Rotational Motion Angular Displacement, Velocity, Acceleration Rotation w/constant angular acceleration Torque Moment of Inertia Angular Displacement (π) β’ Angular = Rotational β’ Measured in Radians β’ 1 complete rotation = 2π radians β’ Easy to convert from angular to translational π β’ π = ππ π Examples A dog (r = 0.12m) makes 3 complete rotations as it rolls down a hill. β’ What is the angular displacement of the dog? β’ What is the linear displacement of the dog? β’ A cat runs 7m on wheel with a radius of 0.75 m. How many rotations does the wheel make? Angular Velocity = (π) β’π£= π π‘ = π π‘ = πππΞ€π ππ β’ rpm = rotations per minute β’ πππ β 2π 60 = π β’ π£ = ππ EXAMPLE β’ Convert 45rpm to rad/s β’ What is the translation Velocity on the outer edge of the record? (diameter = 30cm) Angular Acceleration = (Ξ±) β’πΌ= βπ π‘ ππ‘ Translational Acceleration β’ ππ‘ = πΌπ Centripetal Acceleration β’ ππ = π£2 π = Ο2π πΌ ππ Example: A Washing machine motor accelerates from 0 -200rpm in 5 seconds. β’ What is the angular acceleration of the motor in rad/s2? β’ A small pony is in the machine (r = 0.30m), what is the translational acceleration of the pony as the motor accelerates? β’ What is the centripetal acceleration of the pony? Motion with Constant Acceleration π£= π π‘ π π= π‘ d=vt +di π = ππ‘ + ππ π= βπ£ π‘ βπ πΌ= π‘ vf =at + vi π = πΌπ‘ + ππ 1 2 π = ππ‘ + π£ππ‘ + ππ 2 1 π = (π£1 + π£2)π‘ 2 π£π2 = 2ππ + π£π2 1 2 π = πΌπ‘ + πππ‘ + ππ 2 1 π = (π1 + π2)π‘ 2 ππ2 = 2πΌπ + ππ2 Example: A biker accelerates from rest to 8 m/s in 5 seconds. The radius of the bicycle wheels is 0.25 m. β’ What is the angular acceleration of the wheels? β’ How far does the biker travel? β’ How many rotations does the wheel make? β’ If he gets tired and comes to rest in 10m, what is the angular acceleration of the wheels? Example: A centrifuge accelerates from rest to 1500 rpm in 150 seconds. β’ How many rotations does the centrifuge make? β’ What is the angular acceleration in rad/s2 Example: A fan moving at 250 rev/minutes. It turns through 120 revolutions as it comes to rest. β’ What is the angular acceleration of the fan? β’ How long did take to come to rest? Example: π =? πππ π = 0.04π π = 90πππ π = 0.10π r = 0.35m V= TORQUE β’ Causes a change in angular motion β’ Product of Force and Lever Arm β’ Lever Arm = Perpendicular Distance from axis of rotation to line through which the force acts. π = πΉππ πππ = πΉπ‘πππ TORQUE A wrench has length of 30.0cm. A force of 200N is applied at the end of the lever A) What is the maximum torque than can be applied? B) What is the torque when the force is applied at an angle of 30o to the wrench? TORQUE β’ Vector cross product π =π×πΉ β’ Direction of Torque determined Right Hand Rule TORQUE β’ Determine the torque exerted on the wheel below. r1=0.40m, r2=0.20m, F1=30.0 N, F2=40.0 N, TORQUE β’ Causes a change in angular motion β’ π =πΉβπ β’ πΉ = ππ β π = ππ π β’ π = πΌπ β π = π πΌπ π = ππ2πΌ β’ ππ2 = πΌ = ππππππ‘ ππ πΌππππ‘ππ β’ π = π°πΆ β πππ€π‘πππ 2ππ πΏππ€ πππ π ππ‘ππ‘πππ Moment of Inertia β’ Rotational Analog of Mass β’ Depends on distribution of mass β’ Chart on Pg. 208 TORQUE π = π°πΆ β πππ€π‘πππ 2ππ πΏππ€ πππ π ππ‘ππ‘πππ Ξ± =? r=0.30m m=4.0m Solid disc F=20N TORQUE F=20N ΞΌ=? Ξ±=3.0rad/s2 m=2.0kg r=0.40m TORQUE β’ The 2 Helicopter blades accelerate from 0 -500rpm in 30 seconds β’ Mass of blade = 150kg β’ Length of blade = 5.0m β’ What torque is required? Moment of Inertia determine the moment of Inertia around the center of the object below Length=0.5 Mass = 2.0kg Length=0.5 Mass = 2.0kg Radius=0.5 Mass = 12.0kg Review Radius of wheel = 0.15m Position of running mouse = .05m Velocity of running mouse=0.80m/s Angular Velocity of wheel? Linear Velocity of outer mouse? Centripetal Acceleration of outer mouse? mass of wheel = 0.250kg Radius of wheel = 0.15m Time to reach max speed = 1.50s Torque required? Force exerted by inner mouse? Number of Rotations?
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