INTRODUCTION:

Things move. They sometimes move in random fashion, sometimes

along a path of some sort, sometimes in a repeating motion. Let’s first

explore the latter, and narrow that further to things moving back and

forth. Indeed, if you look at something going around and around from

the side, it appears to be going back and forth anyway. The simplest back  and forth, or vibrating motion is called simple harmonic motion.

In your imagination if not actually, take a string about 3 feet long and

tie a reasonably large weight (two or three pounds) to one end, the

other to a hook at the top center of your doorway or some other place

where it can swing freely. You have made a pendulum, which is one

example of something moving in Simple Harmonic Motion (See fig.

Figure 2-1

SIMPLE HARMONIC MOTION

REVEIW OF LITERATURE

1.FRIDAY’S PHYSICAL LAW -
SIMPLE HARMONIC MOTION

First we know that simple harmonic motion is a function of a sin/cosine . But Friday gives in his law that it is also a restoring force.

2. SUNIL KUMAR SINGH: He given some suggestions about the simple harmonic motion.

3.Dr.HAYWARD

4.Dr.STEPHEN

5.Sc.SWIFF

THEORY

Simple Harmonic Motion

Idea: Any object that is initially displaced slightly from a stable equilibrium point will oscillate about its equilibrium position. It will, in general, experience a restoring force that depends linearly on the displacement x from equilibrium:

Hooke’s Law:

where the equilibrium position is chosen to have x -coordinate x = 0 and k is a constant that depends on the system under consideration. The units of k are:

Definitions:

• Amplitude ( A ): The maximum distance that an object moves from its equilibrium position. A simple harmonic oscillator moves back and forth between the two positions of maximum displacement, at x = A and x = – A .
• Period ( T ): The time that it takes for an oscillator to execute one complete cycle of its motion. If it starts at t = 0 at x = A , then it gets back to x = A after one full period at t = T .
• Frequency ( f ): The number of cycles (or oscillations) the object completes per unit time.

• The unit of frequency is usually taken to be 1 Hz = 1 cycle per second.
• Simple Harmonic Oscillator: Any object that oscillates about a stable equilibrium position and experiences a restoring force approximately described by Hooke’s law. Examples of simple harmonic oscillators include: a mass attached to a spring, a molecule inside a solid, a car stuck in a ditch being “rocked out” and a pendulum.

Note:

• The negative sign in Hooke’s law ensures that the force is always opposite to the direction of the displacement and therefore back towards the equilibrium position (i.e. a restoring force).
• The constant k in Hooke’s law is traditionally called the spring constant for the system, even when the restoring force is not provided by a simple spring.
• The motion of any simple harmonic oscillator is completely characterized by two quantities: the amplitude, and the period (or frequency

A simple harmonic motion can be conceived as a “to and fro” motion along an axis (say x-axis). In order

to simplify the matter, we choose origin of the reference as the point about which particle oscillates. If we

start our observation from positive extreme of the motion, then displacement of the particle “x” at a time

“t” is given by

x=Acoswt

where “w” is angular frequency and “t” is the time. The figure here shows the positions of the particle

executing SHM at an interval of “T/8”. The important thing to note here is that displacements in different

intervals are not equal, suggesting that velocity of the particle is not uniform. This also follows from the

nature of cosine function. The values of cosine function are not equally spaced with respect to angles.

Simple harmonic motion

A simple harmonic motion can be conceived as a “to and fro” motion along an axis (say x-axis). In order

to simplify the matter, we choose origin of the reference as the point about which particle oscillates. If we

start our observation from positive extreme of the motion, then displacement of the particle “x” at a time

“t” is given by :

http://cnx.org/content/m15572/latest/ (5 of 17)11/18/2008 9:04:29 PM

Simple harmonic motion

x=Acoswt

where “w” is angular frequency and “t” is the time. The figure here shows the positions of the particle

executing SHM at an interval of “T/8”. The important thing to note here is that displacements in different

intervals are not equal, suggesting that velocity of the particle is not uniform. This also follows from the

nature of cosine function. The values of cosine function are not equally space with respect to angles. The solution of the differential equation of simple harmonic motion is:

A general equation describing simple harmonic motion is:

CHARACTERSTICS OF SHM

1 THE MOTION OF THE BODY EXECUTING SHM IS PERODIC AND ‘TO AND FRO’

.

THE GRAPH REPRESENTING AMPLITUDE OF SIMPLE HARMONIC MOTION

APPLICATIONS

1. The pendulum -

Show the movement of a simple pendulum bob and explain

SHM. When the bob is hanging downward it is in

equilibrium position. When it is disturbed, it

executes SHM. Gravity pulls it back to equilibrium

position but momentum carries it past that position to

another point. It relies on acceleration due to

gravity to deep swinging. Try different lengths of

string and different weights of bobs and see what

happens. Which moves slower, a heavy bob or a light

bob? What difference in acceleration is observed due

to length of string? What else did you observe?

2. The metronome -

This inverted compound pendulum executes SHM when set

into motion by a force. The period of swing can be

altered by varying the position of the small weight on

the arm of the metronome. How does the weight set near

the top of the arm affect the swing of the arm, faster

or slower? What happens when the weight is moved to

the bottom of the arm? Is the swing equal on each side

of the position of equilibrium?

3. A diving board -

This oscillates with SHM after the diver has started

into his dive. Before this when the diver is bouncing

to gain maximum height, the board undergoes forced

oscillation. Observe a diver as he makes a dive, or

set up a simulation of a dive in the classroom.

SUMMARY

They sometimes move in random fashion, sometimes

along a path of some sort, sometimes in a repeating motion. Let’s first

explore the latter, and narrow that further to things moving back and

forth. Indeed, if you look at something going around and around from

the side, it appears to be going back and forth anyway. The simplest back and forth, or vibrating motion is called simple harmonic motion.

Suggestions:

As a student I am not able to give any suggestions about the simple harmonic motion.