Chapter 12 Simple Harmonic Motion

Photo by Mark Tippens A TRAMPOLINE exerts a restoring force on the jumper that is directly proportional to the average force required to displace the mat. Such restoring forces provide the driving forces necessary for objects that oscillate with simple harmonic motion.

After finishing this section, you should be able to: Write and apply Hooke’s Law for objects moving with simple harmonic motion.Write and apply Hooke’s Law for objects moving with simple harmonic motion. Describe the motion of pendulums and calculate the length required to produce a given frequency.Describe the motion of pendulums and calculate the length required to produce a given frequency. Write and apply formulas for finding the frequency f, period T, velocity v, or acceleration a in terms of displacement x or time t.Write and apply formulas for finding the frequency f, period T, velocity v, or acceleration a in terms of displacement x or time t.

Periodic Motion Simple periodic motion is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time. Amplitude A Period (seconds,s) Period, T, is the time for one complete oscillation. (seconds,s) Frequency Hertz (s -1 ) Frequency, f, is the number of complete oscillations per second. Hertz (s -1 )

Example 1: The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? xF Period: T = s Frequency: f = 2.00 Hz

Simple Harmonic Motion, SHM Simple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx xF

Hooke’s Law When a spring is stretched, there is a restoring force that is proportional to the displacement. F = -kx The spring constant k is a property of the spring given by: k = FxFx F x m

Hooke's law is the relationship between the force exerted on the mass and its position x

Example 2: A 4-kg mass suspended from a spring produces a displacement of 20 cm. What is the spring constant? F 20 cm m The stretching force is the weight (W = mg) of the 4-kg mass: F = (4 kg)(9.8 m/s 2 ) = 39.2 N Now, from Hooke’s law, the force constant k of the spring is: k = = FFxxFFxx  0.2 m k = 196 N/m

Period and Frequency as a Function of Mass and Spring Constant. For a vibrating body with an elastic restoring force: Recall that F = ma = -kx Recall that F = ma = -kx: The frequency f and the period T can be found if the spring constant k and mass m of the vibrating body are known. Use consistent SI units.

Example 3: The frictionless system shown below has a 2-kg mass attached to a spring (k = 400 N/m). The mass is displaced a distance of 20 cm to the right and released. What is the frequency of the motion? m x = 0 x = +0.2 m x v a x = -0.2 m f = 2.25 Hz

Summary Simple harmonic motion (SHM) is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time. F x m The frequency (rev/s) is the reciprocal of the period (time for one revolution).

Summary (Cont.) F x m Hooke’s Law: In a spring, there is a restoring force that is proportional to the displacement. The spring constant k is defined by:

Summary: Period and Frequency for Vibrating Spring. m x = 0x = +Ax = -A x v a

Summary: Simple Pendulum L