Midterm 3 Review (Ch 9-14)

Midterm 3 Review (Ch 9-14) PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Ch 9 Overview: Rotational Motion Take our results from linear physics and do the same for angular physics Analogue of Position Velocity Acceleration Force Mass Momentum Energy Chapters 1-3 Chapters 4-7

Ch 9: Rotational Kinematics Kinematic equation just like before: θ like displacement x ω like velocity v α like acceleration a Relation between linear quantities and angular quantities (by geometry) l R v tan l t R a tan v tan t R a rad v2 tan R R 2 Kinetic energy of a rotating object about a fixed (stationary) axis: KE rot 1 2 I 2 I m i r i 2 i Moment of inertia or angular mass (DEPENDS ON THE AXIS CHOSEN) KE 1 2 mv 2 cm 1 2 I cm 2 Sum of translational and rotational energies

Rolling without Slipping In reality, car tires both rotate and translate They are a good example of something which rolls (translates, moves forward, rotates) without slipping Is there friction? What kind? SM Flag: 30% Relation between translational and rotational velocities in no slip condition

A Rolling Wheel A wheel rolls on the surface without slipping with velocity V (your speedometer) What is the velocity of the center of the wheel (point C)? What is the velocity of the lowest point (point P) w.r.t. the ground? Does it make sense to you?

Try Differently: Paper Roll A paper towel unrolls with velocity V Conceptually same thing as the wheel What s the velocity of points: A? B? C? D? D C B A C Point C is where rolling part separates from the unrolled portion Both have same velocity there B A

Bicycle comes to Rest A bicycle with initial linear velocity V 0 decelerates uniformly (without slipping) to rest over a distance d. For a wheel of radius R: a) What is the angular velocity at t 0 =0? b) Total revolutions before it stops? c) Total angular distance traversed by wheel? d) The angular acceleration? e) The total time until it stops?

An athlete throwing the discus A discus thrower moves the discus in a circle of radius 80.0 cm. At a certain instant the thrower is spinning at an angular speed of 10.0 rad/s and the angular speed is increasing at 50.0 rad/s2. At this instant find the acceleration of the discus and its magnitude a tan r (0.800 m)(50.0 rad/s 2 ) = 40.0 m/s 2 a rad r 2 (10.0 rad/s) 2 (0.800 m) = 80.0 m/s 2 a a tan 2 a rad 2 89.4 m/s 2

Finding the moment of inertia for common shapes: Moment of Inertia will be provided for the exam

What is the velocity of the block when it hits the ground? The work done by the cable is zero since the two tension forces cancel each other out so energy is conserved KE i PE i KE f PE f 0 mgh 1 2 mv2 1 2 I 2 0 0 mgh 1 2 mv2 1 1 2 2 MR2 v 2 R 0 v= 2mgh (m +M/2)

Rolling without slipping When rolling without slipping then This is the condition to roll v cm R without slipping. Then, if you are rolling without slipping the kinetic energy is KE 1 2 mv 2 cm 1 2 I cm 2 1 2 mv 2 cm 1 2 I cm R 2 v2 Also note that when one is rolling without slipping (i.e. rolling down an incline) the friction force is static so no work is done by it and energy is conserved in this case.

Consider the speed of a yo-yo toy What is the speed of the Yo-yo at the bottom (use conservation of energy) Why conservation of energy: the hand is not moving so it does no work on the system. You may be confused about the tension but keep in mind that it is an internal force so the sum of the upper and lower tension is zero. E i E f 0 Mgh 1 2 Mv 2 cm 0 Mgh 1 2 Mv 2 cm 1 2 I cm 2 1 1 2 2 MR2 v 2 cm R Mgh 3 4 Mv 2 cm v cm 4 3 gh

Ch 10: Rotational Dynamics Torque results from force applied at a distance from a pivot point r r F r rf sin The direction of the torque is given by the RHR

Write Torque as Torque r F sin r F To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist If the axis is fixed, what is the net Torque on the wheel?

Torque and Moment of Inertia Forcevs. Torque F=ma and = I Mass vs. Moment of Inertia m I mr 2 or I r 2 dm

Flywheel problem from Ch 9 (using work energy theorem) The cable is wrapped around a cylinder. If it unwinds 2.0 m by pulling it with a force of 9.0 N and it starts at rest, what is its final angular velocity and velocity of the cable? (use work energy theorem) W total KE f KE i F x 1 2 I 2 0 2F x I 4F x mr 2 20 rad/s v R (20 rad/s)(0.060 m) =1.2 m/s

Flywheel problem using torque (using work energy theorem) The cable is wrapped around a cylinder. If it unwinds 2.0 m by pulling it with a force of 9.0 N and it starts at rest, what is its final angular velocity and velocity of the cable? (use work energy theorem) I 1 2 MR2 FR (9.0 N)(0.06 m) I I 2FR MR 2 2F MR 6.0 rad/s2 Use α to get acceleration of the cable: a tan R (0.06m)(6.0 rad/s 2 ) 0.36 m/s 2 Then use kinematics v 2 v 0 2 2a tan (x x 0 ) v 0 2(0.36 m/s 2 )(2 m) 1.2 m/s

What is the velocity of the block when it hits the ground? The work done by the cable is zero since the two tension forces cancel each other out so energy is conserved KE i PE i KE f PE f 0 mgh 1 2 mv2 1 2 I 2 0 0 mgh 1 2 mv2 1 1 2 2 MR2 v 2 R 0 v= 2mgh (m +M/2)

Another look at the unwinding cable What is the linear acceleration of the block? These are two coupled objects; one rotates and the other moves linearly For the rotating wheel we have: I TR 1 a 2 MR2 T 1 R 2 Ma For the block we have: mg T ma Combine the two equations to get mg 1 2 Ma ma a mg m M /2 SM Flag: 55% Connection between rotational and translational accln of coupled objects

SM Flag: 50% Angular momentum both rxp and Iw

Conservation of angular momentum Recall that torque was defined as r r F r Similarly the angular momentum of a particle is r L r p r r m v r Before we saw that if the external forces on a system are zero then linear momentum is conserved. Similar for angular momentum. If the external torques on a system are zero then the TOTAL angular momentum of the systems is conserved. Note that in this case the TORQUE has to be zero. There can be still forces acting on the system but they do not generate any torque (e.g. force due to gravity on the cat which acts at the center of mass) I 1 1 I 2 2

How a car s clutch work The clutch disk and the gear disk is pushed into each other by two forces that do not impart any torque, what is the final angular velocity when they come together? L z before L z after I A A I B B (I A I B ) final final I A A I B B (I A I B )

Ch 11: Conditions for equilibrium Net Force is zero (x,y,z) Net torque is zero.

Strategy F x 0 F y 0 0 1 st Draw the free body diagram and the location where the forces are applied 2 nd Choose a Pivot Point which will delete the largest amount of torques 3 rd Compute the torques from each force 4 th Solve the equations

Problem 11.27 The horizontal beam in the figure weighs 150 N, and its center of gravity is at its center. Find the tension in the cable and on the beam at the hinge. SM Flag: 30% Contact forces

Ch 13: Newton s Law of Gravitation The gravitational force is always attractive and depends on both the masses of the bodies involved and their separations. F g Gm 1m 2 r 2 G 6.6742 10-11 N m 2 kg 2

Weight (skip Weight Watchers, just climb upward) Gravity (and hence, weight) decreases as altitude rises.

Gravitational potential energy Objects changing their distance from earth are also changing their potential energy with respect to earth. PE G m 1m 2 r 12 This is the true potential energy. The zero level is set when they are very far apart

Satellite circular motion The force is radial so Newton s 2 nd law reads: r : G M m E s v 2 m R 2 s R m 4 2 R s T 2 v G M E R T 4 2 R 3 GM E T R 3/2 SM Flag: 40% Gravitational force provides centripetal accln

Escaping from the Earth What is the velocity you need to shoot straight up from the surface of the earth an not come back (conservation of energy)? KE 1 PE 1 KE PE 1 2 m v2 G mm E R E 0 0 SM Flag: 40% Energy conservation v 2GM E R E 1.12 10 4 m/s = 25,000 mph

Ch 14: Simple Harmonic Motion The spring drives the glider back and forth on the air-track and you can observe the changes in the free-body diagram as the motion proceeds from A to A and back. The pendulum is another example Elements of harmonic motion: Frequency: f=1/t Angular frequency: ω=2πf Amplitude: maximum deflection/stretch/compression

Simple harmonic motion An ideal spring responds to stretch and compression linearly, obeying Hooke s Law. F x kx The restoring force towards the equilibrium point is LINEAR from the displacement from equilibrium Let s look at Newton s 2 nd law: F x ma x kx ma x m d 2 x dt 2 d 2 x dt 2 d 2 x dt 2 k m x 2 x

Watch variables change for a glider example As the glider undergoes SHM, you can track changes in velocity and acceleration as the position changes between the turning points. x(t) Acos( t ) v(t) Asin( t ) a(t) A 2 cos( t )

Energy in SHM Energy is conserved during SHM and the forms (potential and kinetic) interconvert as the position of the object in motion changes. E 1 2 mv 2 x 1 2 kx 2 1 2 ka2 1 2 mv 2 max SM Flag: 37% Energy at different points

Energy in SHM II

The simple pendulum Ok when Θ<<1 F tan mgsin mg F tan ma tan ml d 2 dt 2 ml d 2 dt 2 mg d 2 dt 2 g L g L f 1 2 g L T 2 L g

The physical pendulum A physical pendulum is any real pendulum that uses an extended body in motion. This illustrates a physical pendulum. T 2 I mgd