Dr. Jie Zou PHY 1151G Department of Physics1 Chapter 13 Oscillations About Equilibrium (Cont.)

Dr. Jie Zou PHY 1151G Department of Physics2 Outline Energy conservation in oscillatory motion The simple pendulum The period of a simple pendulum Driven oscillations and resonance

Energy conservation in oscillatory motion In an ideal system with no friction or other nonconservative forces, the total energy (E) is conserved. For a mass on a horizontal spring: Total energy E = K + U K = kinetic energy = (1/2)mv 2 U = elastic potential energy = (1/2)kx 2 The kinetic (K) and potential (U) energy change with time, but the total energy E is conserved. It can be shown that E = (1/2)kA 2, where A is the amplitude of oscillation. Dr. Jie Zou PHY 1151G Department of Physics3

Energy vs. position in simple harmonic motion Dr. Jie Zou PHY 1151G Department of Physics4

Example A kg block slides on a frictionless, horizontal surface with a speed of 1.32 m/s. The block encounters an unstretched spring with a force constant of 245 N/m. (a) How far is the spring compressed before the block comes to rest? (b) How long is the block in contact with the spring before it comes to rest? Answers: (a) m (b) s Dr. Jie Zou PHY 1151G Department of Physics5

6 The pendulum Simple pendulum: consists of a mass m suspended by a light string or rod of length L. Period of a pendulum: The period of a simple pendulum depends on the length of the pendulum and the acceleration of gravity. It is independent of the mass and the amplitude. L m

Dr. Jie Zou PHY 1151G Department of Physics7 Example A pendulum is constructed from a string m long attached to a mass of kg. The pendulum completes one oscillation every 1.59 s. Find the acceleration of gravity, g. “Gravity maps” are valuable tools for geologists attempting to understand the underground properties of a given region, such as density of rocks.

Dr. Jie Zou PHY 1151G Department of Physics8 Driven oscillations and resonance Natural frequency, f 0 : The frequency at which the system oscillates when it is not driven. In general, driving any system at a frequency near its natural frequency results in large oscillations. Resonance: This type of large response, due to frequency matching, is known as resonance. Tacoma Narrows bridge, 1940

Dr. Jie Zou PHY 1151G Department of Physics9 Homework See online homework on