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: The kinetic and potential energy are: Substituting into K and U, we get

The energy “sloshes” back and forth from kinetic to potential

Energy in Mass-Spring System Kmax Umax Total E is CONSTANT. Amplitude A can also be written x0, which is the turning point.

The pendulum has a similar energy relationship. 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: 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… What is the spring constant for Block 2? HINT: What is the total force?

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.) Combining:

Problem 13-33 (slightly modified) A mass M is attached to a spring, which drops a distance D. Then, it oscillates with angular frequency w. What was D? D 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. M g Oscillations Spring forces.