Physics 31 Lectue 17 Main points of today s lectue: Centipetal acceleation: a c = a c t Rotational motion definitions: Δω Δω α =, α = limδ t 0 Δt Δt Δ s= Δ θ;t = ω;at = α Rotational kinematics equations: Δθ = ωt ω = ω + αt ω = 1 1 ( ω + ω ) Δθ = ( ω + ω ) 0 1 Δθ = ω0t + αt ω = ω0 + αδθ Rolling motion: Δx = s = Δθ = ω; a = α 0 0 t
Foces and unifom cicula motion Conside the mass undegoing hoizontal cicula motion at the ight. At any point, the instantaneous elocity is tangential to the cicle. Because the diection of the tangent changes, howee, the diection of the elocity changes. Δ = f 0 i This means thee is an acceleation pointing inwad towads the cente of the cicle. If we look at the component of the acceleation in this diection: a Δ Δθ = lim = lim = ω = Δt Δt c Δ t 0 Δ t 0 Hee we hae used: a c t ω = a c = / is called the centipetal acceleation
Reading Quiz 1. Fo unifom cicula motion, the acceleation A. is paallel to the elocity. B. is diected towad the cente of the cicle. C. is lage fo a lage obit at the same speed. D. is always due to gaity. E. is always negatie. Slide 6-6
Reading question. When a ca tuns a cone on a leel oad, which foce poides the necessay centipetal acceleation? A. Fiction B. Tension C. Nomal foce D. Ai esistance E. Gaity Slide 6-9
Motion in a hoizontal cicle: A ca going though a bend m/s 150 m a c? m/s? μ s 0.5 m A ca is going down a winding oad with a speed of m/s and is going aound a cue with a adius = 150 m. What is the centipetal acceleation? a) 0.16 m/s b) 3.3 m/s a c = / =(m/s) /150 c) 9.8 m/s =3.3 m/s The coefficient of static fiction between oad and ca is 0.5. What is the minimum adius the bend can hae without the ca sliding of the oad? f s = ma c = m / also f s μ s mg s c At the minimum adius f s = mμ s g=m / = /(μ s g )=(m/s ) /(0.5 9.81m/s)=99m
Conceptual question You ae a passenge in a ca and not weaing you seat belt.without inceasing o deceasing its speed, the ca makes a shap left tun, and you find youself colliding with the ight-hand doo. Which is the coect analysis of the situation? a) Befoe and afte the collision, thee is a ightwad foce pushing you into the doo. b) Stating ti at the time of collision, i the doo exets a leftwad foce on you. c) both of the aboe d) neithe of the aboe
Checking Undestanding: Cicula Motion Dynamics When the ball eaches the beak in the cicle, which path will it follow? Slide 6-1
Conceptual question 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? a) b) c) d) e)
Checking Undestanding When a ball on the end of a sting is swung in a hoizontal cicle: What is the diection of the acceleation of the ball? A. Tangent to the cicle, in the diection of the ball s motion B. Towad the cente of the cicle Slide 6-15
Example Conside the motion o of a 1 kg mass constained to moe in a etical cicle of adius = m by a massless wie. What is the tension in this wie if the mass is at the side of the cicle (wie) is hoizontal and is moing with an instantaneous speed of 3 m/s? a) 1.5 N b).3 N c) 4.5 N d) 0 (3m / s) t T= m = 1kg = 4.5N 3 m/s m m ac m 1k kg T?
Example Conside the motion o of a 1 kg mass constained to moe in a etical cicle of adius = m by a massless wie. a) What is the tension in this wie if the mass is at the bottom of the cicle and is moing with a instantaneous speed of 3 m/s? a) 4.5N b) 9.8 N T mg= mac = m c) 14.3N d) 4.3N T= m(g + ) ( ) 3m / s T= 1kg(9.8m /s + ) T = 14.3N m t a c 3 m/s Hint: thee ae two foces in the y diection. i m 1 kg T? m
Example Conside the motion o of a 1 kg mass constained to moe in a etical cicle of adius = m by a massless wie. What is the tension in this wie if the mass is at the top of the cicle and is moing with an instantaneous speed of 3 m/s? a) 5.3N b) 9.8 N c) T + mg = ma c = m c) 5.3N d) 0 ( 3m / s) T = m( g) = 1kg( 9.8m /s ) m t a a c T = 5.3N Result is impossible, tension cannot be negatie T = 0 3 m/s m m 1 kg T?