Ch13 - Chapter 13: Oscillatory Motions Simple harmonic...

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Chapter 13: Oscillatory Motions Simple harmonic motion ± Spring and Hooke’s law • When a mass hanging from a spring and in equilibrium, the Newton’s 2 nd law says: This means the force due to the spring is equal to the force by gravity and opposite in direction when the spring is stretched. = = = g s g s y F F F F ma F 0 • Hooke’s law states that increasing the weight by equal amounts increases the stretch of the spring by equal amount. Therefore, the force due to the spring must be proportional to the stretch of the spring. constant. spring the is where k x k F y s r r = x is deviation from the spring w/o weight This is also true when the spring shrinks
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Simple harmonic motion ± Simple harmonic motion (SHM) • Let’s study a motion of the mass m. When the mass is attached to the spring, the spring stretches by x 0 . Then lift the mass by A and release it. = = ma F F ma F g s y ma mg x x k = ) ( 0 • Since in equilibrium , 0 0 mg kx F F g s = = d and ) / ( 2 2 dt x dt dx dt d dt dv a x m k a ma kx = = = = = x x m k dt x d 2 2 2 ω = = y yy • The initial stretch is x 0 -x and from Hooke’s law: Equation for SHM
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Simple harmonic motion ± Simple harmonic motion (SHM) (cont’d) x x m k dt x d 2 2 2 ω = = • Solution: ) cos( ) ( φ ω+ = t A t x A x x A x 0 0 arccos , cos ) 0 ( As = = = (rad/s) frequency angular : m k = phase constant A f T f : amplitude , / 1 : period , 2 / : Frequency = = π Hz s x x a t A dt t dv t a x A x v t A dt t dx t v 2 2 2 2 ) ( ), cos( ) ( ) ( ) ( ), sin( ) ( ) ( = + = = ± = + = = velocity acceleration
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Simple harmonic motion ± Simple harmonic motion (SHM) (cont’d) • Solution: ) cos( ) ( φ ω+ = t A t x • What is SHM/SHO? A simple harmonic motion is the motion of an oscillating system which satisfies the following condition: 1. Motion is about an equilibrium position at which point no net force acts on the system. 2. The restoring force is proportional to and oppositely directed to the displacement.
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Ch13 - Chapter 13: Oscillatory Motions Simple harmonic...

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