If a small body or a particle vibrates or moves to and fro along a straight line under the influence of a force so that its acceleration toward a fixed point (or it equilibrium position) is proportional to its distance or displacement from that point, the body is said to be performing simple harmonic motion
Acceleration of simple harmonic motion
A body performing simple harmonic motion experiences a constant acceleration directed toward the center of the motion
The Force influences the bodies motion in such a way that it always direct the body back to its equilibrium position, such a force is called restoring force since it tends to restore the body to its equilibrium position
Examples of simple harmonic motion
1.A motion of a body fastens to the end of a spring.
When a mass is hung in equilibrium from the lower end of spring and the other end of the spring is firmly clamped from a rigid support is pulled down it begins to move up and down. This type of motion is called simple harmonic motion though Hooke’s law must be obeyed that means that elastic limit of the spring must not be exceeded.
2. The simple pendulum.
This consists of a small lead weight or bob suspended from a rigid support by means of a thread. When it is displaced through a small angle the bob perform simple harmonic motion by moving to and fro and always trying to return to equilibrium position when it has reached maximum height call amplitude.
It is observed that the motion of the pendulum does not continue indefinitely, this is because frictional force and air resistance gradually reduce the motion until the amplitude becomes zero
Period in simple pendulum
The period T of simple pendulum of length l is given as
Where g=acceleration due to gravity and the L=length of the thread
3. Loaded test tube in a liquid
Another example of simple harmonic motion can be demonstrated in laboratory using a test tube loaded with sand. Put just enough sand into the test tube so that it can stand vertically erect in a liquid, depress the tube into the water and release it. The tube is seen to move up and down in the water or any liquid, you have used. That motion is called the simple harmonic motion.
Relationship between Simple harmonic motion and circular motion
Let us imagine that at every instant perpendicular line can be drawn from Q to meet the diameter AB of the circle at P. As Q moves once round the circle, the foot of the perpendicular P move from B through C to A and back B.
Thus the point P will move along AB with simple harmonic motion while the point Q moves round the circle. The velocity of P along AB changes continuously as a the point moves along AB. It has its greatest velocity at C, but a zero velocity at A and B. This means the point P is at rest at A and B before changing its direction of motion
As Q moves around the circle the angle Ѳ between QC and CP also changes. This angle change from 0o to 360o (2π radians) as the point Q move around the circle once
We define the angular velocity of the point Q as
And also angular acceleration which is defined as the rate of change of angular velocity is given as
Where α=angular acceleration
The relationship between linear velocity, linear acceleration and angular velocity
where V= velocity, r =radius and ω = angular velocity
where a =linear acceleration, α angular acceleration and r =radius
Energy of simple harmonic motion
since force and displacement are involved in simple harmonic motion, energy is also involved. At any instant of motion the system may contain some energy as kinetic or potential or both.
We consider a mass suspended from the end of spring which is made to execute a simple harmonic motion. The force needed describe the spring a distance is given by
Where k= elastic constant of the spring
The work done in stretching the spring is given by average force x displacement
= 1/2 kx2.
The energy stored in the stretched spring is 1/2 kx2 therefore the potential energy in stretched spring is given by
P.Emax = 1/2 kx2
Where x = maximum displacement or amplitude of the motion
The kinetic energy k.E at any instant of the motion is given by
m =mass of the body and v is its velocity at that instant.
At maximum displacement v=zero, therefore at this point the kinetic energy equals zero. And also the total energy is inform
of potential energy.
At the center or at equilibrium position the speed is at its maximum, and displacement = zero therefore the potential energy is also zero why the total energy is being represented by kinetic energy
At any position x the total energy is given by
1/2 mv2 + 1/2kx2
Consider the Simple harmonic motion of simple pendulum in picture below
At A or B where the bob momentarily comes to rest, the total energy is potential energy giving as Mgh
Where h=the height of B above the equilibrium points C, m = mass of Bob and g is acceleration due to gravity
At C where the speed of the bob is maximum, and height = 0, all energy is giving as
K E = 1/2mv2
At other positions of motion the kinetic energy and potential energy each contribute energy
Ideal simple harmonic motion is the one which continue to vibrate with maximum amplitude forever but practically it is not possible, because the amplitude of oscillation continues to decrease gradually until the vibration eventually stops. A simple harmonic motion does not continue to vibrate forever. Such a motion in which the amplitude decrease with time is said to be a damped oscillation. This happens because of a resistance which gradually damps the motion.
The important features of simple harmonic motion
The period of the motion is independent of the amplitude
2. At the equilibrium position C the displacement is zero, the speed of the body is maximum and acceleration is zero
3. As the body moves the speed increase toward the centre and the decease as it moves out toward the maximum displacement4. When the displacement is maximum in either direction the speed is zero the acceleration is maximum but directed opposite to displacement
Forced oscillation or vibration In order to maintain an oscillating system in a constant and continuous motion, some external periodic force needs to be applied to the system. For instance the simple harmonic motion of a simple pendulum oscillating at its natural frequency will surely be dumped. So when a bob is subjected to external force, the oscillatory frequency is no longer the natural frequency but that of external force. Such a vibration that results from external periodic force acting on a system is called force oscillationor vibration
Examples of forced oscillation or vibration
1. Vibration of bridge under the influence of marching soldiers
2. Vibration of turning fork when exposed to the periodic force of a sound wave