What is simple harmonic motion?
Simple harmonic motion is defined as:”When the net force is directly proportional to the displacement from the mean position and is always directed towards mean position.” A body is said to be vibrating if it moves back and forth or to and fro about a point.Another term for vibration is oscillation.A special kind of vibratory or oscillatory motion is called the simple harmonic motion(SHM).
Examples of Simple harmonic motion(SHM):
- Motion of mass attached to spring
- Ball and Bowl system
- Motion of simple pendulum
- Atoms vibrating in molecules
- The vibration of the string of a violin
Conditions for simple Harmonic motion:
- The acceleration of the oscillator should be directly proportional to its displacement and always be directed towards mean position.
- Total energy must remain conserved in simple harmonic motion.
- There must be elastic restoring force acting on the system.
- The restoring force must be directly proportional to the displacement from the mean position.(The system must obey Hook’s law).
- The oscillator in simple harmonic motion must have inertia.
Simple harmonic Oscillator:
“A body executing simple harmonic motion is called simple harmonic oscillator.” OR “A vibrating body is said to be simple harmonic oscillator,if the magnitude of restoring force is directly proportional to the magnitude of its displacement from mean position.Vibration of simple harmonic oscillator will be linear when frictional forces are absent.’ Examples:
- Motion of mass attached to spring
- Motion of simple pendulum
- Atoms vibrating in molecules
We will discussed few examples in detail:
Motion of mass attached to a spring:
One of the simplest types of oscillatory motion is that of horizontal mass spring system.If the spring stretched or compressed through a small displacement x from its mean position,it exerts a force F on the mass.According to Hooke’s law this force is directly proportional to the change in length x of the spring i.e.,
F =-Kx ………….(1)
Where x is the displacement of the mass from its mean position O,and k is a constant called spring constant defined as:
The value of k is a measure of the stiffness of the spring.Stiff springs have large value of k and soft springs have small value of k. As we know:
If (k/m) is constant then:
a ∝ -x …………..(2)
It means that the acceleration of a mass attached to a spring is directly proportional to its displacement from the mean position.Hence,the horizontal motion of a mass spring system is an example of simple harmonic motion.The negative sign in equation (2) means that the force exerted by the spring is always directed opposite to the displacement of the mass.Because the spring force always acts towards the mean position,it is sometimes called a restoring force.Which is defined as:”A restoring force always pushes or pulls the object performing oscillatory motion towards the mean position.”
Initially the mass m is at rest in mean position O and the resultant force on the mass is zero as shown in figure(a).Suppose the mass is pulled through a distance x up to extreme position A and then released as shown in figure (b).The restoring force exerted by the spring on the mass will pull it towards the mean position O.Due to the restoring force magnitude of the restoring force decreases with the distance from the mean position and becomes zero at O.However,the mass gains speed as it moves towards the mean position and its speed becomes maximum at O.Due to inertia the mass does not stop at the mean position O but continues its motion and reaches the extreme position B. As the mass moves from the mean position O to the extreme position B,the restoring force acting on it towards the mean position steadily increases in strength.Hence the speed of the mass decreases as it moves towards the extreme position B.The mass finally comes briefly to rest at the extreme position B as shown in figure (c).Ultimately the mass returns to the mean position due to the restoring force. This process is repeated,and the mass continues to oscillate back and forth about the mean position O.Such motion of a mass attached to a spring on a horizontal frictionless surface is known as simple harmonic motion (SHM). The time period T of the simple harmonic motion of a mass ‘m’ attached to a spring is given by the following equation:
A simple pendulum also exhibits SHM. It consists of a small bob of mass ‘m’ suspended from a light string of length ‘L’ fixed at its upper end.In the equilibrium position O,the net force on the bob is zero and the bob is stationary.Now if we bring the bob to extreme position A,the net force is not zero as shown in fig.There is no force acting along the string as the tension in the string cancels the component of the weight mg cosθ.Hence there is no motion along this direction. The component of the weight mg sinθ is directed towards the mean position and acts as a restoring force.Due to this force the bob starts moving towards the mean position O.At O,the bob has got the maximum velocity and due to inertia,it does not stop at O rather it continues to move towards the extreme position B.During its motion towards point B,the velocity of the bob decreases due to restoring force.The velocity of the bob becomes zero as it reaches the point B. The restoring force mg sinθ still acts towards the mean position O and due to this force the bob again starts moving towards the mean position O.In this way,the bob continues its to and fro motion about mean position O. It is clear from the above discussion that the speed of the bob increases while moving from point A to O due to the restoring force which acts towards O.Therefore,acceleration of the bob is also directed towards O.Similarly,when the bob moves from O to B ,its speed decreases due to restoring force which again acts towards O.Therefore,acceleration of the bob is again directed towards the mean position O.Hence the motion of a simple pendulum is SHM .
Motion of simple pendulum is SHM:
Consider the bob at the position B during its vibratory motion as shown in figure.Let ‘m’ be the mass of the bob and x be the displacement of the bob from the mean position at position B.There are two forces are acting on the bob at this position.
- Weight mg of the bob acting vertically downward.
- Tension T of the string acting along the direction of the string.
The weight mg of the bob can be resolved into two rectangular components:
- mg cosθ=component of weight along the string
- mg sinθ=component of the weight perpendicular to the string.
As there is no motion of the bob in the direction of the string,the component mg cosθ balances the tension in the string.So:
T = mg cosθ ……………(1)
So these two cancel each other.Therefore,the component mg sinθ is responsible for the motion of the bob which brings the bob back towards its mean position.Thus mg sinθ represents restoring force.So:
F =-mg sinθ …………….(2)
If ‘a’ is the acceleration of the bob at point B then according to Newton’s second law of motion:
F =ma ……………………(3)
Comparing equation (2) and (3),we have:
ma = -mg sinθ
⇒ a=-g sinθ ……….(4)
According to trigonometry if θ is small,the sinθ ≈θ,then equation (4) will become
⇒ a=-gθ …………..(5)
Using relation S=r θ
⇒ θ=S/r ……………………(6)
Now when θ is small ,s=arc AB =OB =x and r =l
Equation (6) now will become
⇒ θ = arc AB/l =OB/l=x/l
Now putting the value of θ in equation (5),we have
a = -g x/l …………….(7)
Since g/l is constant ,therefore,
⇒ a ∝ – x ……………..(8)
Equation (8) shows that the acceleration a of the bob is directly proportional to the displacement x and negative sign shows that it is directed towards the mean position.Hence the motion of simple pendulum is simple harmonic.
Time period of simple pendulum:
The acceleration of the body is given by:
⇒ a = -ω2x …………………..(9)
But the acceleration of simple pendulum is given by
⇒ a = -g x/l ……………………..(10)
Comparing equation (9) and (10),we have
-ω2x = -g x/l
⇒ ω =√g/l …………….(11)
The time period of a body executing SHM is given by:
Putting the value of ω from equation (11) ,we have
⇒ T =2π √l/g ……………….(12)
Since frequency is related to time period as: f=1/T
Energy conservation in S.H.M:
Let us consider the case of a vibrating mass spring system.When the mass m is pulled slowly,the spring is stretched by an amount x0 against the elastic restoring force F.It is assumed that stretching is done slowly so that acceleration is zero.According to Hook’s law
When displacement =0 force=0
When displacement=x0 force=kx0
Average force is:
Work done in displacing the mass m through x0 is:
This work appears as elastic potential energy of the spring.Hence
The above equation gives the maximum P.E at the extreme position.Thus
At any instant t,if the displacement is x,then P.E at that instant is given by:
The velocity at that instant is given by equation:
Thus kinetic energy is maximum when x=0,i.e .when the mass is at equilibrium or mean position:
For any displacement x,the energy is partly P.E and partly K.E.Hence:
Thus the total energy of the vibrating mass and spring is constant.When the K.E of the mass is maximum ,the P.E of the spring is zero.Conversely ,when the P.E of the spring is maximum ,the K.E of the mass is zero.The interchange occurs continuously from one form to the other as the spring is compressed and released alternately.The variation of P.E and K.E with displacement is essential for maintaining oscillations.This periodic exchange of energy is a basic property of all oscillatory systems.In the case of simple pendulum gravitational P.E of the mass ,when displaced ,is converted into K.E at the equilibrium position .The K.E is converted into P.E as the mass raises to the top of the swing.Because of the frictional forces,energy is dissipated and consequently ,the systems do not oscillate indefinitely.
Watch also video about simple harmonic motion: