# Chapter 8 Rotational Motion Rotational Motion n Angular

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Chapter 8 Rotational Motion

Rotational Motion n Angular Distance (q) o Replaces distance for rotational motion o Measured in n Degrees n Radians n Revolutions q

Radian Measure r r 1 rad = 57. 3 degrees 2 p rad in one circle

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Rotational Motion n Speed of Rotation (w) o w = Angle covered/Time required o o = Dq/Dt Note similarity to v = Dx/Dt o Measured in degrees/second n radians/second n revolutions/second n w

Rotational Motion v Angular Acceleration - Measures how angular velocity is changing (a) o a = Dw/Dt Note similarity to a = Dv/Dt o Measured in … degrees/s 2 n radians/s 2 n revolutions/s 2 n

Rotational Inertia n 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 n 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 Bars v Ring and Disk on Incline v Metronome v People walking v Weighted Stick - Bare Stick v

Torque n Product of Force and Lever Arm o Torque = Force X Lever Arm n Examples: o Balance o See-Saw o Wrench

W 1 d 1 = W 2 d 2

Sample Torque Problem (0. 5 kg)(9. 8 m/s 2)(0. 1 m) = (0. 2 kg)(9. 8 m/s 2)d

F Line of Action Lever Arm

Torque Examples

Torque n n Just as unbalanced forces produce acceleration, unbalanced torques produce angular acceleration. Compare: SF = ma St = Ia

Center of Mass n 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

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Stability n n 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 n Which object is most stable?

Centripetal Force n n 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 2 = mv /r 2 = mrw

Centrifugal force n Fictitious center fleeing force o Felt by object in an accelerated reference frame n Examples: o Car on a circular path o Can on a string

Space Habitat (simulated gravity) w r

Space Habitat (simulated gravity) n n “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 n Linear o Impulse n Rotational o Rotational Impulse SF = Dp/Dt Dp = SF Dt St = DL/ Dt DL = St Dt

Conservation of Momentum n Linear o If SF = 0, then p is constant. n Angular o If St = 0, then L is constant.

Conservation of Angular Momentum n n n 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/s 2) a (rad/s 2) m (kg) F (N) I (kg·m 2) t (N·m) p (N·s) L (N·m·s)

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