Chapter 10. Simple Harmonic Motion and Elasticity

Transcription

1 Chapter 10 Simple Harmonic Motion and Elasticity

2 10.1 The Ideal Spring and Simple Harmonic Motion F Applied x = k x spring constant Units: N/m FF SSSSSSSSSSSS = kkkk

3 10.1 The Ideal Spring and Simple Harmonic Motion Example 1 A Tire Pressure Gauge The spring constant of the spring is 30 N/m and the bar indicator extends.0 cm. What force does the air in the tire apply to the spring? 6.4 N

4 10.1 The Ideal Spring and Simple Harmonic Motion Conceptual Example Are Shorter Springs Stiffer? A 10-coil spring has a spring constant k. If the spring is cut in half, so there are two 5-coil springs, what is the spring constant of each of the smaller springs?

5 10.1 The Ideal Spring and Simple Harmonic Motion HOOKE S LAW: RESTORING FORCE OF AN IDEAL SPRING The restoring force on an ideal spring is F x = k x

6 10. Simple Harmonic Motion and the Reference Circle DISPLACEMENT x = Acos θ = Acosωt

7 10. Simple Harmonic Motion and the Reference Circle amplitude A: the maximum displacement period T: the time required to complete one cycle frequency f: the number of cycles per second (measured in Hz) f = 1 T ω = π f = π T

8 10. Simple Harmonic Motion and the Reference Circle VELOCITY ACCELERATION Radius = A v x = v T sinθ = A ω sinω t v max a x = a c A cosθ = ω cosω t a max = ωω xx

9 10. Simple Harmonic Motion and the Reference Circle Example 3 The Maximum Speed of a Loudspeaker Diaphragm The frequency of motion is 1.0 KHz and the amplitude is 0.0 mm. (a) What is the maximum speed of the diaphragm? (b) Where in the motion does this maximum speed occur? v x = v T sinθ = A ω sinω t v max ω = π f rad/s vv mmmmmm = 0.68 mm/ss Where?

10 10. Simple Harmonic Motion and the Reference Circle For a mass, m attached to a spring and set in vibration on frictionless surface FREQUENCY OF VIBRATION x = Acosωt a x = Aω cosωt F = ma x F = kx kx = ma x ka = maω ω = k m f = 1 π k m

11 10.3 Energy and Simple Harmonic Motion W elastic ( F ) s = ( kx + kx ) cos(0 ( x x ) 1 θ = cos o f ) f o W elastic = 1 1 kx f kxo

12 10.3 Energy and Simple Harmonic Motion DEFINITION OF ELASTIC POTENTIAL ENERGY The elastic potential energy is the energy that a spring has by virtue of being stretched or compressed. For an ideal spring, the elastic potential energy is PEelastic = 1 kx SI Unit of Elastic Potential Energy: joule (J)

13 10.3 Energy and Simple Harmonic Motion Example 8 Adding a Mass to a Simple Harmonic Oscillator A 0.0-kg ball is attached to a vertical spring. The spring constant is 8 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring? kkkk = mmmm xx = 0.07 mm

14 10.3 Energy conservation EE = KKKK TT + KKKK RR + PPPP gg + PPPP ee In absence of any external force E f = E o 1 mv + Iω + mgh + kx = mv + Iω + mgh + f 1 f f 1 f 1 o 1 o o 1 kx o

15 Simple Pendulum xx ττ = IIII II = mmrr LL θθ ττ = mmmmmm II = mmll xx aa TT = LLLL αα = aa TT LL mmmm mmmmmm = mmll aa TT LL aa TT = AAωω cos ωωωω = ωω xx ωω = gg LL ωω = ππππ ff = ωω ππ = 1 ππ gg LL TT = 1 ff TT = ππ LL gg Period of a simple pendulum is independent of mass of pendulum

16 10.7 Elastic Deformation STRETCHING, COMPRESSION, AND YOUNG S MODULUS F = L Y Lo A Young s modulus has the units of pressure: N/m

17 10.7 Elastic Deformation

18 10.7 Elastic Deformation SHEAR DEFORMATION AND THE SHEAR MODULUS F = x S Lo A The shear modulus has the units of pressure: N/m

19 10.7 Elastic Deformation

20 10.7 Elastic Deformation Example 14 J-E-L-L-O You push tangentially across the top surface with a force of 0.45 N. The top surface moves a distance of 6.0 mm relative to the bottom surface. What is the shear modulus of Jell-O? F = x S Lo A S = FLo A x

21 10.7 Elastic Deformation FLo S = A x S ( 0.45 N)( m) = = 460 N m ( ) ( m m)

22 10.7 Elastic Deformation VOLUME DEFORMATION AND THE BULK MODULUS P V = B V o The Bulk modulus has the units of pressure: N/m

23 10.7 Elastic Deformation

24 10.8 Stress, Strain, and Hooke s Law In general the quantity F/A is called the stress. The change in the quantity divided by that quantity is called the strain: V V o L L o x L o HOOKE S LAW FOR STRESS AND STRAIN Stress is directly proportional to strain. Strain is a unitless quantity. SI Unit of Stress: N/m

25 For Practice Ch 10 FOC Questions:, 4, 11, 14 and 18. Problems: 11, 17, 7, 30, 44, 54, 56 and 9.