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Chapter 8 Rotational Motion Rotational Motion Angular Distance (q) o Replaces distance for rotational motion o Measured in Degrees Radians Revolutions q Radian Measure r r 1 rad r 1 rad = 57.3 degrees 2p rad in one circle Windows Calculator Rotational Motion Speed of Rotation (w) w = Angle covered/Time required o o = Dq/Dt Note similarity to v = Dx/Dt o Measured in degrees/second radians/second revolutions/second w Rotational Motion Angular Acceleration - Measures how angular velocity is changing (a) a = Dw/Dt Note similarity to a = Dv/Dt o Measured in … degrees/s2 radians/s2 revolutions/s2 Rotational Inertia Property of an object that resists changes in rotation • For linear motion mass was a measure of inertia • For rotational motion Moment of Inertia (I) is the measure of rotational Inertia Moments of Inertia Depends on … o Mass of the Object o Axis of Rotation o Distribution of Mass in the Object Moments of Inertia Standard Shapes Moment of Inertia Inertia Bars Ring and Disk on Incline Metronome People walking Weighted Stick - Bare Stick Torque Product of Force and Lever Arm o Torque = Force X Lever Arm Examples: o Balance o See-Saw o Wrench W1d1 = W2d2 Sample Torque Problem (0.5 kg)(9.8 m/s2)(0.1 m) = (0.2 kg)(9.8 m/s2)d 0.5 kg d (0.1 m) 0.2 kg d 0.25 m F Line of Action Lever Arm Torque Examples Torque Just as unbalanced forces produce acceleration, unbalanced torques produce angular acceleration. Compare: SF = ma St = Ia Center of Mass Average position of the mass of an object o Newton showed that all of the mass of the object acts as if it is located here. o Find cm of Texas/USA Finding the Center of Mass Line of action Pivot point Lever arm Torque weight No Torque High Jumper Stability In order to balance forces and torques, the center of mass must always be along the vertical line through the base of support. Demo • Coke bottle • Chair pick-up Stability Base of Support Stability Which object is most stable? Centripetal Force Any force that causes an object to move in a circle. Examples: • • • • • Carousel Water in a bucket Moon and Earth Coin and hanger Spin cycle Centripetal force F = mac = mv2/r 2 = mrw Centrifugal force Fictitious center fleeing force o Felt by object in an accelerated reference frame Examples: o Car on a circular path o Can on a string Space Habitat (simulated gravity) w r Space Habitat (simulated gravity) “Down” is away from the center The amount of “gravity” depends on how far from the center you are. Angular Momentum L = (rotational inertia) X (angular velocity) L = Iw Compare to linear momentum: p = mv Linear Momentum and Force Angular Momentum and Torque Linear o Impulse Rotational o Rotational Impulse SF = Dp/Dt Dp = SF Dt St = DL/ Dt DL = St Dt Conservation of Momentum Linear o If SF = 0, then p is constant. Angular o If St = 0, then L is constant. Conservation of Angular Momentum Ice Skater Throwing a football Rifling Helicopters Precession Rifling Football Physics L Helicopter Physics Rotation of Rotor Body Rotation Tail rotor used to produce thrust in opposite direction of body rotation Precession Age of Aquarius Linear - Rotational Connections Linear x (m) Rotational q (rad) v (m/s) w (rad/s) a (m/s2) a (rad/s2) m (kg) F (N) I (kg·m2) t (N·m) p (N·s) L (N·m·s)