Table of content
Table of content:
2-1 Simple Harmonic Motion and Hooke’s Law
2-1-1 Harmonic Motion and Hooke’s Law
2-1-2 Mass-spring Oscillator
2-1-3 Simple Pendulum
2-1-4 Simple Harmonic Motion
2-1-5 Energy in Simple Harmonic Motion
2.3 Forced Oscillations and Resonance
3- Wave motion
3-1 Definition of Wave
3-2 Types of Waves
3-3 Mathematical Description of a Wave
3-4 Speed and Energy in Wave motion
3-5 Principle of Superposition
4- Solved Examples on Oscillations and Wave Motion
5- Applications on Oscillations and Wave motion
It is the repetitive variation in time of some measure about a central value which is often a point of equalitarian, or between different states. The term vibrating is used to specifically describe mechanical oscillations.
2.1 Simple Harmonic Motion and Hooke’s Law:
– 2-1-1 Harmonic Motion and Hooke’s Law:
Hooke’s law and the principle of simple harmonic motion are important for understanding physics associated with elastic objects. Hooke’s law implies that in order to deform an elastic object, there must be a force applied to overcome the restoring force in which exerted by the object. The elastic object stores the energy that has the ability to do the work which is known as the potential energy of the spring.
To hold a spring in its compressed or stretched position requires exerting force on the spring this force is directly proportional to the displacement of the spring. On the other hand, the spring will exert a force having the same value but in the opposite direction. This restoring force is the product of both the elasticity constant K of the object and the displacement ?y but in the opposite direction of the displacement or the applied force. This leads to the following equation which is known as Hooke’s law.
F = -k ?x, (Equation 1)
– 2-1-2 Mass-spring Oscillator:
It is a system in which when it undergoes displacement form its equilibrium position, it experiences a restoring force F that is proportional to the displacement. The simplest system that can have periodic motion is a frictionless horizontal guide system with a mass rests on it, attached to a spring of negligible mass that is possible to be compressed and stretched. Hooke’s law holds only for relatively small extensions of the spring and therefore, only small masses can be attached. This is because the displacement of the mass cannot be made too large.
– 2-1-3 Simple Pendulum:
For small displacements, a pendulum is a simple harmonic oscillator. Simple pendulum is defined to have a pendulum bob which is an object with a small mass, suspended from string that has a very small mass but is strong enough not to stretch appreciably. The linear displacement from equilibrium is arc length s. when a pendulum is displaced sideways from its equilibrium position, the restoring force due to gravity will accelerate it back to its equilibrium position. This restoring force acting on the pendulum’s mass causes it to oscillate about its equilibrium position, when releasing it will swing back and forth. The time spent for completing one cycle is called the period. When the restoring force is directly proportional to the displacement #, it is considered as a simple harmonic oscillator. For small angles ( less than 15) the expression of the restoring force is :
This expression is in the form:
F= -K?x ; Where k=mg/l and the displacement x=s.
– 2-1-4 Simple Harmonic Motion
It is a special type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and act on the opposite direction of the displacement. In other words, it is the motion of a particle moving along a straight line with an acceleration that has a specific direction to a fixed point on the line and whose magnitude proportional to the distance from a fixed point. The simplest mechanical oscillating system is weight that is attached to a linear spring subject only to weight and tension. It is in an equilibrium state when the spring is static but when the system is displaced there is a restoring force that acts on the mass bringing it back to equilibrium position. The time taken for an oscillation to occur is often called the oscillation period. Oscillations occur because when the spring is static at the equilibrium position the mass has a kinetic energy which is converted to a potential energy stored in the spring at the extremes of its path.
In Newtonian mechanics, for simple harmonic motion in one dimension, the equation of motion can be obtained by means of Newton’s second law for a mass on a spring.
F =m = -kx ; where m is the mass of the body. X is the displacement from the equilibrium, and k is the spring constant. Therefore,
= -kx solving the differential equation gives the following sinusoidal function:
X(t)=a cos(?t+?) where, ?=2?f =2?/ T ,thus, the period T can be given by:
T = 2? (m/k)1/2 this ilustrat that the period T is proportional to the square root of the weight attached.
Mass on a spring, uniform circular motion and mass of a simple pendulum are all examples of simple harmonic oscillator.
– 2-1-5 Energy in Simple Harmonic Motion
It is the total energy that the body possesses while performing a simple harmonic motion. When the object moves toward the extreme position it is in motion and at the moment it reaches the extreme position it comes to rest again. Therefore, to calculate energy in a simple harmonic motion, both kienitic and potential energy that the particle possesses. Kinetic energy is the possessed energy by the object when it is moving (in motion). While potential energy is the one possessed by the object when it is at rest (no motion). The total energy in a simple harmonic motion is the sum of its potential and kinetic energy.
The formula used to calculate the kinetic energy is:
Kinetic energy= 1/2 mv2 = 1/2 m ?2 ( a2 – x2) , and since k/m = ?2 , Then, Kinetic energy= 1/2 k ( a2 – x2)
The formula used to calculate the potential energy is:
Total work done = 1/2 K x2 = 1/2 m ?2×2 , the total work done is stored as a potential energy, therefore, Potential Energy= 1/2 kx2 = 1/2 m ?2×2
Since that total energy in a S.H.M as mentioned above is the sum of its kinetic energy and potential energy, it can be calculated using the following equation:
E = 1/2 m ?2a2, Thus, the total energy in the simple harmonic motion of an object is proportional directly to its mass and the square of both the frequency and amplitude of the oscillation. The total energy in simple harmonic motion will always be a constant, this is resulted from the law of conservation of energy that says energy can neither be created nor destroyed.
2-2 Damped Oscillations:
It is known that in reality an oscillating object won’t oscillate for ever since that frictional forces will cause the system to eventually come back to rest. A damped oscillation means that oscillations are fading away with time. In this case frictional force will be added to the mass and the spring. So instead of the amplitude being constant in S.H.M, the amplitude is decaying with time. The damping force is often proportional to the velocity of the body that is oscillating but in the opposite direction.
2-3 Forced Oscillations and Resonance
When applying a periodic driving force to the system on a simple harmonic oscillator, it will add energy into the system at a certain frequency not necessarily the same as the natural frequency. The natural frequency is the frequency that the system will oscillate with no driving and no dumping force. For example, when applying an external force to a body attached to a spring vertically by moving the support of the spring up and down. In this case the hand that apply this external force will move with a driving frequency and the object attached will response at the same frequency, but it is possible to have a larger amplitude. In this case equation of motion of the mass :
X= Bsin(? t+?) where ? is the angular frequency of the driving force, and B amplitude of the oscillations. The amplitude B has a maximum value when ?= ? which is called the resonance condition. The resonance condition is when the driving force has the same frequency as the natural frequency of the system, and the system that is driven is said to be resonate. Whenever the driving frequency get higher than the natural frequency, the amplitude of oscillations will get smaller until oscillations are less and close to disappear.
3- Wave motion:
3-1 Definition of Wave:
A wave is known as a disturbance that transfer energy through a matter or space. Waves consist of oscillations of a medium around a fixed location. Waves contribute in everything in real life such as light, sound, magnet, water and earthquakes. They can be characterized by wavelength, frequency and speed at which they move. The lowest point on the wave is called trough, while the higher point is called the peak, both of them are separated by a distance that is twice the amplitude of the wave.
The medium in which a certain wave move through may be impossible for other waves to move through. For example, light waves can freely move through empty space while, on the other hand, sound waves cannot. Still, all waves transport energy from one place to another.
3-2 Types of Waves: Transverse wave and Longitudinal wave :
– Transverse wave:
Where the direction in which the wave is traveling makes a 90-degree angle with the vibrations of a particle. In Transverse wave while travelling crests and troughs move along. For example, shaking a rope up and down.
– Longitudinal wave:
Are waves that have both direction of travel and vibrations of particles parallel to each other. In the travelling wave compressions and rarefactions move along. For example, when dropping a stone on water, the surface will form longitudinal waves.
3-3 Mathematical Description of a Wave
When the wave is travelling through a medium, the particles of that medium will undergo displacement from their initial positions( undisturbed positions). In order to calculate this displacement in terms of time as the wave travel we should use the simple harmonic motion of the source for periodic waves. The following equation will express the displacement of a particle resulted from the wave travelling in the positive x direction, with a frequency f, wavelength ?, and an amplitude A:
Y= Asin(2? f-(2?/ ?)), and for motion towards the negative side of the x axis: Y= Asin(2? f+(2?/ ?) those equations are used in transverse or longitudinal waves assuming that when the time is equal to zero both y and x will also be equal to zero.
3-4 Speed and Energy in Wave motion
The speed of any object describes how fast it is moving and is usually considered as distance traveled per time. When a point on the wave such as crest, is travelling along a medium it can clearly show the distance covered. Therefore, in case of waves, the speed is the distance traveled by a given point on the wave an a specific time interval and is given by:
Speed=distance/time , also given by : v=???? T), v=(?? f). And wave speed depends only on the medium that the wave is moving through only.
As mentioned all waves carry energy, and the amount of energy in a wave is dependent on its amplitude. A wave is a displacement that is resisted by a restoring force, when the displacement is larger it will need a larger force to create it. Since the energy is put into the wave by the work done to create it, wave’s energy is directly proportional to the amplitude squared since W is proportional to Fx. In addition, the effects of energy also depends on time, for example when using ultrasound deep-heat for along time more energy will be transformed. Furthermore, changing the area that the wave covers has an important effect in terms of energy, like using the sun light for starting camp fire. All the previous factors are part of the defining intensity I as power per unit area.
3-5 Principle of Superposition:
This principle can be applied whenever having two or more waves that are travelling through the same medium at the same time. It is related only to waves that have finite length and with continues sine waves. Those waves would pass through each other without any disturbance in any of them. The net displacement of the medium at any point is the sum of the individual wave displacements.
The interference of the two waves can either be constructive or destructive. When the waves are travelling in the same direction (having same amplitude, frequency, and wavelength) and using the principle of superposition, they will result in wave with a higher amplitude that is twice the amplitude of the individual wave which means that the waves have interfered constructively. On the other hand, when the two have opposite directions they would interfere destructively.
4-Solved Examples on Oscillations and Wave Motion
A mass of 2 kilograms is oscillating on a spring that has the constant equals to 4 N/m passes through the equilibrium point with velocity = 8m/s, what is the energy of the system at the equilibrium point? Derive the maximum displacement of the mass.
At the equibilirium point, there is no potential energy stored in the spring, and therefore, all of the energy in the system is kinetic:
K=(1/2) mv =(1/2)(2)(8) = 64 J
Using this answer we can calculate the maximum displacement of the mass, since this is the total energy of the system. When the block has a maximum displacement, it is at rest and all of the energy of the system is stored as the potential enery in the spring. Because of the conservation of energy law and since the energy is conserved in the system the answer can be related to the energy of one position at another:
E = E
(1/2)kxm =(1/2)mv =64
xm equals the squre root of 64/k , since k=4, calculating the square root of 16
then the final answer will be = 4 meters
At what point during the oscillation of a spring is the force on the mass the greatest?
F=-kx, therefore, the force on the mass will be the greatest when x the displacement of the block is maximum.
5-Applications on Oscillations and Wave motion:
The collapse of the Tacoma Narrows Bridge that is located in Washington state back in 1940. The heavy winds caused the bridge to drive into oscillations at its resonate frequency leaving harmful effects. The damping force decreased when the cables that were supporting the bridge broke, allowing to increase amplitudes even more until the structure failed.
Also known as seismic waves, have many important applications one of them is in early estimation of earthquakes. Also, to visualize the propagation of seismic waves from historic earthquakes through the earth’s interior and around the surface of earth. In addition to controlling the speed or reverse the wave propagation.
Summary and conclusion: