Physics 201, Review 3

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1 Physics 0, Reiew Important Notes: This reiew does not replace your own preparation efforts Exercises used in this reiew do not form a test problem pool. Please practice more with end of chapter problems. About Exam q When and where onday No 0th, 5:0-7:00 pm Same location as for the preious ones. q Format Closed book One 8x formula sheet allowed, must be self prepared. (Absolutely no sample problems, examples, class lectures, HW etc.) 0 multiple choice questions. Bring a calculator (but no computer). Only basic calculation functionality can be used. q Special needs/ conflicts: All requests for alternatie test arrangements should hae been settled by now. (except for medical emergency) All alternatie test sessions are in our lab room, only for approed requests. Chapters Coered q Chapter 9: Linear momentum and collision q Chapter 0: Rotation about fixed axis q Chapter : Angular omentum q Chapter : Static equilibrium and elasticity note: Section 7.9: Energy diagram/equilibrium is also coered at conceptual leel. Your TAs will offer again a Super Friday tomorrow (Also please note that the current homework assignment has a two week due time. BUT please make an effort to work it out Before the midterm ) Basic Concepts and Quantities q omentum, Angular omentum Linear omentum, Impulse Angular omentum (magnitude and direction) Impulse, Torque q Collisions: Elastic, Inelastic, Perfectly Inelastic q Center of ass / Center of Graity q Rotational otion (-axis) Angular displacement (Δθ) / Velocity(ω)/Acceleration (α). oments of Inertia Rotational Kinetic Energy Rolling w/o slipping q Conseration Laws Energy, omentum, Angular omentum q Static Equilibrium q Elasticity Young s, Shear, Bulk odulus.

2 Basic Techniques q Calculate translational and rotational kinetic energy. q Calculate moment of inertia using Ι Σm i r i for discrete masses q Calculate using basic rotational kinematical equations q Relate linear and rotational kinematic quantities q Deal with applications such as: oing/rolling on slope, hill, cured track (energy approach) Rolling without slipping Static equilibrium. (seesaw, ladder on wall, etc..) Elastic or perfectly inelastic collisions, simple explosions Simple Pulley System with non-zero pulley mass. ω change on erry-go-round, figure skating (I f ω f I i ω i ) q Vector product: agnitude and direction. Cross products of unit ectors. q Use Right Hand Rules Reiew: Linear omentum and omentum Conseration q Linear omentum p p + p + p +... m + m + m +... p q Impulse-omentum theorem Δp p p F q omentum Conseration: p f f i ext Δt I pi, if Fext (impulse) 0 j Reiew: omentum And Kinetic Energy q Recall: KE ½ m q That is: and p m Kinetic Energy KE m (m) m p m For same momentum, the larger the mass, the smaller the KE Reiew: Collisions q Collision: An eent in which two particles come close and interact with each other by force. omentum is consered in collision: P f P i (Per Impulse approx.) Kinetic Energy of the system may or may not be consered: Elastic: KE f KE i Inelastic: KE f KE i Two extreme cases: Elastic and Perfectly Inelastic. f m m m + m i + m m m m + m i i f m m + m i + m m m m m + m i i Formulas for -D only, will not exam -D collision beyond conceptual. f f m + m m + m Be sure to understand and be able to calculate energy loss in a Perfectly Inelastic Collision

3 A Simple but not Triial Quiz q A ery massie particle, initially at speed, is making an elastic (-D) collision with a ery light particle m (at rest). What is the speed of the particle m after the collision? Reiew Exercise: Two Body System q Consider a system of two.0 kg balls as shown.0 m/s.0 m/s A B,, ery large (~/m), ery large (~ /m ) Q.a: what is total momentum? A:.0kgm/s, B: 4.0 kgm/s, C: zero, D: -4.0 m/s p A A A.0x.0.0 kgm/s, p B B B.0x(-.0)-.0 kgm/s pp A +p B.0 + (-.0) 0 Q.b: what is total kinetic energy? A:0.0, B: 4.0 J, C: 8.0 J, D: 6.0 J E A ½ A A ½.0x.0.0 J, E B ½ B B.0 J also EE A +E B J Exercise : omentum Change A ball with original momentum +4.0 kg m/s hits a wall and bounces straight back without losing any kinetic energy. The change in momentum of the ball is: A. zero B kgm/s C kgm/s D kgm/s E kgm/s Solution: Δp p f p i (-4.0) (+4.0) -8.0 kgm/s (why p f -4.0 kgm/s?) Reiew: Center of ass For a multi-particle system: m, m, m,... at r, r, r,... one can define: Ø Total mass: Σ m j m + m + m +... Ø Center of ass (C) position: m r + mr + mr +... r C Ø C Velocity and Acceleration C d r C dt a C d C dt m + m + m +... m a + m a + m a +... Ø Now think of C as a irtual particle, it has, r,, a F ext a C r m r C C r m m r P j C

4 Reiew Exercise: Find Center of ass q Find the C for these object system. (all masses same) Exercise : Conceptual Q q At t0, the center-of-mass (C) elocity of a multi-particle system is zero. Which of the following statement is true: The system s kinetic energy must be zero The system s momentum must be zero The system s angular momentum must be zero The total external force on the system must be zero None of aboe mx + mx + mx m 0 + m 0 + m L L x C m + m + m m m y + m y + m y m L + m 0 + m 0 L y C m + m + m m C Δ r C Δt m + m + m +... Σ p j Exercise : C and Collision q Consider two particles in -D collision. Before collision, the left particle, of mass m kg, moes to the right at m/s, the right particle, of mass m kg, moes to the left at -m/s. Ignore all external forces (Impulse approximation) Ø What is the system s C elocity before collision? C m + m + ( ).m /s m + m + Reiew: oments Of Inertial I m i r i ( r dm) Ø What is the system s C elocity after collision? (answer in 0 sec) Ø Answer: still -. m/s (quiz: why?) Ø Suppose the collision is elastic (or perfectly inelastic), what are the elocities of the particles after collision? (Exercise after class) See demo for rolling of wheels with different I 4

Reiew Quiz Order the following objects, all haing the same R and, according

( largest) Linear and Rotational otion q Obsere the similarity in math between

Linear (D) Rotational (-axis) x θ ω a α m I 4 KE½ m KE½ Iω Fma momentum pm τια

translation of C + rotation about C Especially, total KE KE C + KE rot

acceleration) Reminder: Angular omentum And Angular omentum conseration q

fixed axis: L Iω recall: Pm q Angular omentum and Torque: dl/dt Στ q τ Fsinφ r

5 Reiew Quiz Order the following objects, all haing the same R and, according their moments of inertia around there respectie axis as shown. ( largest) Linear and Rotational otion q Obsere the similarity in math between linear and rotational motion. Linear (D) Rotational (-axis) x θ ω a α m I 4 KE½ m KE½ Iω Fma momentum pm τια angular momentum L Iω (rxp) Fdp/dt τdl/dt q For a rigid object, its motion translation of C + rotation about C Especially, total KE KE C + KE rot Reminder: Rotational Kinematics and Dynamics q Rotational Kinematics (constant acceleration) Reminder: Angular omentum And Angular omentum conseration q Angular momentum L rxp (LΣL j if multiple particle) For a rigid object about a fixed axis: L Iω recall: Pm q Angular omentum and Torque: dl/dt Στ q τ Fsinφ r q Rotational Dynamics: Στ I α I : oment of inertia Σ m i r i q echanical Equilibrium ΣF 0 and Στ 0 q Angular momentum conseration: if Στ0 L f L i, (I f ω f I i ω i for -axis) 5

Reiew: Angular omentum And Rotational Kinetic Energy q Recall: KE rot ½ I ω q That is: and L I ω Rotational Kinetic Energy KE rot Iω (Iω) For same angular momentum, the larger the moment of inertia,

6 Reiew: Angular omentum And Rotational Kinetic Energy q Recall: KE rot ½ I ω q That is: and L I ω Rotational Kinetic Energy KE rot Iω (Iω) For same angular momentum, the larger the moment of inertia, the smaller the KE rot I L I Reiew Exercise : Right Hand Rule q A uniform disc is rotating about the z axis as shown. What is the direction of its angular momentum? A. +x B. +y C. +z D. None of aboe. Use your right hand to erify it is actually z. q Further questions: Ø For the aboe disc, assuming R, ω, are known, are you able to get its (linear) momentum and its kinetic energy? Exercise 4: Climbing Up a Ladder q An 80 kg man is one fourth of the way up a 0 m ladder that is resting against a smooth, frictionless wall. If the ladder has a mass of 0 kg and it makes an angle of φ60 with the ground, find the force of friction of the ground on the foot of the ladder Ø What is the friction from ground. N W A: 780N, B:00N, C:50N, D: 4N Solution: Identify external forces (N F,N W, m g,m g, f s ), draw FBD Choose ground support as the reference point τ NW N W sinφ L τ mg - m gsin(90 o -φ)l/4, τ mg - m gsin(90 o -φ)l/ τ NF τ fs 0 Στ0 N W sinφ L (m /+m )gcosφl/0 N W (m /+m )g/(tanφ) 4N also: 0 ΣF x F friction -N W F friction N W 4 N to the right N F m g m g F s Exercise 5: Hanging a Sign q A sign of mass is hung m from the end of a 4 m long beam (mass m) as shown in the diagram. The beam is hinged at the wall. What is the tension in the wire? wire θ 0 ο SIGN m m mg θ 0 ο Solution: Identify external forces: (N,F, mg,g, T), draw FBD Choose the hinge as the reference point τ mg - mg, τ g - g, τ T 4Tsin(0 o ) T, τ F τ N 0 Στ0 -mg -g + T 0 T (m+.5)g N F m 4 m T g 6

Reiew Exercise: Jumping On erry-go-round q A freely spinning erry-go-round of mass m mgr 00 kg and radius R mgr m has an initial angular speed ω i 6 re/min.

Solution: free spinning no torque L f L i L i I mgr ω i ½ m mgr R mgr ω i L f (I mgr + I child )ω f (½ m mgr R mgr + m c R mgr ) ω f à ω f ½ m mgr R mgr / (½ m mgr R mgr + m c R mgr ) ω i ½ m mgr /

7 Reiew Exercise: Jumping On erry-go-round q A freely spinning erry-go-round of mass m mgr 00 kg and radius R mgr m has an initial angular speed ω i 6 re/min. After a child of mass m c 5kg jumps on it at the edge as shown, what is the new ω (I disc ½ mr )? Solution: free spinning no torque L f L i L i I mgr ω i ½ m mgr R mgr ω i L f (I mgr + I child )ω f (½ m mgr R mgr + m c R mgr ) ω f à ω f ½ m mgr R mgr / (½ m mgr R mgr + m c R mgr ) ω i ½ m mgr / (½ m mgr + m c ) ω i 4 re/min Exercise 6: Rolling w/o Slipping Down a Slope q A uniform wheel (or disc, or sphere) of mass, radius R, and moment of inertia IR is rolling down a slope without slipping as shown. (θ0 o ) Calculate its C acceleration. q Solution: Ø Step : FBD as shown Ø Step : Set up axis as shown Ø Step : Dynamics for C (x direction): mgsinθ f s ma C Ø Step 4: Dynamics for rotation: -f s R - Iα Ø Step 5: rolling w/o slipping: Rαa C Ø Sole for unknowns: a C g/4, f s mg/4 fs mg N θ x Same Exercise in lecture 9: Rolling w/o Slipping Down a Slope q A uniform disc (or wheel, or sphere) of mass, radius R, and moment of inertia I is rolling down a slope without slipping as shown. Calculate its C acceleration. q Solution: Ø Step : FBD as shown Ø Step : Set up axis as shown Ø Step : Dynamics for C (x direction): mgsinθ f s ma C Ø Step 4: Dynamics for rotation: -f s R - Iα Ø Step 5: rolling w/o slipping: Rαa C Ø Sole for unknowns: a C gsinθ mgsinθ + I, f s mr + mr I fs mg N θ x Exercise 7: Rolling w/o Slipping Down a Slope q Consider two spheres A and B of the same radius and mass rolling down the same slope without slipping. Sphere A is uniformly solid while sphere B is a uniform shell (i.e. hollow inside). Which one rolls faster? (Ignore air resistance) A, B, Same 7

Exercise 8: Rotational and Linear otion A solid cylinder (I R /) has a string wrapped around

When I release the cylinder, holding on to the string, the cylinder falls and spins without

Ø What is the downward acceleration of the cylinder as it falls? (g9.

Strain Stress / (Elastic modulus) Ø There are three general types of stress/strain:

8 Exercise 8: Rotational and Linear otion A solid cylinder (I R /) has a string wrapped around it many times. When I release the cylinder, holding on to the string, the cylinder falls and spins without slipping as the string unwinds. Ø What is the downward acceleration of the cylinder as it falls? (g9.8m/s ) Reiew: Deformation and Elasticity q Small deformation (strain under small stress): Strain Stress / (Elastic modulus) Ø There are three general types of stress/strain: Solution: Rotational: τtr Iα ½ R α àt ½ a (Note: αra is used) Translational: g- T a Sole: a / g 6.5 m/s tensile shear bulk tensile stress Y tensile strain F / A ΔL /L shear stress S shear strain F / A Δx /h olume stress B olume strain ΔF / A ΔV /V ΔP ΔV /V Good Luck! 8

Review: Angular Momentum. Physics 201, Lecture 20

Review: Angular Momentum. Physics 201, Lecture 20 q Physics 01, Lecture 0 Today s Topics More on Angular Momentum and Conservation of Angular Momentum Demos and Exercises q Elasticity (Section 1.4. ) Deformation Elastic Modulus (Young s, Shear, Bulk)