IIT JEE and NEET Physics @ Venkats Academy

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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.

Time Period of Simple pendulum

Problems on Simple Pendulum with Solutions

Loaded spring in simple harmonic motion

Damped Oscillations and Forced Oscillation

Oscillations Problems with Solution One

Oscillations Problems with Solution Two

Units and Dimensions an Introduction Video Lesson

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.

Simple Harmonic Motion Displacement ,Velocity and Acceleration

Time Period,Frequency and Phase of a body in SHM

Problems on SHM displacement,velocity and Acceleration

Energies of Body in Simple Harmonic Motion

Thermodynamics Problems with Solutions Five

Thermodynamics Problems with Solutions One

Thermodynamics Problems with Solutions Two

Thermodynamics Problems with Solutions Three

Waves and Oscillations Complete Lessons

The body can have broadly three different types of motions. If all particles of the body are having a similar kind of displacement,the motion is called translatory motion.

If the body particles are rotating about a given axis then the body is said to be in rotatory motion.

If the particle is oscillating about the mean position then it is called as a vibratory motion or oscillatory motion.

The energy transformation from one place to another place can take place in the format of waves in so many situations.

The transmission of the sound in the air simply travels like a longitudinal wave.

So here in this chapter we are going to discuss regarding the different issues of waves and oscillations, corresponding problems with their respective solutions.

Oscillations

Simple Harmonic Motion Displacement ,Velocity and Acceleration

Time Period,Frequency and Phase of a body in SHM

Problems on SHM displacement,velocity and Acceleration

Energies of Body in Simple Harmonic Motion

Time Period of Simple pendulum

Problems on Simple Pendulum with Solutions

Loaded spring in simple harmonic motion

Damped Oscillations and Forced Oscillations

Waves

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

Other Complete Lessons in this blog includes

Units and Dimensions

One and Two Dimensional Motion

Mechanical Properties of Matter

Heat and Thermodynamics

Ray Optics

Wave Optics

Semi Conductor Devices

Dual nature of Radiation and Matter

Loaded spring in simple harmonic motion

A mass less spring suspended at a rigid support and having a heavy mass suspended at the bottom can execute simple harmonic motion under a slight disturbance.

We can derive the question for the time period of a loaded spring. The restoring force on the spring is directly proportional to the extension of the spring and the proportionality constant is called Force constant or spring constant.

The force constant of the spring is inversely proportional to its length. More than length , Les the spring constant value.

Problem and solution

a spring of certain length is divided into two parts having lengths in the ratio of 2 : 3 .What is going to be the spring constant of the longer part?

We can use the concept that spring constant of this spring is inversely proportional to the length as shown below.

Expression for the time period of a loaded spring

The force that is acting towards the mean position is the restoring force and the force acting in the opposite direction is the weight of the load that is suspended. At an equilibrium position we can equate this two forces therefore we can get time period of a loaded spring as shown.

The time period of a simple pendulum is dependent of acceleration due to gravity.But the time period of loaded spring the independent of this gravitational forces. Therefore loaded spring can be operated even in the space and vacuum where as a simple pendulum cannot be operated because of the absence of the acceleration due to gravity.

Effective spring constant when they are connected in series and in parallel

When the springs are connected in parallel the force applied on the combination will be shared across them but each of them will have the similar kind of extension.

When the springs are connected in series the force acting on both of them is going to be the same but each of them is going to extend differently basing on the nature.

Using this concept we can derive a question for the effective spring constant when they are connected in series and in parallel as shown below.

Problem and solution

Find the effective spring constant of the system as shown below?

We can solve the problem simply by applying the concept that the effective spring constant is decreasing when they are connected in series and it is increasing when there are connected in parallel as per the derivation that is made in the previous diagram.