Principles of Physics I

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1 Pinciples of Physics I J. M. Veal, Ph. D. vesion Contents Linea Motion 3. Two scala equations Anothe scala equation Constant acceleation Homewok Execises Vectos 3 2. Basics Unit vectos Addition: geometic, components Multiplication: scala & vecto poducts Homewok Execises Thee-Dimensional Motion 3 3. Basics Pojectiles: tajectoy, ange Cicles: centipetal acceleation Galilean elativity Homewok Execises Newton s Laws of Motion 4 4. st law nd law d law: nomal & tension foces Mass vs. weight What is mass? Homewok Execises Thee Paticula Foces 4 5. Fiction: static, kinetic Dag Teminal speed Centipetal Homewok Execises Exam Wok 4 6. Basics Kinetic enegy Gavity Vaiable foce Spings & Hooke s law Kinetic enegy Powe Homewok Execises Enegy 5 7. Potential enegy Path independence Gavity, sping Implied foce Homewok Execises Consevation of Enegy 5 8. Mechanical enegy Themal enegy of sliding Total enegy Homewok Execises Linea Momentum 6 9. Basics Cente of mass Solid bodies Newton s 2 nd law

2 J. M. Veal, Pinciples of Physics I Intenal enegy Homewok Execises Consevation of Linea Momentum 6 0. Basics Vaiable mass Homewok Execises Collisions 6. Bouncing a ball Impulse Inelastic Elastic Cente of mass Two dimensions Homewok Execises Exam Rotation 7 2. Basics Constant angula acceleation Kinetic enegy Moment of intetia Paallel axis theoem Fomulae Homewok Execises Toque 8 3. Basics Newton s angula 2 nd law Wok Vecto Homewok Execises Rolling 8 4. Basics Kinetic enegy Ramp Homewok Execises Angula Momentum 8 5. Basics Newton s angula 2 nd law Homewok Execises Consevation of Angula Momentum 8 6. Basics Pecession Coiolis effect Consevation of linea momentum Homewok Execises Similaities 9 7. Tanslation vs. otation Static Equilibium 9 8. Basics Homewok Execises Oscillations 9 9. Simple hamonic motion Diff. eq Hooke s law Enegy Thee pendulums Damping Resonance Homewok Execises Gavitation Newton s law Shell theoems Potential enegy Keple s laws Obital enegy Thee masses Einstein s theoy Homewok Execises Exam Final Exam A All Fomulae 0

3 J. M. Veal, Pinciples of Physics I 3 Linea Motion Given the definition of a deteminant, show that. Two scala equations a b pa y b z a z b y qî pa x b z a z b x qĵ pa x b y a y b x qˆk. Given the definition a 9v, show that Given v v 0 v v 0 at. at and the definition v 9x, show that x x 0.2 Anothe scala equation v 0 t 2 at2. Given v v 0 at and x x 0 v 0 t at 2 {2, show that.3 Constant acceleation v 2 v 2 0 2apx x 0q. Given v v 0 at and x x 0 v 0 t at 2 {2, show that x x 0 2 pv 0 vqt. Given v v 0 at and x x 0 v 0 t at 2 {2, show that.4 Homewok Execises x x 0 vt 2 at2. View The Mechanical Univese and Beyond, 2. The Law of Falling Bodies. Read you text, chapte 2: Motion Along a Staight Line. 2 Vectos 2. Basics 2.2 Unit vectos 2.5 Homewok Execises View The Mechanical Univese and Beyond, 5. Vectos. Read you text, chapte : Units, Physical Quantities, and Vectos. 3 Thee-Dimensional Motion 3. Basics 3.2 Pojectiles: tajectoy, ange Given the equations fo linea motion with constant acceleation, choose two of them appopiately and show that the tajectoy of a pojectile is given by y x tan θ 0 2 gx2 pv 0 cos θ 0 q 2, whee x and y ae the hoizontal and vetical coodinates, espectively, of the pojectile s position elative to the launching point, θ 0 is the pojectile s launch angle, and g is the acceleation due to gavity. Given that a pojectile s tajectoy equation is given by y x tan θ 0 pgx 2 {2qpv 0 cos θ 0 q 2, show that the ange is given by R v2 0 g sin 2θ Cicles: centipetal acceleation Conside unifom cicula motion with adius and constant speed v in the xy plane. Assume the cicle is centeed at the oigin. If a is the acceleation and px, yq is the position of the moving paticle, show that 2.3 Addition: geometic, components 2.4 Multiplication: scala & vecto poducts Given a a x î a y ĵ a zˆk and a b ab cos θ, show that a b a x b x a y b y a z b z. Futhemoe, show that a v2 2 pxî a v2. yĵq.

4 J. M. Veal, Pinciples of Physics I Galilean elativity 5.2 Dag v pa v pb v BA D 2 CρAv2 3.5 Homewok Execises View The Mechanical Univese and Beyond, 4. Inetia. Read you text, chapte 3: Motion in Two o Thee Dimensions. 4 Newton s Laws of Motion 4. st law nd law F net m a A foce F F x î given by F y ĵ acts on a mass m. Show that the acceleation is a F x m î F y m ĵ. 5.3 Teminal speed Given that the dag foce can be expessed as D 2 CρAv2, show that the teminal speed fo an object falling though a fluid is given by d 2F g v t CρA. 5.4 Centipetal F c mv2 5.5 Homewok Execises View The Mechanical Univese and Beyond,. Intoduction. Read you text, chapte 5: Applying Newton s Laws d law: nomal & tension foces 4.4 Mass vs. weight 4.5 What is mass? 4.6 Homewok Execises View The Mechanical Univese and Beyond, 6. Newton s Laws. Read you text, chapte 4: Newton s Laws of Motion. 5.6 Exam 6 Wok 6. Basics W F d Exam coves mateial up to hee. 5 Thee Paticula Foces 5. Fiction: static, kinetic 6.2 Kinetic enegy K 2 mv2 f s µ s N f k µ k N Given v 2 v 2 0 2a x, F ma, and K 2 mv2, show that W K.

5 J. M. Veal, Pinciples of Physics I Gavity 6.4 Vaiable foce W» f i F d 6.5 Spings & Hooke s law F k l Given W is given by ³ f i F d and F k l, show that the wok done by a sping W s 2 kpx2 i x2 f q. Include a justification of the esult of the dot poduct. 6.6 Kinetic enegy Given W 6.7 Powe P dw dt ³ f i 6.8 Homewok Execises F pq d and F ma, show that, in geneal, W K. Read you text, chapte 6: Wok and Kinetic Enegy. 7 Enegy 7. Potential enegy U W 7.2 Path independence 7.3 Gavity, sping Given U W, show that gavitational potential enegy is given by U pyq mgy. Given U W, show that a sping s potential enegy is given by 7.4 Implied foce U pxq 2 kx2. Given U W, show that potential enegy implies a foce accoding to 7.5 Homewok Execises du pxq F pxq dx. View The Mechanical Univese and Beyond, 3. Consevation of Enegy. Read you text, chapte 7: Potential Enegy and Enegy Consevation (sections - 3). 8 Consevation of Enegy 8. Mechanical enegy E mec K U Given E mec K U, W K, and U W, show that mechanical enegy is conseved: E mec Themal enegy of sliding E th f k d 8.3 Total enegy E E mec E th E int 8.4 Homewok Execises View The Mechanical Univese and Beyond, 4. Potential Enegy. Read you text, chapte 7: Potential Enegy and Enegy Consevation (sections 4-5).

6 J. M. Veal, Pinciples of Physics I 6 9 Linea Momentum 9. Basics p m v F d p dt 9.2 Cente of mass com com V M ņ i m i i» dv 9.3 Solid bodies 9.4 Newton s 2 nd law F net M a com 9.5 Intenal enegy 9.6 Homewok Execises Read you text, chapte 8: Momentum, Impulse, and Collisions (Sections & 5). 0 Consevation of Linea Momentum 0. Basics P Vaiable mass Conside a vaiable-mass system such as an acceleating ocket. With no net extenal foce, it s tue that P 0. Let M and a be the mass and acceleation of the ocket, espectively, let v exh be the speed of the exhaust elative to the ocket, and let 9M be the fuel consumption ate. a) Show that the thust of the ocket is descibed by Ma 9 Mv exh. b) As the ocket consumes some amount of fuel, m M i M f, its speed inceases. Show that this incease in speed is given by 0.3 Homewok Execises v v exh ln p m{m f q. View The Mechanical Univese and Beyond, 5. Consevation of Momentum. Read you text, chapte 8: Momentum, Impulse, and Collisions (Sections 2, 3, & 6). Collisions. Bouncing a ball.2 Impulse J» tf t i p J F ptqdt.3 Inelastic.4 Elastic In an elastic collision, thee is no themal enegy: E th 0. Conside such a collision in one dimension with two masses m and m 2 taveling at initial speeds v i and v 2i. Show that the speeds afte the collision ae given by and v m m 2 f v i v 2m 2f v i 2m 2 v 2i, m 2 m v 2i.

7 J. M. Veal, Pinciples of Physics I 7.5 Cente of mass P v com.6 Two dimensions.7 Homewok Execises View The Mechanical Univese and Beyond, 0. Fundamental Foces. Read you text, chapte 8: Momentum, Impulse, and Collisions (Section 4)..8 Exam 2 2 Rotation 2. Basics θ s ω v α a t a v2 ω 2πf Exam 2 coves mateial between Exam and hee. 2.2 Constant angula acceleation ω ω 0 θ θ 0 ω 2 ω 2 0 θ θ 0 αt ω 0 t 2 αt2 2αpθ θ 0q 2 pω 0 ωqt θ θ 0 ωt 2 αt2 2.3 Kinetic enegy Given K mv 2 {2 and the definition of moment of inetia: show that 2.4 Moment of intetia» I 2 ρdv I ņ i m i 2 i, K 2 Iω2. annula cylinde paxisq I 2 mp2 2 2 q solid cylinde pdiameteq I 4 m2 2 ml2 solid sphee pdiameteq hollow sphee pdiameteq hoop pdiameteq I 2 m2 I 2 5 m2 I 2 3 m2 slab pcente, pependiculaq I 2 mpa2 b 2 q 2.5 Paallel axis theoem Given the definition of moment of inetia:» I 2 dm, deive the paallel-axis theoem: 2.6 Fomulae 2.7 Homewok Execises I I com mh 2. View The Mechanical Univese and Beyond, 9. Moving in Cicles. Read you text, chapte 9: Rotation of Rigid Bodies.

8 J. M. Veal, Pinciples of Physics I 8 3 Toque 3. Basics τ F sin φ 3.2 Newton s angula 2 nd law τ Iα 3.3 Wok W» θf θ i τdθ 3.4 Vecto τ F 3.5 Homewok Execises Read you text, chapte 0: Dynamics of Rotational Motion (Sections, 2, & 4). 4 Rolling 4. Basics 4.2 Kinetic enegy Show that the kinetic enegy of a olling object is given by 4.3 Ramp K 2 I comω 2 2 mv2 com. Show that an object of adius, mass m, and moment of inetia I com olling down a amp of inclination θ has an acceleation of magnitude a com g sin θ I com {m Homewok Execises Read you text, chapte 0: Dynamics of Rotational Motion (Section 3). 5 Angula Momentum 5. Basics l p L Iω 5.2 Newton s angula 2 nd law Given the definitions of momentum, angula momentum, and toque, show that Newton s second law of angula motion is given by 5.3 Homewok Execises τ d l dt. View The Mechanical Univese and Beyond, 9. Angula Momentum. Read you text, chapte 0: Dynamics of Rotational Motion (Section 5). 6 Consevation of Angula Momentum 6. Basics L Pecession 6.3 Coiolis effect 6.4 Consevation of linea momentum 6.5 Homewok Execises View The Mechanical Univese and Beyond, 20. Toques and Gyoscopes.

9 J. M. Veal, Pinciples of Physics I 9 Read you text, chapte 0: Dynamics of Rotational Motion (Sections 6 & 7). 7 Similaities 7. Tanslation vs. otation 8 Static Equilibium 8. Basics F net 0 τ net Homewok Execises Take a cusoy glance at you text chapte : Equilibium and Elasticity. 9.4 Enegy A fictionless mass is attached to a sping. Given that the wok done by a sping is W s kpx 2 i x2 f q{2, use W U to show that the mechanical enegy of the system is given by 9.5 Thee pendulums c κ ω I E 2 kx2 m. Given τ F, show that a physical pendulum of length l and otational inetia I undegoing small oscillations has an angula fequency given by c mgl ω I. ω a g{l 9 Oscillations 9. Simple hamonic motion xptq x m cos pωt φq Given that the position of an object undegoing simple hamonic motion is xptq x m cos pωt φq, show that the acceleation is given by 9.6 Damping c k xptq x m e bt{2m cos t 9.7 Resonance m 9.8 Homewok Execises b2 4m 2 φ 9.2 Diff. eq Hooke s law aptq ω 2 xptq. View The Mechanical Univese and Beyond, 6. Hamonic Motion. Read you text, chapte 4: Peiodic Motion. Given that Hooke s law is F kd and simple hamonic motion is descibed by xptq x m cos pωt φq, show that the angula fequency of oscillation is given by c k ω m. 20 Gavitation 20. Newton s law F G m m 2 2

10 J. M. Veal, Pinciples of Physics I Shell theoems 20.0 Final Exam a g GM` R 2` 20.3 Potential enegy ³ f Given W i F pq d and W U, show that the gavitational potential enegy fo an object of mass m above sea level at a distance fom the Eath s cente (Eath s mass is M) is given by v esc c 2GM R 20.4 Keple s laws 4π T 2 2 a 3 GM 20.5 Obital enegy U GMm. Given F ma and U GMm show that the enegy of an object of mass m in an obit of semi-majo axis a about some mass M is given by 20.6 Thee masses 20.7 Einstein s theoy 20.8 Homewok Execises E GMm 2a. View The Mechanical Univese and Beyond, 8. The Apple and the Moon, and 2. Keple s Thee Laws. Read you text, chapte 3: Gavitation Exam 3 A All Fomulae The final exam is cumulative up to this point. Exam 3 coves mateial between Exam 2 and hee.

11 J. M. Veal, Pinciples of Physics I Linea Motion x x 0 v 2 v 2 0 x x 0 x x 0 v v 0 v 0 t at 2 at2 2apx x 0q 2 pv 0 vqt vt 2 at2 Vectos a b a x b x a y b y a z b z a b pa y b z a z b y qî pa x b z a z b x qĵ Thee-Dimensional Motion y x tan θ 0 2 gx2 pv 0 cos θ 0 q 2 R v2 0 g sin 2θ 0 a v2 2 pxî a v2 v pa v pb yĵq v BA pa x b y a y b x qˆk Thee Paticula Foces f s µ s N f k µ k N D 2 CρAv2 v t d 2F g CρA F c mv2 Wok W F d K 2 mv2 W W K» f i F k d P dw dt Enegy F d U W U pyq mgy Consevation of Enegy E mec K E th f k d U E E mec E th E int Linea Momentum com p m v F d p dt M com V ņ i m i i» dv F net M a com Consevation of Linea Momentum P 0 dm v exh Ma dt v v exh ln M 0 M J Collisions» tf t i p J F ptqdt Newton s Laws U pxq 2 kx2 v m m 2 f v i m 2m 2 v 2i m 2 F net m a du pxq F pxq dx v 2m 2f v i P v com m 2 m v 2i

12 J. M. Veal, Pinciples of Physics I 2 Rotation Toque Oscillations θ s ω v α a t a v2 ω 2πf ω ω 0 αt θ θ 0 ω 0 t 2 αt2 ω 2 ω0 2 2αpθ θ 0q θ θ 0 2 pω 0 ωqt θ θ 0 ωt 2 αt2 τ F sin φ W τ Iα» θf θ i τ F Rolling τdθ K 2 I comω 2 2 mv2 com a com,x g sin θ I com {m 2 Angula Momentum xptq x m cos pωt aptq ω 2 xptq ω c k m E 2 kx2 m ω c κ I ω a mgl{i a ω g{l c k xptq x m e bt{2m cos t m φq b2 4m 2 φ K 2 Iω2» 2 ρdv I l p L Iω Gavitation annula cylinde paxisq I 2 mp2 2 2 q solid cylinde pdiameteq I 4 m2 2 ml2 solid sphee pdiameteq hollow sphee pdiameteq I 2 5 m2 I 2 3 m2 τ d l dt Consevation of Angula Momentum L 0 F G m m 2 2 a g GM` R 2` U GMm c 2GM hoop pdiameteq I 2 m2 slab pcente, pependiculaq I 2 mpa2 b 2 q I I com mh 2 Static Equilibium F net 0 τ net 0 v esc T 2 R 4π 2 a 3 GM E GMm 2a

b) (5) What average force magnitude was applied by the students working together?

b) (5) What average force magnitude was applied by the students working together? Geneal Physics I Exam 3 - Chs. 7,8,9 - Momentum, Rotation, Equilibium Nov. 3, 2010 Name Rec. Inst. Rec. Time Fo full cedit, make you wok clea to the gade. Show fomulas used, essential steps, and esults