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Harmonic Motion Chapter 13 4-8 Critical Concepts • In the absence of friction, mechanical energy is conserved • Mechanical energy is the sum of the potential and kinetic energy • In oscillatory motion, the kinetic and potential energy are “traded” back and forth • When the kinetic energy is a maximum, the potential energy is a minimum • The total mechanical energy remains constant Conservation of energy in mass-spring system X0 We have learned that: x x0 cos t v x0 sin t Substituting into K and U, we get The kinetic and potential energy are: 1 2 mv 2 1 U Kx 2 2 K E K U 1 1 m 2 x02 sin 2 t Kx02 cos 2 t 2 2 1 Kx02 sin 2 t cos 2 t 2 1 Kx02 U MAX 2 The energy “sloshes” back and forth from kinetic to potential E K U 1 1 m 2 x02 sin 2 t Kx02 cos 2 t 2 2 1 Kx02 sin 2 t cos 2 t 2 1 Kx02 U MAX 2 Energy in Mass-Spring System Umax Kmax Total E is CONSTANT. Amplitude A can also be written x0, which is the turning point. The pendulum has a similar energy relationship. E Turning points Problem 13-28 Given spring constant K and mass m, what is the period of oscillation for Block 1 and Block 2? Solution: The period is related to the angular frequency by: T 1 2 f So, find the angular frequency and you have the period T. The angular frequency is obtained from K and M: The only tricky part is figuring out the spring constant. We learned for springs in parallel that… K effective 2k What is the spring constant for Block 2? HINT: What is the total force? K M Problem 13-45 A bullet of mass m and speed Vo is embedded in a block of mass M attached to spring K. If A is measured, what was the initial speed Vo? How long does it take to compress the spring? Strategy: First use conservation of momentum to find the speed of the bullet-block just after collision. Use this to find the intial kinetic energy. Use conservation of energy to find A. Then figure out the period of oscillation, and take ¼ of that to find the time to compress the spring. (You have to think about this part to see that it is ¼ of T that answers the question.) mv0 M mV E K max Combining: v0 1 M TOTV 2 2 KM TOT A m E U max 1 KA2 2 K 2 M TOT T Problem 13-33 (slightly modified) A mass M is attached to a spring, which drops a distance D. Then, it oscillates with angular frequency . What was D? D M g Strategy: From the angular frequency, and the known mass, we can get the spring constant K. From the spring force law and K, we can get D. Oscillations K M 2 Spring forces. KD Mg Mg g D 2 K