Physics Complete

Problems on SHM displacement,velocity and Acceleration

Progressive Wave Representation and Problems

Resonance in Physics

Every body will have some frequency with which it vibrates and it is called its natural frequency.Under the influence of external force body vibrates with a frequency different from its natural frequency and it is called forced vibration. Resonance is a special case of forced vibration.If the frequency of external force is equal to the frequency of body which is going to be vibrated then the body virates with the maximum frequency and the phenomenon is called Resonance.

A person swinging in a swing without anyone pushing it or a simple pendulum, displaced and released, are examples of free oscillations. In both the cases, the amplitude of swing will gradually decrease and the system would, ultimately, come to a halt. Because of the ever present dissipative forces, the free oscillations cannot be sustained in practice.

However, while swinging in a swing if you apply a push periodically by pressing your feet against the ground, you find that not only the oscillations can now be maintained but the amplitude can also be increased. Under this condition the swing has forced, or driven, oscillations.

The maximum possible amplitude for a given driving frequency is governed by the driving frequency and the damping, and is never infinity. The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance.

Example for Resonance

Consider a set of five simple pendulums of assorted lengths suspended from a common rope as shown in figure. The pendulums 1 and 4 have the same lengths and the others have different lengths. Now let us set pendulum 1 into motion. The energy from this pendulum gets transferred to other pendulums through the connecting rope and they start oscillating. The driving force is provided through the connecting rope. The frequency of this force is the frequency with which pendulum 1 oscillates. If we observe the response of pendulums 2, 3 and 5, they first start oscillating with their natural frequencies of oscillations and different amplitudes, but this motion is gradually damped and not sustained.

Their frequencies of oscillation gradually change and ultimately they oscillate with the frequency of pendulum 1, i.e. the frequency of the driving force but with different amplitudes. They oscillate with small amplitudes. The response of pendulum 4 is in contrast to this set of pendulums. It oscillates with the same frequency as that of pendulum 1 and its amplitude gradually picks up and becomes very large. A resonance-like response is seen. This happens because in this the condition for resonance is satisfied, i.e. the natural frequency of the system coincides with that of the driving force.

SHM related topics

Damped simple harmonic motion
Simple Pendulum
What is periodic and Oscillatory Motion is ? Displacement in Oscillatory motion
Simple Harmonic Motion
Velocity and acceleration of SHM
Energy of particle in SHM

Wave Motion an introduction

Frequency of Stationary wave in a stretched String

Stretched String Problems and Solutions

Vibrations of Sound in Closed Pipe and Open Pipe

Beats and Its Applications

Doppler Effect and Its Applications

Simple Hormonic Motion Damped

The motion of a simple pendulum, swinging in air, dies out eventually.This is because the air drag and the friction at the support oppose the motion of the pendulum and dissipate its energy gradually. The pendulum is said to execute damped oscillations.

In damped oscillations, although the energy of the system is continuously dissipated, the oscillations remain apparently periodic. The dissipating forces are generally the frictional forces. To understand the effect of such external forces on the motion of an oscillator,let us consider a system as shown in figure where a block of mass m oscillates vertically on a spring with spring constant k.

The block is connected to a vane through a rod . The vane is submerged in a liquid. As the block oscillates up and down, the vane also moves along with it in the liquid. The up and down motion of the vane displaces the liquid, which in turn, exerts an inhibiting drag force (viscous drag) on it and thus on the entire oscillating system. With time, the mechanical energy of the block spring system decreases, as energy is transferred to the thermal energy of the liquid and vane.

Let the damping force exerted by the liquid on the system be Fd. Its magnitude is proportional to the velocity v of the vane or the block. The force acts in a direction opposite to the direction of v. This assumption is valid only when the vane moves slowly. Then for the motion along the x-axis Fd = –b v

where b is a damping constant that depends on the characteristics of the liquid and the vane. The negative sign makes it clear that the force is opposite to the velocity at every moment.

When the mass m is attached to the spring and released, the spring will elongate a little and the mass will settle at some height. This position is the equilibrium position of the mass. If the mass is pulled down or pushed up a little, the restoring force on the block due to the spring is

FS = –kx, where x is the displacement of the mass from its equilibrium position. Thus the total force actingon the mass at any time t is F = –k x –b v.

If a(t) is the acceleration of the mass at time t, then by Newton’s second law of motion for force components along the x-axis, we have m a(t) = –k x(t) – b v(t)

If the oscillator is damped, the mechanical energy is not constant but decreases with time.

SHM related topics

Simple Pendulum
What is periodic and Oscillatory Motion is ? Displacement in Oscillatory motion
Simple Harmonic Motion
Velocity and acceleration of SHM
Energy of particle in SHM

Wave Motion an introduction

Progressive Wave Representation and Problems

Frequency of Stationary wave in a stretched String

Stretched String Problems and Solutions

Vibrations of Sound in Closed Pipe and Open Pipe

Beats and Its Applications

Doppler Effect and Its Applications

Simple Pendulum and SHM

A point sized heavy mass suspended with a inextensible string from a rigid support is called simple pendulum. For less angle of oscillation it executes simple harmonic motion.

The forces acting on the bob are the force T, tension in the string and the gravitational force Fg (= m g), as shown in figure. The string makes an angle θ with the vertical. We resolve the force Fg into a radial component Fg cos θ and a tangential component Fg sin θ.

The radial component is canceled by the tension, since there is no motion along the length of the string. The tangential component produces a restoring torque about the pendulum’s pivot point. This torque always acts opposite to the displacement of the bob so as to bring it back towards its central location. The central location is called the equilibrium position ( θ = 0), because at this position the pendulum would be at rest if it were not swinging.

The restoring torque τ is τ = –L (Fg sin θ) where the negative sign indicates that the torque acts to reduce θ, and L is the length of the moment arm of the force Fg sin θ about the pivot point. For rotational motion , τ = I α where I is the pendulum’s rotational inertia about the pivot point and α is its angular acceleration about that point.

and further we can prove that time period of a pendulum is 2(pi) square root of (L/g).

SHM related topics

Simple Harmonic Motion Displacement ,Velocity and Acceleration

Problems on SHM displacement,velocity and Acceleration

Time Period of Simple pendulum

Simple Harmonic Motion Displacement ,Velocity and Acceleration

What is periodic and Oscillatory Motion is ? Displacement in Oscillatory motion
Simple Harmonic Motion
Velocity and acceleration of SHM
Energy of particle in SHM

Topics of Heat and Thermodynamics

Heat engine
Internal Energy
Zeroth law of thermodynamics
Thermodynamics Introduction

Energy of Simple Hormonic Motion

According to law of conservation of energy total energy of a system always remains constant and the particle in simple harmonic motion is not a exception from it.

A particle executing simple harmonic motion has kinetic and potential energies, both varying between the limits, zero and maximum. Here we will use the equations of a particle's displacement and velocity in SHM.

The velocity of a particle executing SHM, is a periodic function of time. It is zero at the extreme positions of displacement. Therefore, the kinetic energy (K) of such a particle, which is defined as

is also a periodic function of time, being zero when the displacement is maximum and maximum when the particle is at the mean position.Since the sign of v is immaterial in K, the period of K is T/2.
The spring force F = –kx is a conservative force, with associated potential energy U = 1/2 K x^2.
Hence the potential energy of a particle executing simple harmonic motion is,

Thus the potential energy of a particle executing simple harmonic motion is also periodic, with period T/2, being zero at the mean position and maximum at the extreme displacements.

Total Energy

The total mechanical energy of a harmonic oscillator is thus independent of time as expected for motion under any conservative force. The time and displacement dependence of the potential and kinetic energies of a linear simple harmonic oscillator are shown in figure.

In a linear harmonic oscillator, all energies are positive and peak twice during every period. For x = 0, the energy is all kinetic and for x = ± A it is all potential.

In between these extreme positions, the potential energy increases at the expense of kinetic energy. The former stores its potential energy and the latter stores its kinetic energy.

SHM related topics

Problems on SHM displacement,velocity and Acceleration

Time Period of Simple pendulum

Simple Harmonic Motion Displacement ,Velocity and Acceleration

What is periodic and Oscillatory Motion is ?
Displacement in Oscillatory motion
Simple Harmonic Motion
Velocity and acceleration of SHM

Topics of Heat and Thermodynamics

Heat engine
Internal Energy
Zeroth law of thermodynamics
Thermodynamics Introduction

Velocity and Acceleation of SHM

A body in oscillatory motion whose acceleration is directly proportional to displacement and is always directed towards mean position is called simple harmonic motion. Here we are going to derive equations for velocity and acceleration of body in SHM.

Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion takes place.

A particle in SHM will have a displacement y represented as y = a cos(wt) whose initial phase is equal to zero. Here the particle is in horizontal circular motion. If it is in vertical circular motion instead of sin function we have to use cosine function.By differentiating the above equation we will get velocity and by differentiating once again we will get the equation for acceleration.

Acceleration can be determined further with equation – ω^2 A cos ( ωt + φ) or – ω^2 (t)

Its graphical method we can represent displacement ,velocity and acceleration as shown below.

SHM related topics

Problems on SHM displacement,velocity and Acceleration

Time Period of Simple pendulum

Simple Harmonic Motion Displacement ,Velocity and Acceleration

What is periodic and Oscillatory Motion is ?
Displacement in Oscillatory motion
Simple Harmonic Motion

Topics of Heat and Thermodynamics

Heat engine
Internal Energy
Zeroth law of thermodynamics
Thermodynamics Introduction
Heat transfer by radiation
Heat transfer by convection
Heat transfer and conduction
Heat and Temperature

Simple Hormonic Motion

For a body to be in simple harmonic motion it shall satisfy some conditions.They are

1.The motion shall be oscillatory.
2.Acceleration of motion shall be directly proportional to displacement.
3.Acceleration shall be always directed to words its mean position.

If this three conditions are met then the oscillatory motion is said to be in simple harmonic motion.

Example :

Let us consider a particle vibrating back and forth about the origin of an x-axis between the limits +A and –A . In between these extreme positions the particle moves in such a manner that its speed is maximum when it is at the origin and zero when it is at ± A. The time t is chosen to be zero when the particle is at +A and it returns to +A at t = T.

Let us record its positions as a function of time by taking ‘snapshots’ at regular intervals of time.The position of the particle with reference to the origin gives its displacement at any instant of time. For such a motion the displacement x(t ) of the particle from a certain chosen origin is found to vary with time as,

x (t) = A cos ( ωt + φ) in which A, ω, and φ are constants.

The motion represented by equation is called simple harmonic motion (SHM); a term that means the periodic motion is a sinusoidal function of time. Equation in which the sinusoidal function is a cosine function.

The previous post is about What is periodic and Oscillatory Motion is ? Displacement in Oscillatory motion

Problems on SHM displacement,velocity and Acceleration

Time Period of Simple pendulum

Simple Harmonic Motion Displacement ,Velocity and Acceleration

Topics of Heat and Thermodynamics

Heat engine
Internal Energy
Zeroth law of thermodynamics
Thermodynamics Introduction
Heat transfer by radiation
Heat transfer by convection
Heat transfer and conduction
Heat and Temperature

Displacement in Oscillatory Motion

Displacement of a particle as the change in its position vector. It refers to change with time of any physical property under consideration. For example, in case of rectilinear motion of a steel ball on a surface, the distance from the starting point as a function of time is its position displacement. The choice of origin is a matter of convenience. Consider a block attached to a spring, the other end of which is fixed to a rigid wall as shown in figure below.

It is convenient to measure displacement of the body from its equilibrium position. For an oscillating simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable as shown in figure below. The term displacement is not always to be referred in the context of position only.

The displacement can be represented by a 0mathematical function of time. In case of periodic motion, this function is periodic in time. One of the simplest periodic functions is given by

f (t) = A cos ωt

Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients.

The previous post is about What is periodic and Oscillatory Motion is ?

Problems on SHM displacement,velocity and Acceleration

Time Period of Simple pendulum

Simple Harmonic Motion Displacement ,Velocity and Acceleration

Periodic and Oscillatory Motion

Oscillatory motion is a to and fro motion about a mean position and periodic motion repeats at regular intervals of time. All oscillatory motions are periodic and all periodic motions are not oscillatory.

If the body is given a small displacement from the position, a force comes into play which tries to bring the body back to the equilibrium point, giving rise to oscillations or vibrations. For example, a ball placed in a bowl will be in equilibrium at the bottom. If displaced a little from the point, it will perform oscillations in the bowl. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory. Circular motion is a periodic motion, but it is not oscillatory.

When the frequency is small, we call it oscillation (like the oscillation of a branch of a tree), while when the frequency is high, we call it vibration (like the vibration of a string of a musical instrument). Simple harmonic motion is the simplest form of oscillatory motion. This motion arises when the force on the oscillating body is directly proportional to its displacement from the mean position, which is also the equilibrium position. At any point in its oscillation, this force is directed towards the mean position.

A motion that repeats itself at regular intervals of time is called periodic motion.The following are the examples of periodic motion.

In practice, oscillating bodies eventually come to rest at their equilibrium positions, because of the damping due to friction and other dissipative causes. However, they can be forced to remain oscillating by means of some external periodic agency.

Any material medium can be pictured as a collection of a large number of coupled oscillators. The collective oscillations of the constituents of a medium manifest themselves as waves. Examples of waves include water waves, seismic waves, electromagnetic waves.

Topics of Heat and Thermodynamics

Problems on SHM displacement,velocity and Acceleration

Time Period of Simple pendulum