Presentation on theme: "Chapter 10:Rotation of a rigid object about a fixed axis"— Presentation transcript:
1Chapter 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, don’t worry about integral calculation), 10.6 to 10.9Homework: 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, 59Due date: Monday, March 28Rotational motion,Angular displacement, angular velocity, angular accelerationRotational energyMoment of InertiaTorqueMidterm 2 coming up on Wednesday, March 30; (chapter 1-10)
2Rotational motion Look at one point P: Arc length s: Thus: Planar, rigid object rotating about origin O.q is measured in degrees or radians (SI unit: radian)Full circle has an angle of 2p radians.Thus, one radian is 360°/2p = 57.3Radian degrees2p 360°p 180°p/2 90°1 57.3°
3Define quantities for circular motion (note analogies to linear motion!!)Angular displacement:Average angular speed:Instantaneous angular speed:Average angular acceleration:Instantaneous angular acceleration:
4Angular velocity is a vector Right-hand rule for determining the direction of this vector.Every particle (of a rigid object):rotates through the same angle,has the same angular velocity,has the same angular acceleration.q, w, a characterize rotational motion of entire object
5Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration, a.
6Black board example 10.1A 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/s2)?A. 0B. 1C. 2D. 3E. 42. Through what angle does the wheel rotate in these 3 sec (in rad)?A. 18B. 24C. 30D. 36E. 483. Through what angle does the wheel rotate between 2 and 3 sec (in rad)?A. 5B. 10C. 15D. 20E. 25
7Relation between angular and linear quantities Arc length s:Tangential speed of a point P:Tangential acceleration of a point P:Note: This is not the centripetal acceleration arThis is the tangential acceleration at
8Black board example 10.2vtA 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.Calculate the (tangential) speed of the fly.What are the tangential and angular speeds of another fly sitting half way in?Starting from rest it takes the motor 20 seconds to reach this speed. What is the angular acceleration?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 won’t fall off?(Note difference between angular and centripetal acceleration).
9Both sticks have the same weight. Demo:Both sticks have the same weight.Why is it so much more difficult to rotate the blue stick?
10Rotational energyA rotating object (collection of i points with mass mi) has a rotational kinetic energy ofWhere:Moment of inertia or rotational inertia
11Black board example 10.3 i-clicker 2 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 m314What 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/sWhat is the rotational energy if the system is rotated about the y-axis?i-clicker for question b):A) 281 J B) 291 J C) 331 J D) 491 J E) 582 J
12Moment 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.
13Calculation of Moments of inertia for continuous extended objects Refer to Table10.2Note that the moments of inertia are different for different axes of rotation (even for the same object)
15Black board example 10.4 Rotational energy earth. The earth has a mass M = 6.0×1024 kg and a radius of R = 6.4×106 m. Its distance from the sun is d = 1.5×1011 m What is the rotational kinetic energy ofits motion around the sun?its rotation about its own axis?
16Parallel axis theorem h Rotational inertia for a rotation about an axis that is parallel to an axis through the center of masshBlackboard example 10.5What 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 w = 2 rad/sec?
17Conservation 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!
18Connected cylinders. Black board example 10.6 Two masses m1 (5.0 kg) and m2 (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.What will happen when the masses are released?Find the velocity of the masses after they have fallen a distance of 0.5 m.What is the angular velocity of the pulley at that moment?
19TorquefA force F is acting at an angle f 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 t on the lever
20Black 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 F1 = 80°; the other pulls at the middle of wrench with the same force and at an angle F2 = 90°.What is the net torque the two mechanics are applying to the screw?A. 742 Nm B. 750 Nm C. 900 Nm D Nm E Nm
21angular acceleration a. Torque t andangular acceleration a.Particle of mass m rotating in a circle with radius r.Radial force Fr to keep particle on circular path.Tangential force Ft accelerates particle along tangent.Torque acting on particle is proportional to angular acceleration a:
22Work in rotational motion: Definition of work:Work in linear motion:Component of force F along displacement s. Angle g between F and s.Work in rotational motion:Torque t and angular displacement q.
23Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration, a.
24Summary: Angular and linear quantities Linear motionRotational motionKinetic Energy:Kinetic Energy:Force:Torque:Momentum:Angular Momentum:Work:Work:
25Superposition principle: Rolling motionSuperposition principle:Rolling motion = Pure translation Pure rotationKinetic energyof rolling motion:
26Black board example 10.8 Demo 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?All the sameThe diskThe ringThe sphere