Chapter 10. Simple Harmonic Motion and Elasticity. Example 1 A Tire Pressure Gauge
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1 0. he Ideal Spring and Simple Harmnic Mtin Chapter 0 Simple Harmnic Mtin and Elasticity F Applied x k x spring cnstant Units: N/m 0. he Ideal Spring and Simple Harmnic Mtin 0. he Ideal Spring and Simple Harmnic Mtin Example A ire Pressure Gauge he spring cnstant the spring is 30 N/m and the bar indicatr extends.0 cm. What rce des the air in the tire apply t the spring? F Applied x k x ( 30 N m)( 0.00 m) 6.4 N
2 0. he Ideal Spring and Simple Harmnic Mtin Cnceptual Example Are Shrter Springs Stier? A 0-cil spring has a spring cnstant k. I the spring is cut in hal, s there are tw 5-cil springs, what is the spring cnstant each the smaller springs? 0. he Ideal Spring and Simple Harmnic Mtin HOOKE S AW: RESORING FORCE OF AN IDEA SPRING he restring rce n an ideal spring is F x k x 0. Simple Harmnic Mtin and the Reerence Circle 0. Simple Harmnic Mtin and the Reerence Circle DISPACEMEN x Acs θ Acsωt x Acs θ Acsωt
3 0. Simple Harmnic Mtin and the Reerence Circle 0. Simple Harmnic Mtin and the Reerence Circle VEOCIY v x v sinθ { A ω sinω t v max amplitude A: the maximum displacement perid : the time required t cmplete ne cycle requency : the number cycles per secnd (measured in Hz) π ω π 0. Simple Harmnic Mtin and the Reerence Circle 0. Simple Harmnic Mtin and the Reerence Circle Example 3 he Maximum Speed a udspeaker Diaphragm he requency mtin is.0 KHz and the amplitude is 0.0 mm. (a) What is the maximum speed the diaphragm? (b) Where in the mtin des this maximum speed ccur? v x v sinθ { A ω sinω t v max 3 3 (a) v Aω A( π ) ( m)( π )(.0 0 Hz) max.3m s (b) he maximum speed ccurs midway between the ends its mtin.
4 0. Simple Harmnic Mtin and the Reerence Circle ACCEERAION 0. Simple Harmnic Mtin and the Reerence Circle FREQUENCY OF VIBRAION x Acsωt a x Aω csωt a x a { A t c csθ ω csω a max F kx ma x ka maω ω k m 0. Simple Harmnic Mtin and the Reerence Circle 0. Simple Harmnic Mtin and the Reerence Circle Example 6 A Bdy Mass Measurement Device he device cnsists a spring-munted chair in which the astrnaut sits. he spring has a spring cnstant 606 N/m and the mass the chair is.0 kg. he measured perid is.4 s. Find the mass the astrnaut. k ω m ttal k ω m ttal k m ttal m chair + m π ( ) astr π ω π m k astr ( π ) ( 606 N m)(.4s) 4π m chair.0 kg 77. kg
5 0.3 Energy and Simple Harmnic Mtin A cmpressed spring can d wrk. 0.3 Energy and Simple Harmnic Mtin W elastic ( F θ ) s ( kx + kx ) cs0 ( x x ) cs W elastic kx kx 0.3 Energy and Simple Harmnic Mtin DEFINIION OF EASIC POENIA ENERGY he elastic ptential energy is the energy that a spring has by virtue being stretched r cmpressed. Fr an ideal spring, the elastic ptential energy is PEelastic kx 0.3 Energy and Simple Harmnic Mtin Cnceptual Example 8 Changing the Mass a Simple Harmnic Oscilatr he bx rests n a hrizntal, rictinless surace. he spring is stretched t xa and released. When the bx is passing thrugh x0, a secnd bx the same mass is attached t it. Discuss what happens t the (a) maximum speed (b) amplitude (c) angular requency. SI Unit Elastic Ptential Energy: jule (J)
6 0.3 Energy and Simple Harmnic Mtin Example 8 Changing the Mass a Simple Harmnic Oscilatr A 0.0-kg ball is attached t a vertical spring. he spring cnstant is 8 N/m. When released rm rest, hw ar des the ball all bere being brught t a mmentary stp by the spring? 0.3 Energy and Simple Harmnic Mtin E E mv + Iω + mgh + ky mv + Iω + mgh + ky kh mgh h mg k ( 0.0 kg)( 9.8m s ) 0.4 m 8 N m 0.4 he Pendulum 0.4 he Pendulum A simple pendulum cnsists a particle attached t a rictinless pivt by a cable negligible mass. Example 0 Keeping ime Determine the length a simple pendulum that will swing back and rth in simple harmnic mtin with a perid.00 s. ω g (small angles nly) π ω π g g 4π ω mg I (small angles nly) g 4π (.00 s) ( 9.80m s ) 0.48 m 4π
7 0.5 Damped Harmnic Mtin 0.5 Damped Harmnic Mtin In simple harmnic mtin, an bject scillated with a cnstant amplitude. In reality, rictin r sme ther energy dissipating mechanism is always present and the amplitude decreases as time passes. his is reerred t as damped harmnic mtin. ) simple harmnic mtin &3) underdamped 4) critically damped 5) verdamped 0.6 Driven Harmnic Mtin and Resnance 0.6 Driven Harmnic Mtin and Resnance When a rce is applied t an scillating system at all times, the result is driven harmnic mtin. Here, the driving rce has the same requency as the spring system and always pints in the directin the bject s velcity. RESONANCE Resnance is the cnditin in which a time-dependent rce can transmit large amunts energy t an scillating bject, leading t a large amplitude mtin. Resnance ccurs when the requency the rce matches a natural requency at which the bject will scillate.
8 0.7 Elastic Dermatin 0.7 Elastic Dermatin Because these atmic-level springs, a material tends t return t its initial shape nce rces have been remved. SRECHING, COMPRESSION, AND YOUNG S MODUUS FORCES AOMS F Y A Yung s mdulus has the units pressure: N/m 0.7 Elastic Dermatin 0.7 Elastic Dermatin Example Bne Cmpressin In a circus act, a perrmer supprts the cmbined weight (080 N) a number clleagues. Each thighbne this perrmer has a length 0.55 m and an eective crss sectinal area m. Determine the amunt that each thighbne cmpresses under the extra weight.
9 0.7 Elastic Dermatin 0.7 Elastic Dermatin SHEAR DEFORMAION AND HE SHEAR MODUUS F Y A F YA ( N)( 0.55 m) m 9 4 ( N m )( m ) x F S A he shear mdulus has the units pressure: N/m 0.7 Elastic Dermatin 0.7 Elastic Dermatin Example 4 J-E---O Yu push tangentially acrss the tp surace with a rce 0.45 N. he tp surace mves a distance 6.0 mm relative t the bttm surace. What is the shear mdulus Jell-O? x F S A F S A x
10 0.7 Elastic Dermatin 0.7 Elastic Dermatin VOUME DEFORMAION AND HE BUK MODUUS S F A x V P B V S ( 0.45 N)( m) 460 N m 3 ( m) ( m) he Bulk mdulus has the units pressure: N/m 0.7 Elastic Dermatin 0.8 Stress, Strain, and Hke s aw In general the quantity F/A is called the stress. he change in the quantity divided by that quantity is called the strain: V V x HOOKE S AW FOR SRESS AND SRAIN Stress is directly prprtinal t strain. Strain is a unitless quantitiy. SI Unit Stress: N/m
11 0.8 Stress, Strain, and Hke s aw
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