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Rotational motion, Angular displacement, angular velocity, angular acceleration Rotational energy Moment of Inertia Torque Chapter 10:Rotation of a rigid object about a fixed axis Reading assignment: Chapter 10.1 to10.4, 10.5 (know concept of moment of inertia, dont worry about integral calculation), 10.6 to 10.9 Homework:CQ1, CQ8, CQ13, QQ3, QQ4, AE1, AE3, OQ8, 2, 3, 6, 7, 12, 13, 15, 19, 26, 29, 35, 36, 38, 43, 49, 55, 56, 59 Due date: Monday, March 28 Midterm 2 coming up on Wednesday, March 30; (chapter 1-10)

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Planar, rigid object rotating about origin O. Rotational motion Look at one point P: Arc length s: Thus: is measured in degrees or radians (SI unit: radian) Full circle has an angle of 2 radians. Thus, one radian is 360°/2 Radian degrees 2 360° 180° 90° 157.3°

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Define quantities for circular motion (note analogies to linear motion!!) Angular displacement: Average angular speed: Instantaneous angular speed: Average angular acceleration: Instantaneous angular acceleration:

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Angular velocity is a vector Right-hand rule for determining the direction of this vector. rotates through the same angle, has the same angular velocity, has the same angular acceleration. Every particle (of a rigid object): characterize rotational motion of entire object

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Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration,

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Black board example 10.1 A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of 12.0 rad/s in 3.00 s. 1. What is the magnitude of the angular acceleration of the wheel (in rad/s 2 )? A. 0 B. 1 C. 2 D. 3 E Through what angle does the wheel rotate in these 3 sec (in rad)? A. 18 B. 24 C. 30 D. 36 E Through what angle does the wheel rotate between 2 and 3 sec (in rad)? A. 5 B. 10 C. 15 D. 20 E. 25

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Relation between angular and linear quantities Tangential speed of a point P: Tangential acceleration of a point P: Note: This is not the centripetal acceleration a r This is the tangential acceleration a t Arc length s:

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A fly is sitting at the end of a ceiling fan blade. The length of the blade is 0.50 m and it spins with 40.0 rev/min. a)Calculate the (tangential) speed of the fly. b)What are the tangential and angular speeds of another fly sitting half way in? c) Starting from rest it takes the motor 20 seconds to reach this speed. What is the angular acceleration? d)At the final speed, with what force does the fly (m = 0.01 kg, r = 0.50 m) need to hold on, so that it wont fall off? (Note difference between angular and centripetal acceleration). Black board example 10.2 vtvt

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Demo: Both sticks have the same weight. Why is it so much more difficult to rotate the blue stick?

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Rotational energy A rotating object (collection of i points with mass m i ) has a rotational kinetic energy of Where: Moment of inertia or rotational inertia

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a)What is the rotational energy of the system if it is rotated about the z-axis (out of page) with an angular velocity of 5 rad/s b)What is the rotational energy if the system is rotated about the y-axis? i-clicker for question b): A) 281 JB) 291 JC) 331 JD) 491 JE) 582 J Black board example 10.3 i-clicker Four small spheres are mounted on the corners of a weightless frame as shown. M = 5 kg; m = 2 kg; a = 1.5 m; b = 1 m

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Moment of inertia (rotational inertia) of an object depends on: - the axis about which the object is rotated. - the mass of the object. - the distance between the mass(es) and the axis of rotation.

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Calculation of Moments of inertia for continuous extended objects Refer to Table10.2 Note that the moments of inertia are different for different axes of rotation (even for the same object)

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Moment of inertia for some objects Page 287

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Rotational energy earth. The earth has a mass M = 6.0×10 24 kg and a radius of R = 6.4×10 6 m. Its distance from the sun is d = 1.5×10 11 m What is the rotational kinetic energy of a)its motion around the sun? b)its rotation about its own axis? Black board example 10.4

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Parallel axis theorem Rotational inertia for a rotation about an axis that is parallel to an axis through the center of mass h What is the rotational energy of a sphere (mass m = 1 kg, radius R = 1m) that is rotating about an axis 0.5 away from the center with = 2 rad/sec? Blackboard example 10.5

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Conservation of energy (including rotational energy): Again: If there are no non-conservative forces energy is conserved. Rotational kinetic energy must be included in energy considerations!

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Connected cylinders. Two masses m 1 (5.0 kg) and m 2 (10 kg) are hanging from a pulley of mass M (3.0 kg) and radius R (0.10 m), as shown. There is no slip between the rope and the pulleys. (a)What will happen when the masses are released? (b)Find the velocity of the masses after they have fallen a distance of 0.5 m. (c)What is the angular velocity of the pulley at that moment? Black board example 10.6

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Torque A force F is acting at an angle on a lever that is rotating around a pivot point. r is the distance between the pivot point and F. This force-lever pair results in a torque on the lever

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Black board example 10.7 i-clicker Two mechanics are trying to open a rusty screw on a ship with a big ol wrench. One pulls at the end of the wrench (r = 1 m) with a force F = 500 N at an angle = 80°; the other pulls at the middle of wrench with the same force and at an angle = 90°. What is the net torque the two mechanics are applying to the screw? A. 742 NmB. 750 NmC. 900 NmD NmE Nm

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Particle of mass m rotating in a circle with radius r. Radial force F r to keep particle on circular path. Tangential force F t accelerates particle along tangent. Torque and angular acceleration Torque acting on particle is proportional to angular acceleration :

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Work in rotational motion: Definition of work: Work in linear motion: Component of force F along displacement s. Angle between F and s. Torque and angular displacement.

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Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration,

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Summary: Angular and linear quantities Kinetic Energy: Torque: Linear motion Rotational motion Kinetic Energy: Force: Momentum:Angular Momentum: Work:

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Superposition principle: Rolling motion = Pure translation + Pure rotation Rolling motion Kinetic energy of rolling motion:

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A ring, a disk and a sphere (equal mass and diameter) are rolling down an incline. All three start at the same position; which one will be the fastest at the end of the incline? Black board example 10.8 Demo A.All the same B.The disk C.The ring D.The sphere

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