Showing posts with label Oscillations. Show all posts
Showing posts with label Oscillations. Show all posts

Oscillations Problems with Solution Three

We are solving series of problems based on the concept of oscillation and simple harmonic motion. If the acceleration of a body is directly proportional to the displacement and opposite to it, then the motion is said to be simple harmonic motion. This can be done by many type of systems and if they are obeying the above mentioned condition, we can get the time period or frequency and it is constant all over the oscillatory motion. A loaded spring do oscillate when it is slightly disturbed. In practical way, any body won’t continue its oscillatory motion for ever and and due to air resistance, it slowly decreases and finally comes to the state of the rest. This kind of motion is called damped oscillatory motion and it that has to be continued, it shall have some external force support and that kind of motion is called forced oscillation.

Problem

We need to find the time period of pendulum of infinite length and the problem is as shown in the diagram below.


Solution

We need to write the equation for the torque as the product of force and perpendicular distance. Force is nothing but the weight of the body and distance is found as shown in the diagram below. We need to write further torque as the product of moment of inertia and angular acceleration basing on its definition. Thus we can equation for the acceleration and hence the time period as shown in the diagram below.


Problem

Frequency of a particle in SHM is given to us as shown below. We need to find the maximum speed that the particle can reach in the oscillatory motion.


Solution

The restoring force acting on the spring is nothing but the weight of the body and hence we can find the maximum possible displacement by equating them. By comparing that with the standard equation, we can find angular velocity and hence the maximum speed of the particle as shown in the diagram below.


Problem

A particle of known mass is attached to three springs as shown in the diagram below. If the particle is slightly disturbed, we need to know the time period of the system.

 
Solution

The resultant force acting on the mass using the vector laws of addition. We need to resolve the force into components and add as shown in the diagram below. By equating it to the restoring force, we can write the equation for the time period as shown in the diagram below.


Problem

Two blocks are kept one over the other and the lower surface is smooth and the connecting surface is rough as shown in the diagram below. The system is connected to a rigid support with a spring and the time period of the system is given to us in the problem. We need to find the mass of the upper block and the coefficient of friction so that there is no slipping between the two blocks.


Solution

We know the equation for the time period of a loaded spring and using that data, we can find the mass of the upper body as shown in the diagram below. Further equating the frictional force to the restoring force, we can solve the problem as shown in the diagram below.


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Oscillations Problems with Solution Two

We are solving series of problems based on  the concept of oscillations. It is also called vibratory or harmonic motion where there  is to and fro motion that gets repeated at regular intervals of time. The vertical projection of uniform circular motion is simple harmonic and we have derived equations for displacement, velocity and acceleration for the body in simple harmonic motion based on that. The time taken to complete one oscillation is called time period and the number of oscillations per one second is called frequency.

Problem

A body of known mass is connected with two springs and they in tern are connected to rigid support as shown in the diagram below. We need to find the effective time period of the system and the problem is as shown in the diagram below.


Solution

When ever the body is slightly disturbed, it starts oscillating and one spring expands and the other spring contracts by the same magnitude. As the force acting on both of them is same, the two springs behaves as if like they are connected in series and we can find the effective spring constant of the system and time period as shown in the diagram below.


Problem

A spring of spring constant K and length L is cut into two parts and the relation between the lengths of two parts and their ratio is given to us in the problem as shown in the diagram below. We need to measure the spring constant of one part of the spring.


Solution

Spring constant is the measure of nature of spring and it depends on the length of spring in the inverse proportional ratio. Taking that into consideration, we need to solve the problem as shown in the diagram below.


Problem

A piece of wood known dimensions is given to us and its density is also known to us as given in the problem below. It is floating in water with one surface vertical to the surface. It is pushed down and released and it starts oscillating. We need to measure the time period of the system.


Solution

When ever a force is applied on the body, there is buoyant force also and the system starts executing oscillatory motion with a certain restoring force. By comparing that with the standard equation, we can solve the problem as shown in the diagram below.


Problem

A cylindrical piston is used to close a cylinder with a certain gas and when it is slightly distributed, it starts oscillating in simple harmonic motion. We need to find the time period of the system and the problem is as shown in the diagram below.


Solution

By comparing it with the standard equation, we can solve the problem as shown in the diagram below. We need to identity the restoring force and the equation for the acceleration of the body in SHM. The problem is solved as shown in the diagram below. Here in the first case, we get the equation for the force.


We need to equate to the product of mass and acceleration and we further need to substitute the value of acceleration for a body in SHM so that we can compare it with the standard equation. Thus we can get the time period of the system as shown in the diagram below.


Problem

A spherical ball of known mass and radius is rolling with out slipping on a concave surface of known radius as shown in the diagram below. If the oscillations are small, we need to find the time period of the system.


Solution

Let us consider the spherical ball is at a particular position and we can find the angle basing on the definition as shown in the diagram below. As we know the equation for the acceleration of a body sliding on a inclined surface, by writing that equation, we can get that and find the time period as shown in the diagram below.



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Oscillations Problems with Solution One

We are solving series of problems based on the concept of oscillations. Oscillation is a kind of motion where the body oscillates about a fixed point called mean position and all oscillatory motions are periodic. It means oscillatory motion is repeated at regular intervals of time. We need to understand that all oscillatory motions are periodic but all periodic motions are not oscillatory. If oscillatory motion is also satisfying a condition like displacement is directly proportional to acceleration and acceleration is always directed to wards the mean position, we call that kind of oscillatory motion as simple harmonic motion. Simple pendulum is one example that executes simple harmonic motion when it is sightly disturbed from its mean position. We have derived equation for displacement, velocity and acceleration for a body in simple harmonic motion.

Problem

To a body in a simple harmonic motion, velocity is represented as shown in the equation below. We need to measure maximum acceleration that the body can get in the given conditions.



Solution

We have all ready derived equation for the velocity of the particle in simple harmonic motion. We need to get the given equation in the terms of the standard equation and the problem can be solved as shown in the diagram below.



Problem

A particle starts from mean position to a new position and it is as shown in the diagram below. Its amplitude and time period is given to us in the problem. We need to find the displacement where the velocity is half of the maximum velocity.


Solution

We know that the particle in simple harmonic motion has maximum velocity at the mean position. As per the given problem at a given instant, velocity of the particle is half of that maximum. Taking that into consideration and substituting the data in the standard format, we can solve the problem as shown in the diagram below.


Problem

Two different particles are in simple harmonic motion and their displacements are represented  as the given equations of the problem. We need to find the resultant amplitude of the combination. Problem is as shown below.


Solution

When we add to oscillatory motion, we need to get a oscillatory motion. The resultant amplitude can be found using the vector addition equation and the solution is as shown in the diagram below.


Problem

A simple harmonic oscillator starts from extreme position and covers a half the displacement in a given time. We need to measure the further time it is going to take to reach the mean position and the problem is as shown in the diagram below.


Solution

As the particle is here starting from the mean position, we need to know that it has some initial phase that is ninety degree. We know that the particle takes one forth of the time period to reach from extreme to mean position and to measure the remaining time to cover half amplitude to, we need to subtract from it as shown in the diagram below.


Problem

Number of springs are connected in series as shown in the problem to a a given mass and the system is allowed to oscillate. We need to measure the time period of oscillation of that system.


Solution

We know that when the springs are in series, the force acting on all of them is same and the extension in the spring is different and it depends on the nature of the spring. Using the common formula for the time period of the system and further simplify the problem as shown in the diagram below.



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Oscillations Complete Lesson

Oscillatory motion is a to and fro motion and it is one of the basic motion of a body after translatory motion and rotatory motion. The particle in oscillatory motion moves about a fixed point and that fixed point is called mean position. All oscillatory motions gets repeated at regular intervals of time and they are called periodic motion. All oscillatory motions are periodic but all periodic motions are not oscillatory. to express periodic motion, we need to use sin and cos functions as they  also gets repeated  between two fixed values.

If oscillatory motion has displacement directly proportional to acceleration and is directed opposite to it, then it is called simple harmonic motion. Simple pendulum is a famous example who is in simple harmonic motion when slightly disturbed from its mean position. We can derive equations for its displacement, velocity, acceleration and time period as shown in the lessons below. We have also derived equations for its potential energy, kinetic energy and total energy of a body in oscillatory motion and if frictional force is also taken into count, the amplitude of the oscillation decreases and finally it stops. This kind of oscillatory motion is called damped oscillatory.
Here in the following list, all topics of oscillations are explained.


 Time Period of Simple pendulum 

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Damped Oscillations and Forced Oscillations

Damped simple harmonic motion

Theoretically we assume that the simple harmonic motion will always have the same time period with a constant amplitude. But here we have neglected the opposition forces like the viscous force that are acting on the bodies that are in the state of motion.

In real life there will be always viscous forces who are acting the motion and in general their values are directly proportional to the velocity of the bodies. Taking that into consideration a body that starts simple harmonic motion cannot continue its simple harmonic motion forever with a constant amplitude but it will be keep on decreasing and finally they will die out. This kind of oscillations are called damped oscillations. We can derive a equation for the amplitude of a damped oscillation as shown below.



It is very much clearly visible that against the motion there is viscous force as well as the restoring force acting and we can derive the equations basing on the Newton’s second law of motion. 

 It can be clearly visible that the amplitude is varying with respect to time and the angular frequency is also decreasing because of the viscous forces. It can be noticed that if that viscous forces ignored the angular frequency is similar to that of the natural frequency that we had derived for a regular simple harmonic motion.



Graph is drawn taking the time onto the x-axis and displacement onto the y-axis and it is noticeable that as the time progresses displacement decreases and finally the oscillation will die down as shown below. We can also read the equation for the energy of  the body in damped simple harmonic motion which is also decreasing with respect to the time.


Forced Oscillations and Resonance

If the body is allowed to oscillate,it oscillates with a definite frequency depending on its characteristic nature.This kind of oscillation is called as free oscillation and it is going to continue to happen for ever infinite time theoretically with the constant amplitude.This particular frequency is called as natural frequency.

But practically a body cannot continue to vibrate  with its natural frequency forever because of the opposition so  it is damped simple harmonic motion is in real life. Anyway if you want to continue the body to be in the simple harmonic motion with can apply an external force.This kind of a oscillation is called as a forced oscillation.

A simple pendulum generally vibrates with its natural frequency. If that is placed on a wooden table, then pendulum starts vibrating with its natural frequency and the wooden table also starts vibrating with the frequency of the tuning fork. Therefore we can say the table is under the forced vibration of the tuning fork.

We can write a mathematical equation for a forced translation as shown below.


We can further write conditions for forced oscillations as shown below.We can not have a situation where viscous forces can be ignored practically .

In the case of viscous forces are there and if natural frequency is equal to forced frequency then the body is said to be in resonance.In that case the body will vibrate with maximum amplitude. This condition is called resonance.




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Time Period of Simple pendulum 
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