Physics 111 Lectue 14 (Walke: Ch. 6.5) Cicula Motion Centipetal Acceleation Centipetal Foce Febuay 7, 009 Midtem Exam 1 on Wed. Mach 4 (Chaptes 1-6) Lectue 14 1/8 Connected Objects If thee is a pulley, it is easiest to hae the coodinate system follow the sting: Lectue 14 /8
Example: Connected Blocks Block of mass m 1 slides on a fictionless tabletop. It is connected by a sting and pulley to a hanging mass m. Find the acceleation a and sting tension T. Lectue 14 3/8 F = T = ma T = ma 1, x 1 1, x ( ) F = mg T= ma T= m g a ( ) ma= m g a m g= ( m + m ) a 1 1 m mm 1 a = g and T = g m + m m + m 1 1 Lectue 14 4/8
Example: Atwood s Machine Atwood s Machine consists of two masses connected by a sting and pulley. Find the acceleation a. Lectue 14 5/8 F 1, x = T m g ma 1 = 1, x = = T m g = m 1 g T = m a m a ( m m1 ) g = ( m1 + m 1 ) a F m g T m a m m a = g m + m 1 1 Lectue 14 6/8
Unifom Cicula Motion Position: Radius: Speed: = constant = constant Lectue 14 7/8 Paticle in Unifom Cicula Motion T peiod = the time equied fo one complete otation. 1 cicumfeence π = = 1 peiod T Fo a paticle in unifom cicula motion, elocity ecto is constant in magnitude, but continuously changes diection. Paticle has acceleation Thee must be a net foce on paticle Lectue 14 8/8
Example: A Rotating Cankshaft A 4.0 cm diamete cankshaft tuns at 400 pm. What is speed of a point on suface of the cankshaft? 400 e 1 minute = 40 e/s 1 minute 60 s 1 T = = 0.05 s (pe eolution) 40 e/s π π (0.00 m) = = = 5.03 m/s T 0.05 s Lectue 14 9/8 Acceleation in Unifom Cicula Motion Object moing at a constant speed in a cicle of adius has a centipetal (towad cente) acceleation a cp Diection of this acceleation a cp is towads the cente of the cicle. Size of the acceleation is a cp = / Lectue 14 10/8
Centipetal Foce Since object in unifom cicula motion has centipetal acceleation a cp, a foce is needed to poide this acceleation. Foce must be poided by a ope, fiction, gaity, etc. Magnitude of foce f cp equied to keep an object of mass m moing at speed in cicle of adius, called centipetal foce because it points towad cente of cicle, is gien by: fcp = macp = m (6-16) Lectue 14 11/8 Souces of Centipetal Foce This centipetal foce may be poided by the tension in a sting, the nomal foce, o fiction, among othe souces. Lectue 14 1/8
Example: Rounding a Cone A ca ounds a cone of adius. If coefficient of fiction between ties and oad is µ s, what is maximum speed ca can hae without skidding? Fx = fs = µ sn = max = m Fy = 0 = N W = N mg µ smg = m = µ sg = (0.8)(45.0 m)(9.81 m/s ) = 19.0 m/s Question: Did we need to know mass of ca? Lectue 14 13/8 Banked Cues Lectue 14 14/8
If a oad is banked at the pope angle θ, a ca can ound a cue without the assistance of fiction between the ties and the oad and without skidding. What bank angle θ is needed fo a ca of mass m taelling at speed aound a cue of adius? Lectue 14 15/8 F = 0= Ncosθ W = Ncosθ mg y N cosθ = mg Fx = N sinθ = max = macp = m / N sin θ / tanθ m Ncosθ = = mg = g θ = actan = g Notice that thee is only one speed at which gaity exactly poides the needed centipetal foce. Lectue 14 16/8
Centifuge A centifuge is a laboatoy deice used in chemisty, biology, and medicine fo sepaating out solids suspended in a liquid by spinning tubes of the liquid at high speed. The liquid nomal foce cannot poide the needed centipetal foce on the solids, so the solids collect at the bottom of the tubes. Lectue 14 17/8 Tangential & Total Acceleation An object may be changing its speed (speeding up o slowing down) as it moes in a cicula path. In that case, thee is a tangential acceleation as well as a centipetal acceleation. The total acceleation a total is the ecto sum of the centipetal acceleation a cp, which points towad the cente of otation, and the tangential acceleation a t, which points in the diection of speed incease. Lectue 14 18/8
Centipetal Acceleation a cp = (centipetal acceleation) t = π T (tangential elocity) Lectue 14 19/8 Question Which motion has the lagest centipetal acceleation? Lectue 14 0/8
Example: Spinning in a Cicle An enegetic fathe places 0 kg child in a 5.0 kg cat to which is attached a.0 m long ope. He holds the end of the ope and spins the cat and child in a cicle, keeping the ope paallel to the gound. If tension in ope is 100 N, how many eolutions pe minute does cat make? Lectue 14 1/8 m ( Fnet ) = F = T = ; t t T (.0 m)(100 N) = = =.83 m/s; m (5 kg) (.83 m/s) = = 1.41 ad/s = 13.5 pm (.0 m) Lectue 14 /8
Example: Satellite s Motion Satellite moes at constant speed in a cicula obit about cente of Eath and nea the suface of Eath. If magnitude of its acceleation is g = 9.81 m/s and Eath s adius is 6,370 km, find its speed and time T equied fo one complete eolution. a cp = = g = g= = = 3 3 (6,370 10 m)(9.81 m/s ) 7.91 10 m/s 17,700 mi/h T= = = = 3 3 π/ π(6,370 10 m)/(7.91 10 m/s) 5,060 s 84.3 min Lectue 14 3/8 End of Lectue 14 Befoe Monday, ead Walke Chapte 7.1 Homewok Assignment #6d should be submitted using WebAssign by 11:00 PM on Monday, Mach. MIDTERM I on Wednesday Mach 4. (Chaptes 1-6.) Lectue 14 4/8