Understanding Simple Harmonic Motion in Physics

1. Simple Harmonic Motion • Simple harmonic motion (SHM) refers to a certain kind of oscillatory, or wave-like motion that describes the behavior of many physical phenomena: • a pendulum • a bob attached to a spring • low amplitude waves in air (sound), water, the ground • the electromagnetic field of laser light • vibration of a plucked guitar string • the electric current of most AC power supplies

2. SHM Position, Velocity, and Acceleration

3. Simple Harmonic Motion Periodic Motion: any motion of system which repeats itself at regular, equal intervals of time.

4. Simple Harmonic Motion

5. Simple Harmonic Motion • Equilibrium: the position at which no net force acts on the particle. • Displacement: The distance of the particle from its equilibrium position. Usually denoted as x(t) with x=0 as the equilibrium position. • Amplitude: the maximum value of the displacement with out regard to sign. Denoted as xmax or A.

6. The period and frequency of a wave • the periodT of a wave is the amount of time it takes to go through 1 cycle • the frequency f is the number of cycles per second • the unit of a cycle-per-second is commonly referred to as a hertz (Hz), after Heinrich Hertz (1847-1894), who discovered radio waves. • frequency and period are related as follows: • Since a cycle is 2p radians, the relationship between frequency and angular frequency is:

7. Here is a ball moving back and forth with simple harmonic motion (SHM): Its position x as a function of time t is: where A is the amplitude of motion : the distance from the centre of motion to either extreme T is the period of motion: the time for one complete cycle of the motion.

8. Springs and SHM • Attach an object of mass m to the end of a spring, pull it out to a distance A, and let it go from rest. The object will then undergo simple harmonic motion: • What is the angular frequency in this case? • Use Newton’s 2nd law, together with Hooke’s law, and the above description of the acceleration to find:

9. Spring Constant, K The constant k is called the spring constant. SI unit of k = N/m.

10. HOOKE'S LAW The restoring force of an ideal spring is given by, where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in a direction opposite to the displacement of the spring.

11. Simple Harmonic Motion When there is a restoring force, F = -kx, simple harmonic motion occurs.

12. Position VS. Time graph

13. Amplitude Amplitude is the magnitude of the maximum displacement.

14. Period, T For any object in simple harmonic motion, the time required to complete one cycle is the period T.

15. Frequency, f The frequency f of the simple harmonic motion is the number of cycles of the motion per second.

16. Oscillating Mass Consider a mass m attached to the end of a spring as shown. If the mass is pulled down and released, it will undergo simple harmonic motion. The period depends on the spring constant, k and the mass m, as given below,