duing that shap left tun, you found youself hitting the passenge doo. What is the coect desciption of what is actually happening? 1) centifugal foce is pushing you into the doo ) the doo is exeting a leftwad foce on you 3) both of the above 4) neithe of the above
ConcepTest 5.7b Aound the Cuve II Duing that shap left tun, you found youself hitting the passenge doo. What is the coect desciption of what is actually happening? 1) centifugal foce is pushing you into the doo ) the doo is exeting a leftwad foce on you 3) both of the above 4) neithe of the above The passenge has the tendency to continue moving in a staight line. Thee is a centipetal foce, povided by the doo, that foces the passenge into a cicula path.
ConcepTest 5.7c Aound the Cuve III You dive you dad s ca too fast aound a cuve and the ca stats to skid. What is the coect desciption of this situation? 1) ca s engine is not stong enough to keep the ca fom being pushed out ) fiction between ties and oad is not stong enough to keep ca in a cicle 3) ca is too heavy to make the tun 4) a dee caused you to skid 5) none of the above
ConcepTest 5.7c Aound the Cuve III You dive you dad s ca too fast aound a cuve and the ca stats to skid. What is the coect desciption of this situation? The fiction foce between ties and oad povides the centipetal foce that keeps the ca moving in a cicle. If this foce is too small, the ca continues in a staight line! 1) ca s engine is not stong enough to keep the ca fom being pushed out ) fiction between ties and oad is not stong enough to keep ca in a cicle 3) ca is too heavy to make the tun 4) a dee caused you to skid 5) none of the above Follow-up: What could be done to the oad o ca to pevent skidding?
Nonunifom Cicula Motion
Vaying Speed If an object is moving in a cicula path but at vaying speeds, it must have a tangential component to its acceleation as well as the adial one.
Iegula Path This concept can be used fo an object moving along any cuved path, as any small segment of the path will be appoximately cicula.
ConcepTest 5.10 Bael of Fun A ide in a bael of fun finds heself stuck with he back to the wall. Which diagam coectly shows the foces acting on he? 1 3 4 5
ConcepTest 5.10 Bael of Fun A ide in a bael of fun finds heself stuck with he back to the wall. Which diagam coectly shows the foces acting on he? 1 3 4 5 The nomal foce of the wall on the ide povides the centipetal foce needed to keep he going aound in a cicle. The downwad foce of gavity is balanced by the upwad fictional foce on he, so she does not slip vetically. Follow-up: What happens if the otation of the ide slows down?
ConcepTest 5.11a Going in Cicles I You e on a Feis wheel moving in a vetical cicle. When the Feis wheel is at est, the nomal foce N exeted by you seat is equal to you weight mg. How does N change at the top of the Feis wheel when you ae in motion? 1) N emains equal to mg ) N is smalle than mg 3) N is lage than mg 4) none of the above
ConcepTest 5.11a Going in Cicles I You e on a Feis wheel moving in a vetical cicle. When the Feis wheel is at est, the nomal foce N exeted by you seat is equal to you weight mg. How does N change at the top of the Feis wheel when you ae in motion? 1) N emains equal to mg ) N is smalle than mg 3) N is lage than mg 4) none of the above You ae in cicula motion, so thee has to be a centipetal foce pointing inwad. At the top, the only two foces ae mg (down) and N (up), so N must be smalle than mg. Follow-up: Whee is N lage than mg?
ConcepTest 5.11b Going in Cicles II A skie goes ove a small ound hill with adius R. Because she is in cicula motion, thee has to be a centipetal foce. At the top of the hill, what is F c of the skie equal to? 1) F c = N + mg ) F c = mg N 3) F c = T + N mg 4) F c = N 5) F c = mg v R
ConcepTest 5.11b Going in Cicles II A skie goes ove a small ound hill with adius R. Because she is in cicula motion, thee has to be a centipetal foce. At the top of the hill, what is F c of the skie equal to? 1) F c = N + mg ) F c = mg N 3) F c = T + N mg 4) F c = N 5) F c = mg F c points towad the cente of the cicle (i.e., downwad in this case). The weight vecto points down and the nomal foce (exeted by the hill) points up. The magnitude of the net foce, theefoe, is F c = mg N. v mg N R Follow-up: What happens when the skie goes into a small dip?
ConcepTest 5.11c Going in Cicles III You swing a ball at the end of sting in a vetical cicle. Because the ball is in cicula motion thee has to be a centipetal foce. At the top of the ball s path, what is F c equal to? 1) F c = T mg ) F c = T + N mg 3) F c = T + mg 4) F c = T 5) F c = mg v top R
ConcepTest 5.11c Going in Cicles III You swing a ball at the end of sting in a vetical cicle. Because the ball is in cicula motion thee has to be a centipetal foce. At the top of the ball s path, what is F c equal to? 1) F c = T mg ) F c = T + N mg 3) F c = T + mg 4) F c = T 5) F c = mg F c points towad the cente of the cicle (i.e., downwad in this case). The weight vecto points down and the tension (exeted by the sting) also points down. The magnitude of the net foce, theefoe, is F c = T+ mg. v mg T R Follow-up: What is F c at the bottom of the ball s path?
Cicula Motion Poblems
Poblem 5.34 What is the maximum speed with which a 100-kg ca can ound a tun of adius 80.0 m on a flat oad if the coefficient of fiction between the ties and the oad is 0.65? F f F s N m v F f mg s m v v v g s v 3 m/s
Poblem 5.35 A child sitting1.0 m fom the cente of a mey-go-ound moves with a speed of 1.30 m/s. (a) Calculate the centipetal acceleation of the child. a c v (1.30 m/s) 1.0 m/s a c v a c 1.41m/s
Poblem 5.35 (b) Calculate the net hoizontal foce exeted on the child (mass m =.5 kg). Newton s nd Law: F c ma c m v F c v F c 3 N
Poblem 5.47 A jet pilot takes his aicaft on a vetical loop. (a) If the jet is moving at a speed of 100 km/h at the lowest point of the loop, detemine the minimum adius of the cicle so that the centipetal acceleation at the lowest point does not exceed 6.0 g s. v (333.333 m/s) a c a c 58.8 m/s v 1.910 3 m a c v
Poblem 5.47 (b) Calculate the 78-kg pilot s effective weight (the foce with which the seat pushes up on him) at the bottom of the cicle. Applying Newton s nd Law: y F N Fy F c F c F F N mg N F c v m F N 7. 0mg mg F c 5.410 3 w N
Poblem 5.47 (c) Calculate the pilot s effective weight at the top of the cicle (assume the same speed). F y F c y F mg N F c F c w F N mg 6. 0mg F N F N 5. 0mg F N 3.810 3 N
Math Check 04
Math Check 04 1) d [sin(cos( x))] dx ) d [sin( x)cos( x)] dx 3) d dx [ln( e x )] d 4) [ xln( x)] dx