Harmonic Motion (IV) Energy of a simple harmonic oscillator Physics 1D03 - Lecture 34

Energy of a SHO Recall: ETot = K + U = 1/2mv2 + 1/2kx2 for a spring. But we know that: x=A cos(ωt+φ) v=-A ω sin(ωt+φ) Physics 1D03 - Lecture 34

Energy in SHM M Look again at the block & spring Hence: ETot = ½kA2 We could also write E = K+U = ½ m(vmax )2 Physics 1D03 - Lecture 34

Suppose you double the amplitude of the motion, what Concept Quiz Suppose you double the amplitude of the motion, what happens to the maximum speed ? Doubles 4 x Larger Doesn’t change Physics 1D03 - Lecture 34

Suppose you double the amplitude of the motion, what Concept Quiz Suppose you double the amplitude of the motion, what happens to the maximum acceleration? Doubles 4 x Larger Doesn’t change Physics 1D03 - Lecture 34

Suppose you double the amplitude of the motion, what Concept Quiz Suppose you double the amplitude of the motion, what happens to the the total energy? Doubles 4 x Larger Doesn’t change Physics 1D03 - Lecture 34

Energy Since we know the total energy of a SHM, we can calculate the or displacement velocity at any point in time: ETot=1/2kA2 = K+U = 1/2mv2 + 1/2kx2 So, if x=0, all E is in kinetic, and v is at max if x=A, all E is in potential, and v is zero Physics 1D03 - Lecture 34

Example A 100g block is 5cm from the equilibrium position moving at 1.5m/s. a) What is the total energy of the system ? b) What is the amplitude of the oscillations ? Physics 1D03 - Lecture 34

Example A 500g block on a spring is pulled 20cm and released. The motion has a period of 0.8s. What is the velocity when the block is 15.4cm from the equilibrium ? Physics 1D03 - Lecture 34

Example A 1.0kg block is attached to a spring with k=16N/m. While the block is at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40cm/s. what is the amplitude of the subsequent oscillations ? what is the block’s speed at the point where x=A/2 ? Physics 1D03 - Lecture 34