Simple Harmonic Motion

3/5/07 Simple Harmonic Motion 0. The Ideal Spring and Simple Harmonic Motion HOOKE S AW: RESTORING FORCE OF AN IDEA SPRING The restoring force on an ideal spring is F x k x spring constant Units: N/m

3/5/07 0. The Ideal Spring and Simple Harmonic Motion Example 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? 0. The Ideal Spring and Simple Harmonic Motion F Applied x k x 30 N m0.00 m 6.4 N

3/5/07 0. The Ideal Spring and Simple Harmonic Motion Conceptual Example Are Shorter Springs Stiffer? A 0-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? 0. Simple Harmonic Motion and the Reference Circle DISPACEMENT x Acos t 3

3/5/07 0. 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 T f T 0. Simple Harmonic Motion and the Reference Circle VEOCITY v x v T sin A sin t v max 4

3/5/07 0. Simple Harmonic Motion and the Reference Circle Example 3 The Maximum Speed of a oudspeaker Diaphragm The frequency of motion is.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? 0. Simple Harmonic Motion and the Reference Circle v x v T sin A sin t v max 3 3 (a) v A A f 0.00 m.0 0 Hz max.3m s (b) The maximum speed occurs midway between the ends of its motion. 5

3/5/07 0. Simple Harmonic Motion and the Reference Circle ACCEERATION a x a c A cos cos t a max 0. Simple Harmonic Motion and the Reference Circle FREQUENCY OF VIBRATION x Acos t a x A cos t F kx ma x ka ma f k m k m 6

3/5/07 SHM Kinematics m = 0.5 kg What is the Frequency? Period? s f 0.5 Hz What is the Angular Frequency? f k m r /s What is the Amplitude? x m 4 m What function is x(t)? cos x(t) x m cos(t) 4cos(t) What function is v(t)? sin What is V m? vm x m v(t) v m sin(t) 4sin(t) v(t).6sin(t) SHM Kinematics m = 0.5 kg What function is a(t)? What is a m? cos a m x m a(t) a m cos(t) 4cos(t) a(t) 39.5cos(t) What function is F(t)? cos What is F m? F m ma m m x m F(t) F m cos(t) F(t) (0.5) 4cos(t) F(t) 9.7cos(t) 7

3/5/07 0. Simple Harmonic Motion and the Reference Circle Example 6 A Body Mass Measurement Device The device consists of a spring-mounted chair in which the astronaut sits. The spring has a spring constant of 606 N/m and the mass of the chair is.0 kg. The measured period is.4 s. Find the mass of the astronaut. 0. Simple Harmonic Motion and the Reference Circle k mtotal k m total m total k T m chair m astro f T m k astro T 606 N m.4s 4 m chair.0 kg 77. kg 8

3/5/07 0.3 Energy and Simple Harmonic Motion DEFINITION OF EASTIC POTENTIA 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 kx SI Unit of Elastic Potential Energy: joule (J) 0.3 Energy and Simple Harmonic Motion A compressed spring can do work. 9

3/5/07 0.3 Energy and Simple Harmonic Motion W elastic kx o kxf 0.3 Energy and Simple Harmonic Motion Example 8 Changing the Mass of a Simple Harmonic Oscilator 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? 0

3/5/07 0.3 Energy and Simple Harmonic Motion E f E o mv f I mgh f f ky f mv o I mgh o o ky o kho mgh o mg h o k 8 N 0.0 kg9.8m s 0.4 m m Assignment pg. 3 #,3,3 pg. 33-36 #3,4,9,3,6,8,6,3,36,4

3/5/07 0.4 The Pendulum A simple pendulum consists of a particle attached to a frictionless pivot by a cable of negligible mass. g (small angles only) mg I (small angles only) 0.4 The Pendulum Example 0 Keeping Time Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of.00 s. f T g T g 4 T g 4.00 s 9.80m s 0.48 m 4

3/5/07 cos(θ) θ g x m A 0.5 m long pendulum with a mass of kg is released from rest at θ = 0 o. Find x m, v m, a m, τ, f, ω, and the maximum Tension in the string. x m is the maximum arc length: x m 0o 360 0.75 m o v m is the velocity at the bottom: E i E f 0.5 9.8.4 s f 0.70 Hz.4 4.43 r /s.4 v mgh m gh (9.8)0.03 0.77 m/s mv m v m x m 0.77 m/s F x ma m T h = cos(θ) θ mgsin ma a m gsin 3.35 m /s h = 0.03 m mg a m x m 3.4 m/s Maximum tension is at the bottom: T +y +x mg F y ma c m v r T mg m v m T m( v m g) N 0.5 Damped Harmonic Motion In simple harmonic motion, an object oscillated with a constant amplitude. In reality, friction or some other energy dissipating mechanism is always present and the amplitude decreases as time passes. This is referred to as damped harmonic motion. 3

3/5/07 0.5 Damped Harmonic Motion ) simple harmonic motion &3) underdamped 4) critically damped 5) overdamped 0.6 Driven Harmonic Motion and Resonance When a force is applied to an oscillating system at all times, the result is driven harmonic motion. Here, the driving force has the same frequency as the spring system and always points in the direction of the object s velocity. 4

3/5/07 0.6 Driven Harmonic Motion and Resonance RESONANCE Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate. 0.7 Elastic Deformation STRETCHING, COMPRESSION, AND YOUNG S MODUUS F Y A o Young s modulus has the units of pressure: N/m 5

3/5/07 0.7 Elastic Deformation 0.7 Elastic Deformation Example Bone Compression In a circus act, a performer supports the combined weight (080 N) of a number of colleagues. Each thighbone of this performer has a length of 0.55 m and an effective cross sectional area of 7.7 0-4 m. Determine the amount that each thighbone compresses under the extra weight. 6

3/5/07 0.7 Elastic Deformation F Y A o F YA o N0.55 m 540 5 4.0 9 4 9.4 0 N m 7.7 0 m m 0.7 Elastic Deformation SHEAR DEFORMATION AND THE SHEAR MODUUS x F S A o The shear modulus has the units of pressure: N/m 7

3/5/07 0.7 Elastic Deformation 0.7 Elastic Deformation Example 4 J-E---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? x F S A o Fo S Ax 8

3/5/07 0.7 Elastic Deformation S Fo Ax S 0.45 N0.030 m 460 N m 3 0.070 m 6.0 0 m 0.7 Elastic Deformation VOUME DEFORMATION AND THE BUK MODUUS V P B V o The Bulk modulus has the units of pressure: N/m 9

3/5/07 0.7 Elastic Deformation 0.8 Stress, Strain, and Hooke s aw 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 o x o HOOKE S AW FOR STRESS AND STRAIN Stress is directly proportional to strain. Strain is a unitless quantitiy. SI Unit of Stress: N/m 0

3/5/07 0.8 Stress, Strain, and Hooke s aw Assignment pg. 33 #4,7 pg. 36-38 #43,46,49,5,54,60,64,7